The Baker Type-I Model: Theory, Comprehensive Inference, and Empirical Evidence from Complex Reliability and Biomedical Data

Abstract

Recently, two novel extensions of the Weibull distribution have been introduced through Manly’s exponential transformation, offering a flexible mechanism for modeling skewness, tail behavior, and complex hazard rate structures. In this study, we develop a comprehensive theoretical and inferential framework for one of these models, referred to as the Baker–T1 distribution, to establish it as a mature and practically viable lifetime model for reliability and survival analysis. While the Baker–T1 model exhibits remarkable flexibility in capturing skewness, tail behavior, and complex hazard rate shapes, its statistical properties and practical performance have not yet been systematically investigated. To bridge this gap, we derive a wide range of fundamental distributional characteristics, including reliability measures, hazard and reversed-hazard functions, quantiles, moments, skewness, kurtosis, dispersion indices, and order statistics, establishing the model’s analytical tractability and structural richness. An extensive inferential framework is introduced by implementing eight classical estimation techniques, and their finite-sample behavior is rigorously examined through a large-scale Monte Carlo simulation study under diverse parameter configurations. The practical relevance of the Baker–T1 model is further demonstrated using two genuine datasets from biomedical and engineering domains, where it consistently outperforms thirteen competing lifetime distributions according to likelihood-based and information-theoretic criteria.

1. Introduction

The Weibull distribution is one of the most fundamental and widely used models in reliability theory and lifetime data analysis due to its mathematical simplicity, interpretability, and flexibility in modeling monotone failure rates. Its closed-form expressions for key reliability measures, such as the survival and hazard rate, make it particularly attractive for practical applications in engineering, biomedical sciences, and industrial reliability. Despite these advantages, the classical Weibull model is limited by the restrictive nature of its hazard rate, which can only be monotonic, either strictly increasing or strictly decreasing.

In practice, however, lifetime data frequently display more complex failure mechanisms, including non-monotonic, unimodal, bathtub-shaped, and multi-turning-point hazard rate patterns. These limitations of the standard Weibull distribution have motivated extensive research aimed at developing flexible generalizations capable of capturing richer hazard rate dynamics while retaining the desirable analytical properties of the Weibull framework. Recently, Baker [1] introduced a novel approach for constructing two flexible three-parameter Weibull distributions by applying Manly’s transformation (proposed by Manly [2]) to well-established baseline models. These models, referred to as the Y-Weibull and Z-Weibull distributions, are distinguished by their tail characteristics: one is designed to capture heavier right tails, whereas the other accommodates lighter or more rapidly decaying tails, thereby enhancing flexibility in lifetime and reliability modeling.

Both formulations admit closed-form expressions for the probability density and cumulative distribution functions, which facilitates likelihood-based inference, and they reduce to the standard Weibull model as special cases when the transformation parameter is set to zero.

The central idea of this framework is to enhance distributional flexibility—particularly in terms of skewness, tail behavior, and hazard rate shapes—while preserving analytical tractability. By embedding Manly’s exponential transformation within classical survival distributions, the resulting models are capable of accommodating a wide range of empirical behaviors frequently observed in reliability engineering, actuarial science, biomedical, and survival analysis. Compared with existing generalized survival distributions in the literature, such as exponentiated, transmuted, and Marshall–Olkin-type families, the Baker’s framework offers several notable advantages; see Cressie [3]. First, it provides a unified and transparent mechanism for model construction, avoiding the proliferation of ad hoc parameterizations that often complicate inference. Second, the resulting probability density and hazard functions remain mathematically well-behaved and interpretable, facilitating both theoretical analysis and practical implementation. Third, the additional flexibility introduced by Manly’s transform enables the models to capture complex hazard rate shapes, including increasing, decreasing, bathtub, and unimodal forms, which are not simultaneously attainable under many existing generalizations. For additional details, see, for example, Nwakuya and Nkwocha [4].

From an inferential perspective, the Y- and Z-Weibull distributions proposed by Baker [1] are well suited for likelihood-based estimation and model comparison. The presence of closed-form expressions or rapidly convergent series representations for key quantities, such as moments and survival functions, supports efficient numerical implementation. Consequently, these models provide a valuable alternative to existing flexible survival distributions, offering improved goodness-of-fit without excessive computational or conceptual complexity. In the present study, attention is restricted to the Y-Weibull model, hereafter termed the Baker–T1 distribution, which extends the classical Weibull framework through an additional transformation parameter that regulates skewness and tail behavior. In the following section, the structural characteristics of the Baker–T1 model are presented in more detail.

Although Baker [1] introduced two novel extensions of the Weibull distribution based on Manly’s transformation, the proposed models were primarily presented from a constructional perspective. A detailed investigation of their statistical properties was not fully developed, and several important distributional characteristics, such as higher-order moments, skewness, kurtosis, and inferential behavior, were not examined in sufficient depth. Moreover, the estimation framework was limited, leaving open questions regarding the comparative performance of alternative parametric estimation techniques.

To address this important gap, the present study provides a comprehensive theoretical and inferential treatment of the Baker–T1 distribution. Specifically, we derive a wide range of fundamental properties, including moments, central moments, skewness, kurtosis, and reliability measures. In addition, we develop and implement eight parametric estimation methodologies for the model parameters, allowing for a systematic comparison of their efficiency and robustness. Although recent transformation-based approaches, most notably Baker’s construction via Manly’s transformation, have introduced promising extensions to the Weibull family, these models have so far been explored primarily from a structural or constructional perspective.

A rigorous and unified statistical treatment of their theoretical properties, inferential behavior, and practical performance remains largely absent from the literature. In particular, essential characteristics such as higher-order moments, dispersion measures, order statistics, and comparative efficiency of competing estimation procedures have not been systematically investigated. This gap limits the practical adoption of Baker-type models despite their evident flexibility and analytical appeal. Motivated by this shortcoming, the present study undertakes a comprehensive analysis of the Baker–T1 model, aiming to establish it as a mature and practically implementable lifetime distribution by developing its full probabilistic structure, inferential machinery, and empirical relevance through simulation and real-data applications.

Now, the main original contributions of this work can be summarized as follows:

• A complete probabilistic characterization of the Baker–T1 distribution is provided, including closed-form expressions or convergent series representations for the quantile function, reliability measures, hazard and reversed-hazard, skewness, kurtosis, and dispersion indices.
• General expressions for the rth moments are derived and shown to exist for all positive orders.
• Both analytically and graphically, the Baker–T1 model can generate a wide range of hazard rate shapes, including increasing, decreasing, bathtub-shaped, and unimodal forms, within a single parametric framework, a feature not attainable by the classical Weibull distribution.
• Exact closed-form expressions for the densities of order statistics are derived, including the minimum and maximum order statistics, facilitating inference on extremes and reliability-related quantities.
• Eight classical parametric estimation methods are developed and implemented for the Baker–T1 model, namely maximum likelihood, maximum product of spacings, least squares, weighted least squares, percentile estimation, Cramér–von Mises, Anderson–Darling, and right-tail Anderson–Darling estimation.
• A comprehensive simulation study is conducted to compare the finite-sample performance of all proposed estimators under diverse parameter configurations and sample sizes, using multiple accuracy criteria and ranking measures.
• Using two genuine datasets from biomedical and engineering sectors, the Baker–T1 model consistently outperforms thirteen competing lifetime distributions based on nine information-theoretic criteria. This superiority arises from its enhanced flexibility in capturing skewness, tail behavior, and non-monotone hazard rate structures, which competing models fail to accommodate simultaneously. The empirical results confirm that the Baker–T1 distribution provides a more accurate, robust, and interpretable fit across fundamentally different application domains.

The study is organized as follows. Section 2 formally introduces the Baker–T1 distribution. Section 3 presents its fundamental statistical properties. Section 4 describes parameter estimation using eight classical approaches. Section 5 reports a Monte Carlo simulation study. Section 6 illustrates the model’s application to two genuine datasets. Finally, Section 7 concludes with remarks, practical recommendations, and suggestions for future research.

2. The Baker–T1 Model

Baker [1] proposed two distinct three-parameter versions of the Weibull distribution, each characterized by different tail behaviors. This transformation preserves the Weibull distribution as the baseline while inducing additional flexibility in skewness, tail weight, and hazard rate structure. Consequently, the proposed models named Y- and Z-Weibull are capable of capturing numerous failure rates, with one formulation accommodating heavier right tails and the other allowing lighter or more rapidly decaying tails. In this part, we focus on the Baker–T1 lifespan model, which was first mentioned as the Y-Weibull model. Let X be a nonnegative random variable following the Baker–T1 distribution, denoted by $X\sim \mathrm{Baker}\text{–}\mathrm{T}1\left(\xi \right)$, where $\xi ={(\alpha ,\mu ,\theta )}^{\top }$. Subsequently, the resulting cumulative distribution function (CDF), denoted by $F(·)$, and probability density function (PDF), denoted by $f(·)$, are obtained as follows:

$$F(x;\xi )=1-exp\left[-{\left({\displaystyle \frac{e^{\alpha \theta x}-1}{\theta }}\right)}^{\mu }\right],x>0,$$

(1)

and

$$f(x;\xi )=\alpha \mu e^{\alpha \theta x}{\left({\displaystyle \frac{e^{\alpha \theta x}-1}{\theta }}\right)}^{\mu -1}exp\left[-{\left({\displaystyle \frac{e^{\alpha \theta x}-1}{\theta }}\right)}^{\mu }\right],$$

(2)

where $\alpha >0$ governs the time scale, $\mu >0$ controls the shape, and $\theta >0$ determines the distributional tail behavior. The Baker–T1 model extends the standard Weibull (Gompertz) distribution, which can be found as a special case by setting $\theta \to 0$ $(\mu =1,\theta >0)$. In Appendix A, additional derivation steps for the Baker–T1 model are provided.

In the present study, attention is restricted to the Y-Weibull model introduced by Baker [1], which is hereafter referred to as the Baker–T1 distribution. This terminology is adopted to explicitly acknowledge Baker’s transformation-based construction while providing a concise label for the specific model variant analyzed in depth in this work. It is important to note that the Baker–T1 distribution is not a new model in the sense of redefining the original formulation, but rather a comprehensive theoretical and inferential development of the Y-Weibull case within Baker’s framework.

It is important to emphasize that a broad class of lifetime distributions in the literature shares a common exponential-type representation of the form $F\left(x\right)=1-exp\{-\nabla (x\left)\right\}$. These models typically differ in the specific functional form assigned to $\nabla \left(x\right)$, which governs the distribution’s tail behavior and hazard rate structure. Notable examples include:

• The exponential power distribution (Smith and Bain [5]), where $\nabla \left(x\right)=e^{\theta x^{\gamma }}-1$;
• The Gompertz distribution (Gompertz [6]), where $\nabla \left(x\right)=\alpha (e^{\theta x}-1)$;
• The Chen distribution (Chen [7]), where $\nabla \left(x\right)=\alpha (e^{x^{\mu }}-1)$;
• The modified Weibull extension (Xie et al. [8]), where $\nabla \left(x\right)=\alpha \theta (e^{{(x/\alpha )}^{\mu }}-1)$;
• The Weibull–Weibull family (Bourguignon et al. [9]), where $\nabla \left(x\right)=\alpha {(e^{\theta x^{\gamma }}-1)}^{\mu }$.

In contrast, the Baker–T1 distribution arises from Manly’s exponential transformation embedded within Baker’s construction framework, leading to the transformation function $\nabla \left(x\right)={\left(\left(e^{\alpha \theta x}-1\right)/\theta \right)}^{\mu }.$ This formulation introduces a non-trivial interaction among the parameters $\alpha$, $\mu$, and $\theta$, where the parameter $\theta$ simultaneously governs the deformation of the exponential argument and the tail behavior of the distribution. Such a structure differs fundamentally from the aforementioned models and cannot, in general, be reduced to them through simple parameter reparameterization. Moreover, while several existing models achieve flexibility through exponentiation or compounding mechanisms, the Baker–T1 model modifies the argument of the exponential function via a transformation-based approach. This feature yields enhanced adaptability in capturing complex hazard rate shapes, including increasing, decreasing, bathtub-shaped, and unimodal forms within a unified framework. Therefore, although the Baker–T1 distribution shares a superficial structural similarity with other exponential-type models, it should be regarded as a distinct transformation-based extension characterized by its unique parameter interaction and construction methodology.

3. Distribution Characteristics

This section outlines key properties of the Baker–T1 distribution, including its reliability, hazard, moments, order statistics, and related characteristics.

3.1. Reliability Metrics

The reliability function (RF), denoted by $R(·)=1-F(·)$, of the Baker–T1 model evaluated at a time t, where $0<t<\infty$, is expressed as follows:

$$R(t;\xi )=exp\left[-{\left({\displaystyle \frac{e^{\alpha \theta t}-1}{\theta }}\right)}^{\mu }\right],\alpha ,\mu ,\theta >0.$$

(3)

The HRF (symbolized as $h(·)=f(·)/R(·)$) of the Baker–T1 model (at $0<t<\infty$) is

$$h\left(t;\xi \right)=\alpha \mu e^{\alpha \theta x}{\left[{\displaystyle \frac{1}{\theta }}\left(e^{\alpha \theta x}-1\right)\right]}^{\mu -1},\alpha ,\mu ,\theta >0.$$

(4)

The reversed-HRF (say, $h^{\star }(·)=f(·)/F(·)$) of the Baker–T1 model (at $0<t<\infty$) is

$$h^{\star }\left(t;\xi \right)={\displaystyle \frac{\alpha \mu e^{\alpha \theta x}{\left(\frac{e^{\alpha \theta x}-1}{\theta }\right)}^{\mu -1}exp\left[-{\left(\frac{e^{\alpha \theta x}-1}{\theta }\right)}^{\mu }\right]}{1-exp\left[-{\left(\frac{e^{\alpha \theta x}-1}{\theta }\right)}^{\mu }\right]}},\alpha ,\mu ,\theta >0.$$

(5)

Depending on the parameter values of $\mathrm{Baker}\text{–}\mathrm{T}1\left(\xi \right)$, each subplot in Figure 1 displays five density (failure rate) curves. It reveals that:

• Figure 1a illustrates the Baker–T1 density shapes under different parameter settings, highlighting its considerable shape flexibility, which can be symmetric or highly skewed, unimodal with varying peak locations, or strongly right-tailed. Some parameter combinations produce sharply peaked densities with rapid decay. In contrast, others yield flatter or more spread-out shapes, demonstrating the model’s ability to adapt to diverse data patterns commonly observed in lifetime and reliability studies.
• Figure 1b presents the Baker–T1 failure rate shapes and exhibits a wide range of behaviors, including constant, increasing, decreasing, bathtub-shaped, and upside-down bathtub-shaped forms. In particular, the presence of bathtub-shaped and non-monotone hazards is notable, as such patterns are frequently encountered in reliability and survival analysis but cannot be captured by the classical Weibull model in a single formulation.

As a result, the Baker–T1 distribution’s capacity to accommodate diverse hazard rate structures highlights its suitability as a robust and versatile alternative to classical Weibull-type models for analyzing complex lifetime data. Moreover, this variability demonstrates that the additional parameter introduced through the Manly method plays a substantive role in regulating skewness and tail behavior beyond the flexibility of the standard Weibull distribution.

3.2. Quantile and Quartiles

The quantile function of the Baker–T1 distribution constitutes a particularly tractable and valuable feature of the model. By invoking the inverse transform theorem, the quantile function is obtained by inverting the cumulative distribution function $F\left(x\right)$ given in (1). Specifically, the Baker–T1 quantile is defined by solving for x in terms of the probability level p, yielding

$$Q\left(p\right)={\displaystyle \frac{1}{\alpha \theta }}log\left[1+\theta {\left\{-log(1-p)\right\}}^{1/\mu }\right].$$

(6)

An additional advantage of the Baker–T1 model is that its quantile function (6) admits a closed-form expression, which substantially simplifies numerical implementation and reduces computational cost in subsequent analyses. The first three quartiles of the Baker–T1 distribution can be simply developed directly from (6) by setting $p(={\displaystyle \frac{1}{4}},{\displaystyle \frac{2}{4}},{\displaystyle \frac{3}{4}})$.

3.3. Moments

Moments are fundamental descriptors in statistics, capturing essential aspects of a probability distribution such as its location, variability, asymmetry, and tail behavior. They provide valuable insight into the distribution’s shape and dispersion, facilitating effective model interpretation, comparative analysis, and statistical inference.

Theorem 1.

If X follows the Baker–T1(ξ) model, then the r-th moment (for $r=1,2,\dots$) of X, denoted by ${\mathcal{M}}_r^{\prime }$, is given by

$${\mathcal{M}}_r^{\prime }=\mathbb{E}\left[X^r\right]={\displaystyle \frac{1}{{\alpha }^r}}\sum _{j_1=1}^{\infty }\cdots \sum _{j_r=1}^{\infty }{\displaystyle \frac{{(-1)}^{j_1+\cdots +j_r-r}{\theta }^{j_1+\cdots +j_r-r}}{j_1\cdots j_r}}\Gamma \left(1+{\displaystyle \frac{j_1+\cdots +j_r}{\mu }}\right).$$

(7)

Proof. 

See Appendix B.  □

By setting $r=1$ and 2 in (7), the first and second moments of X are obtained, respectively, as:

$${\mathcal{M}}_1^{\prime }={\displaystyle \frac{1}{\alpha }}\sum _{j=1}^{\infty }{\displaystyle \frac{{(-1)}^{j+1}}{j}}\Gamma \left(1+{\displaystyle \frac{j}{\mu }}\right),$$

(8)

and

$${\mathcal{M}}_2^{\prime }={\displaystyle \frac{1}{{\alpha }^2}}\sum _{j_1=1}^{\infty }\sum _{j_2=1}^{\infty }{\displaystyle \frac{{(-1)}^{j_1+j_2}}{j_1{j}_2}}{\theta }^{j_1+j_2-2}\Gamma \left(1+{\displaystyle \frac{j_1+j_2}{\mu }}\right).$$

(9)

Hence, from Equations (8) and (9), the variance of X is therefore

$$\mathrm{Var}\left(X\right)={\mathcal{M}}_2^{\prime }-{\left({\mathcal{M}}_1^{\prime }\right)}^2.$$

(10)

Since $ln(1+\theta t^{1/\mu })$ grows at a logarithmic rate as $t\to \infty$ and the exponential term $e^{-t}$ ensures integrability, the integral in (7) converges for all $r>0$. Consequently, although all positive-order moments of the Baker–T1 distribution exist, the rth moment generally has no closed-form expression and must be evaluated numerically.

In particular, by choosing parameter values $\alpha \in \{0.5,1.0,1.5\}$, $\theta \in \{0.5,1.5\}$, and $\mu \in \{0.1,0.5,1.5,2.0,2.5\}$,

Table 1. Statistics of the Baker–T1 distribution.

$\mathit{\alpha }\downarrow$ $\mathit{\theta }\to$	$\mathit{\mu }$	0.5						1.5
		$\mathcal{M}$	$\mathcal{V}$	$\mathcal{ID}$	$\mathcal{CV}$	$\mathcal{S}$	$\mathcal{K}$	$\mathcal{M}$	$\mathcal{V}$	$\mathcal{ID}$	$\mathcal{CV}$	$\mathcal{S}$	$\mathcal{K}$
0.5	0.1	2.916	22.49	7.711	1.626	2.944	14.316	0.175	0.055	0.315	1.343	2.058	5.835
	0.5	1.829	5.283	2.889	1.257	1.769	6.165	0.203	0.062	0.306	1.229	1.771	4.758
	1.0	1.374	2.351	1.711	1.116	1.386	4.504	0.236	0.069	0.291	1.112	1.440	3.772
	1.5	0.180	0.060	0.335	1.366	2.177	5.859	0.262	0.072	0.276	1.027	1.166	3.148
	2.5	0.210	0.067	0.318	1.231	1.925	4.869	0.272	0.066	0.241	0.940	0.883	2.755
1.0	0.1	0.185	0.064	0.344	1.363	2.326	5.957	0.307	0.064	0.210	0.827	1.351	3.415
	0.5	0.205	0.067	0.326	1.261	2.209	5.413	0.347	0.066	0.190	0.740	0.950	2.766
	1.0	0.233	0.070	0.302	1.139	2.066	4.809	0.363	0.061	0.169	0.682	0.623	2.463
	1.5	0.263	0.073	0.276	1.025	1.922	4.279	0.341	0.049	0.145	0.651	0.512	2.508
	2.5	0.329	0.074	0.224	0.824	1.611	3.399	0.279	0.028	0.099	0.596	0.346	2.410
1.5	0.1	0.177	0.070	0.397	1.497	2.347	5.867	0.405	0.059	0.147	0.602	0.923	2.685
	0.5	0.205	0.075	0.366	1.335	2.199	5.156	0.430	0.055	0.127	0.544	0.468	2.388
	1.0	0.246	0.079	0.322	1.145	2.030	4.417	0.396	0.040	0.102	0.509	0.291	2.477
	1.5	0.292	0.080	0.275	0.970	1.875	3.820	0.349	0.028	0.079	0.475	0.169	2.450
	2.5	0.399	0.072	0.182	0.674	1.554	2.942	0.287	0.015	0.053	0.430	−0.012	2.417

presents the corresponding mean ($\mathcal{M}$), variance ($\mathcal{V}$), index of dispersion ($\mathcal{ID}$), coefficient of variation ($\mathcal{CV}$), skewness ($\mathcal{S}$), and kurtosis ($\mathcal{K}$) for the Baker–T1 distribution. These selected parameter combinations are considered as representative examples to illustrate the behavior of the statistical measures, without any loss of generality.

From Table 1, the following key observations can be drawn:

• Mean ($\mathcal{M}$): Increases moderately with $\alpha$; higher $\mu$ slightly reduces extreme mean values.
• Variance ($\mathrm{Var}\left(X\right)$): Larger for smaller $\mu$; decreases as $\theta$ increases, indicating more concentrated distributions.
• Index of Dispersion ($\mathcal{ID}$): Always above 1, confirming over-dispersion; strongly influenced by $\mu$, with smaller $\mu$ producing higher ID.
• Coefficient of variation ($\mathcal{CV}$): Declines with increasing $\theta$ and $\alpha$, reflecting reduced relative variability; lower for higher $\mu$.
• Skewness ($\mathcal{S}$): Positive in all cases; very high for $\mu =0.5$ (heavy right tail), decreases toward symmetry as $\mu$ or $\theta$ increase.
• Kurtosis ($\mathcal{K}$): Excessive for small $\mu$, indicating heavy tails; stabilizes near mesokurtic range for larger $\mu$ and higher $\theta$.

Overall, Table 1 reveals that the Baker–T1 distribution is a highly flexible family capable of modeling a wide range of dispersion and tail behaviors. The parameter $\mu$ plays a dominant role in controlling tail thickness and asymmetry, while $\theta$ and $\alpha$ fine-tune scale and variability. These properties make the model particularly attractive for applications involving heterogeneous data with varying degrees of skewness and over-dispersion.

A comprehensive visual assessment depicted in Figure 2 illustrates the distributional characteristics of the Baker–T1 model through three-dimensional surface plots of key summary measures based on different configurations of $\alpha$ and $\mu$ when $\theta =1$. These surfaces provide an illustration of how changes in the model parameters influence location, scale, and shape properties of the Baker–T1 distribution. The plots show that the distributional measures are highly sensitive to changes in $\alpha$ and $\mu$. Smaller $\alpha$ and $\alpha$ values lead to higher variability, skewness, and heavy tails, while larger $\alpha$ stabilizes the distribution and reduces relative variability. Overall, the Baker–T1 model can represent a wide range of distributional shapes, from dispersed and skewed to more symmetric and stable forms.

Let X follow the Baker–T1 distribution; differentiating the logarithm of the PDF (2) with respect to x and equating the result to zero yields the mode as follows:

$$1+{\displaystyle \frac{(\mu -1)y}{y-1}}-\mu y{\left({\displaystyle \frac{y-1}{\theta }}\right)}^{\mu -1}=0,y>1,$$

(11)

where $y=e^{\alpha \theta x}$.

It is clear that Equation (11) has no closed-form solution for general $\mu$ and $\theta$ and must therefore be solved numerically. For $\mu >1$, the density $f\left(x\right)$ is unimodal. As $x\to 0^{+}$, the density increases from zero, while as $x\to \infty$, the term $exp\left[-{\left({\displaystyle \frac{1}{\theta }}\left(e^{\alpha \theta x}-1\right)\right)}^{\mu }\right]$ dominates and forces $f\left(x\right)\to 0$. Consequently, the derivative of the log-density changes sign exactly once, ensuring the existence of a unique interior mode. When $0<\mu \le 1$, the density is monotonically decreasing, and the mode occurs at the boundary $x_m=0$.

3.4. Order Statistics

Order statistics offer valuable information about the underlying data structure by facilitating the estimation of quantiles, extreme values, and reliability-related measures in statistical analyses. Let $X_1,X_2,\dots ,X_n$ be a random sample from the Baker–T1 distribution, then the PDF of the kth order statistic $X_{k:n}$ is given by

$$f_{k:n}(x;\xi )=C_k{\left[F(x;\xi )\right]}^{k-1}{\left[R(x;\xi )\right]}^{n-k}f(x;\xi ),x>0,$$

(12)

where $C_k={\displaystyle \frac{n!}{(k-1)!(n-k)!}}$.

Substituting (2) and (1) into (12), we obtain

$$\begin{array}{cc}\hfill f_{k:n}(x;\xi )& =\alpha \mu e^{\alpha \theta x}{A}^{\mu -1}{\left\{1-exp\left[-A(x;\xi )\right]\right\}}^{k-1}exp\left[-(n-k+1)A(x;\xi )\right],\hfill \end{array}$$

(13)

where $A(x;\xi )={\left({\displaystyle \frac{e^{\alpha \theta x}-1}{\theta }}\right)}^{\mu }$.

Applying the following binomial expansion

$${(1-z)}^{k-1}=\sum _{j=0}^{k-1}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{j}}\right)z^j,\left|z\right|<1,$$

(14)

we obtain, upon setting $z=exp[-A(x;\xi \left)\right]$,

$${\left\{1-exp[-A(x;\xi \left)\right]\right\}}^{k-1}=\sum _{j=0}^{k-1}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{j}}\right)exp[-jA(x;\xi )].$$

(15)

Substituting (15) into (12) yields the density of the kth Baker–T1 order statistic as follows:

$$f_{k:n}(x;\xi )=C_k\alpha \mu e^{\alpha \theta x}{A}^{\mu -1}\sum _{j=0}^{k-1}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{j}}\right)exp[-(n-k+1+j)A(x;\xi )].$$

(16)

The respective minimum and maximum order statistics, by setting $k=1$ and n into (16), yield:

$$f_{1:n}\left(x\right)=n\alpha \mu e^{\alpha \theta x}{\left(A(x;\xi )\right)}^{\mu -1}exp\left[-n{\left(A(x;\xi )\right)}^{\mu }\right],x>0.$$

(17)

and

$$f_{n:n}\left(x\right)=n\alpha \mu e^{\alpha \theta x}{\left(A(x;\xi )\right)}^{\mu -1}\sum _{j=0}^{n-1}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{j}}\right)exp\left[-(j+1){\left(A(x;\xi )\right)}^{\mu }\right]x>0.$$

(18)

It is worth mentioning that the expression (16) shows that the kth order statistic density is a finite linear combination of Baker–T1-type kernels with modified exponential weights, which is particularly useful for numerical integration and for deriving further properties of the order statistics. In particular, the minimum order statistic preserves the Baker–T1-type form with the exponential term scaled by the sample size n, whereas the maximum order statistic is represented as a finite alternating sum of Baker–T1-type kernels. These closed-form representations facilitate the study of extreme-value behavior and allow efficient numerical evaluation of tail-related reliability and survival measures.

4. Methods of Estimation

Evaluating and contrasting estimation techniques is crucial for determining which procedures yield the most reliable and efficient parameter estimates from a statistical standpoint. In this section, eight well-established classical estimation methods are applied to derive point estimates of the Baker–T1 model parameters $\alpha$, $\mu$, and $\theta$. Specifically, the maximum likelihood and maximum product of spacings methods are likelihood-based approaches. The least-squares and weighted least-squares methods rely on minimizing the discrepancy between empirical and theoretical distribution functions. The percentile method provides a quantile-based alternative that is easy to implement and less sensitive to certain model assumptions. In addition, the Cramér–von Mises, Anderson–Darling, and right-tail Anderson–Darling methods are minimum distance estimators that differ in how they weight deviations between empirical and fitted distributions. Collectively, these methods offer a comprehensive framework for evaluating estimation performance from multiple theoretical and practical perspectives. For detailed treatments of these estimation methodologies, the reader is referred to Hassan and Alharbi [10], Shafiq et al. [11], Alqasem et al. [12], and Alotaibi et al. [13], among others.

4.1. Maximum Likelihood

The maximum likelihood technique is among the most commonly employed methods in statistical inference, owing to its simplicity and intuitive formulation. In addition, MLEs enjoy a number of attractive theoretical properties (see, e.g., Rohde [14]). Let $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ denote a random sample.

From (2), the log-likelihood function (log-LF), $\mathcal{L}\left(\mathbf{x}\right|\xi )={\sum }_{i=1}^{n}logf(\mathbf{x};\xi )$, for the Baker–T1 distribution parameters is:

$$\begin{array}{c}\hfill \mathcal{L}\left(\mathbf{x}\right|\xi )=n\left[\theta \alpha \overline{x}+log\left(\alpha \mu \right)-(\mu -1)log\left(\theta \right)\right]+\left(\mu -1\right)\sum _{i=1}^{n}log\left({\psi }_i\right)-{\theta }^{-\mu }\sum _{i=1}^n{\psi }_i^{\mu },\end{array}$$

(19)

where ${\psi }_i=e^{\theta \alpha x_i}-1.$

The MLEs ${\widehat{\xi }}_{MLE}$ of $\xi$ can be obtained by solving the following system of score equations:

$${\left.{\displaystyle \frac{n}{\alpha }}+\theta \sum _{i=1}^n{x}_i+\theta (\mu -1)\sum _{i=1}^n{\phi }_i{\psi }_i^{-1}-\mu {\theta }^{-\mu }\sum _{i=1}^n{\psi }_i^{\mu -1}\right|}_{({\widehat{\xi }}_{MLE}=\xi )}=0,$$

(20)

$${\left.{\displaystyle \frac{n}{\mu }}+{\theta }^{-1}\sum _{i=1}^{n}log\left({\psi }_i\right)-{\theta }^{-\mu }\sum _{i=1}^n{\psi }_i^{\mu }\left[log\left({\psi }_i\right)-log\left(\theta \right)\right]\right|}_{({\widehat{\xi }}_{MLE}=\xi )}=0,$$

(21)

and

$${\left.\alpha n\overline{x}-(\mu -1)\left[{\displaystyle \frac{n}{\theta }}-\alpha \sum _{i=1}^n{\phi }_i{\psi }_i^{-1}\right]-{\displaystyle \frac{\mu }{{\theta }^{\mu +1}}}\sum _{i=1}^n{\psi }_i^{\mu -1}\left[\alpha \theta {\phi }_i-{\psi }_i\right]\right|}_{({\widehat{\xi }}_{MLE}=\xi )}=0,$$

(22)

respectively, where ${\phi }_i=x_i{e}^{\alpha \theta x_i}$ (for $i=1,2,\dots ,n$). Since Equations (20)–(22) cannot be expressed explicitly, numerical optimization methods are required. For this purpose, we recommend using optimization iterative procedures, such as Newton–Raphson or BFGS, via the maxLik package v1.5-2.2 (Henningsen and Toomet [15]) to obtain ${\widehat{\xi }}_{MLE}$.

4.2. Maximum Product of Spacings

The maximum product of spacings (MPS) approach is commonly viewed as a strong competitor to the maximum likelihood method in statistical inference. Estimators obtained via MPS (MPSEs) share many of the favorable theoretical characteristics associated with MLEs. In line with Cheng and Amin [16], the objective function to be maximized is defined as the natural logarithm of the MPS criterion. Specifically, let $x_1<x_2<\cdots <x_n$ denote the order statistics of a random sample of size n drawn from the underlying population.

From (1), the log-MPS, $\mathcal{P}\left(\mathbf{x}\right|\xi )\propto {\sum }_{i=1}^{n+1}log\left[F(x_i;\xi )-F(x_{i-1};\xi )\right]$, for the Baker–T1 distribution parameters is:

$$\mathcal{P}\left(\mathbf{x}\right|\xi )=\sum _{i=1}^{n+1}log\left[{\mho }_i\left(\xi \right)\right],$$

(23)

where ${\mho }_i\left(\xi \right)=exp\left(-{\left({\displaystyle \frac{exp\left(\alpha \theta x_{i-1}\right)-1}{\theta }}\right)}^{\mu }\right)-exp\left(-{\left({\displaystyle \frac{exp\left(\alpha \theta x_i\right)-1}{\theta }}\right)}^{\mu }\right).$

The MPSEs ${\widehat{\xi }}_{MPSE}$ of $\xi$, can be obtained by simultaneously solving the following three non-linear equations:

$${\left.\sum _{i=1}^{n+1}{\displaystyle \frac{\mu A_i^{\mu -1}{x}_i{e}^{\alpha \theta x_i}exp(-A_i^{\mu })-\mu A_{i-1}^{\mu -1}{x}_{i-1}{e}^{\alpha \theta x_{i-1}}exp(-A_{i-1}^{\mu })}{exp(-A_{i-1}^{\mu })-exp(-A_i^{\mu })}}\right|}_{({\widehat{\xi }}_{MPSE}=\xi )}=0,$$

(24)

$${\left.\sum _{i=1}^{n+1}{\displaystyle \frac{exp(-A_i^{\mu })A_i^{\mu }log{A}_i-exp(-A_{i-1}^{\mu })A_{i-1}^{\mu }log{A}_{i-1}}{exp(-A_{i-1}^{\mu })-exp(-A_i^{\mu })}}\right|}_{({\widehat{\xi }}_{MPSE}=\xi )}=0,$$

(25)

and

$${\left.\sum _{i=1}^{n+1}{\displaystyle \frac{\mu A_i^{\mu -1}{\Delta }_{i}exp(-A_i^{\mu })-\mu A_{i-1}^{\mu -1}{\Delta }_{i-1}exp(-A_{i-1}^{\mu })}{exp(-A_{i-1}^{\mu })-exp(-A_i^{\mu })}}\right|}_{({\widehat{\xi }}_{MPSE}=\xi )}=0,$$

(26)

respectively, where $A_i={\displaystyle \frac{1}{\theta }}\left(e^{\alpha \theta x_i}-1\right)$ and ${\Delta }_i={\displaystyle \frac{\alpha \theta x_i{e}^{\alpha \theta x_i}-\left(e^{\alpha \theta x_i}-1\right)}{{\theta }^2}}$ (for $i=0,1,\dots ,n+1$). Just like in the MLEs’ scenario, we recommend using the maxLik package (Henningsen and Toomet [15]) to obtain the MPSEs ${\widehat{\xi }}_{MPSE}$ (${\widehat{\zeta }}_{MPSE}$) of $\xi$ ($\zeta$).

4.3. Least-Squares and Weighted Least-Squares

This subsection is devoted to the least-squares estimation (LSE) and weighted least-squares estimation (WLSE) procedures for the Baker–T1 distribution. These methods aim to minimize the sum of squared deviations between the theoretical cumulative distribution function and its empirical counterpart. Let $x_1<x_2<\cdots <x_n$ denote the ordered observations of a random sample; it is well established that

$$\mathbb{E}\left[F\left(x_i;\xi \right)\right]={\displaystyle \frac{i}{n+1}}\mathrm{and}\mathbb{V}\left[F\left(x_i;\xi \right)\right]={\displaystyle \frac{i(n+1-i)}{{(n+1)}^2(n+2)}}.$$

(27)

From (1), the LSE (say, ${\widehat{\xi }}_{LSE}$) and WLSE (say, ${\widehat{\xi }}_{WLSE}$) of $\xi$ can be acquired by minimizing the next two quantities with respect to $\xi$:

$$LS\left(\left.\xi \right|\mathbf{x}\right)=\sum _{i=1}^n{\left\{\left(1-e^{-A_i^{\mu }}\right)-\mathbb{E}\left[F\left(x_i;\xi \right)\right]\right\}}^2$$

(28)

and

$$WS\left(\left.\xi \right|\mathbf{x}\right)=\sum _{i=1}^n{\displaystyle \frac{1}{\mathbb{V}\left[F\left(x_i;\xi \right)\right]}}{\left\{\left(1-e^{-A_i^{\mu }}\right)-\mathbb{E}\left[F\left(x_i;\xi \right)\right]\right\}}^2.$$

(29)

Thus, the ${\widehat{\xi }}_{LSE}$ and ${\widehat{\xi }}_{WLSE}$ expressions of $\xi$ can be solved simultaneously using the following three equations:

$${\left.{\sum }_{i=1}^n{\tau }_i\left\{\left(1-e^{-A_i^{\mu }}\right)-\mathbb{E}\left[F\left(x_i;\xi \right)\right]\right\}{C}_1\right|}_{({\widehat{\xi }}_{LSE}=\xi )}=0,$$

(30)

$${\left.{\sum }_{i=1}^n{\tau }_i\left\{\left(1-e^{-A_i^{\mu }}\right)-\mathbb{E}\left[F\left(x_i;\xi \right)\right]\right\}{C}_2\right|}_{({\widehat{\xi }}_{LSE}=\xi )}=0,$$

(31)

and

$${\left.{\sum }_{i=1}^n{\tau }_i\left\{\left(1-e^{-A_i^{\mu }}\right)-\mathbb{E}\left[F\left(x_i;\xi \right)\right]\right\}{C}_3\right|}_{({\widehat{\xi }}_{LSE}=\xi )}=0,$$

(32)

where ${\tau }_i=1$ (for LSE) and $1/\mathbb{V}\left[F\left(x_i;\xi \right)\right]$ (for WLSE),

$C_1=-\mu A_i^{\mu -1}{x}_i{e}^{\alpha \theta x_i}exp(-A_i^{\mu })$,

$C_2=-exp(-A_i^{\mu })A_i^{\mu }log{A}_i$,

and

$C_3=-\mu A_i^{\mu -1}{\Delta }_{i}exp(-A_i^{\mu })$.

4.4. Percentile

Owing to the availability of an explicit quantile function, the underlying distribution lends itself naturally to percentile-based estimation of its unknown parameters. This approach seeks to minimize the sum of squared discrepancies between the theoretical percentiles of the distribution and their empirical counterparts.

For the Baker–T1 model, from (6), the percentile estimators (PCEs) of $\alpha$, $\mu$, and $\theta$, denoted by ${\widehat{\xi }}_{PCE}$, are obtained by minimizing the following function:

$$P\left(\left.\xi \right|\mathbf{x}\right)=\sum _{i=1}^n{\left\{x_i-Q\left(p_i\right)\right\}}^2,$$

(33)

where $p_i={\displaystyle \frac{i}{n+1}}$ and $Q\left(p_i\right)={\displaystyle \frac{1}{\alpha \theta }}log\left(1+\theta {\left(-log(1-p_i)\right)}^{{\displaystyle \frac{1}{\mu }}}\right)$.

The PCEs ${\widehat{\xi }}_{PCE}$ can be obtained by solving the next equations simultaneously

$${\left.{\sum }_{i=1}^n\left\{x_i-Q\left(p_i\right)\right\}{G}_1\right|}_{({\widehat{\xi }}_{PCE}=\xi )}=0,$$

(34)

$${\left.{\sum }_{i=1}^n\left\{x_i-Q\left(p_i\right)\right\}{G}_2\right|}_{({\widehat{\xi }}_{PCE}=\xi )}=0,$$

(35)

and

$${\left.{\sum }_{i=1}^n\left\{x_i-Q\left(p_i\right)\right\}{G}_3\right|}_{({\widehat{\xi }}_{PCE}=\xi )}=0,$$

(36)

where ${\varrho }_i={[-log(1-p_i)]}^{1/\mu }$

$G_1={\displaystyle \frac{2}{{\alpha }^2\theta }}log(1+\theta {\varrho }_i),$ $G_2=-{\displaystyle \frac{2{\varrho }_{i}log\left[-log(1-p_i)\right]}{\alpha {\mu }^2\left(1+\theta {\varrho }_i\right)}},$ and $G_3=-{\displaystyle \frac{2}{\alpha \theta }}\left[{\displaystyle \frac{{\varrho }_i}{1+\theta {\varrho }_i}}-{\displaystyle \frac{1}{\theta }}log(1+\theta {\varrho }_i)\right].$

4.5. Cramér–Von Mises

The Cramér–von Mises (CvM) estimation method provides a minimum-distance approach for parameter estimation by minimizing the discrepancy between the empirical distribution function and the corresponding theoretical distribution, making it a robust and widely used alternative in distributional inference. From (1), the CvM estimators (CvMEs) for $\alpha$, $\mu$, and $\theta$ of the Baker–T1 distribution, denoted by ${\widehat{\xi }}_{CvME}$, can be obtained by minimizing

$$CR\left(\xi \right|\mathbf{x})={\displaystyle \frac{1}{12n}}+{\sum }_{i=1}^n{\left(F\left(x_i;\xi \right)-{\varsigma }_i\right)}^2,$$

(37)

with respect to $\alpha$, $\mu$, and $\theta$, where ${\varsigma }_i={\displaystyle \frac{2i-1}{2n}}$.

The CvMEs ${\widehat{\xi }}_{CvME}$ of $\xi$ can be obtained by solving the following three non-linear equations:

$${\left.{\sum }_{i=1}^n\left(F\left(x_i;\xi \right)-{\varsigma }_i\right)K_1\right|}_{({\widehat{\xi }}_{CvME}=\xi )}=0,$$

(38)

$${\left.{\sum }_{i=1}^n\left(F\left(x_i;\xi \right)-{\varsigma }_i\right)K_2\right|}_{({\widehat{\xi }}_{CvME}=\xi )}=0,$$

(39)

and

$${\left.{\sum }_{i=1}^n\left(F\left(x_i;\xi \right)-{\varsigma }_i\right)K_3\right|}_{({\widehat{\xi }}_{CvME}=\xi )}=0,$$

(40)

where

$K_1={\displaystyle \frac{1}{6n}}\mu A_i^{\mu -1}{x}_i{e}^{\alpha \theta x_i}exp(-A_i^{\mu })$,

$K_2={\displaystyle \frac{1}{6n}}{A}_i^{\mu }log{A}_{i}exp(-A_i^{\mu })$,

and

$K_3={\displaystyle \frac{1}{6n}}\mu A_i^{\mu -1}{\displaystyle \frac{\alpha x_i\theta e^{\alpha \theta x_i}-(e^{\alpha \theta x_i}-1)}{{\theta }^2}}exp(-A_i^{\mu })$.

4.6. Anderson–Darling and RT Anderson–Darling

The Anderson–Darling (AD) and right-tail AD (RTAD) estimation methods constitute weighted minimum-distance approaches that emphasize tail behavior by assigning greater importance to discrepancies between the empirical and theoretical distribution functions in the tail regions, thereby yielding efficient parameter estimates for distributions with pronounced tail characteristics.

From (1) and (3), the AD estimators (ADEs) for $\xi$ of the Baker–T1 distribution, denoted by ${\widehat{\xi }}_{ADE}$, can be obtained by minimizing

$$AD\left(\xi \right|\mathbf{x})=-n-{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n(2i-1)\left[log\left(F(x_i;\xi )\right)+log\left(R(x_{n-i+1};\xi )\right)\right]$$

(41)

with respect to $\alpha$, $\mu$, and $\theta$.

The ADEs ${\widehat{\xi }}_{ADE}$ of $\xi$ can be obtained by solving the following three non-linear equations:

$${\left.-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{\mu A_i^{\mu -1}{x}_i{e}^{\alpha \theta x_i-A_i^{\mu }}}{F(x_i;\xi )}}-\mu A_{n-i+1}^{\mu -1}{x}_{n-i+1}{e}^{\alpha \theta x_{n-i+1}}\right]\right|}_{({\widehat{\xi }}_{ADE}=\xi )}=0,$$

(42)

$$\begin{array}{c}\hfill {\left.-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{A_i^{\mu }log\left(A_i\right)e^{-A_i^{\mu }}}{F(x_i;\xi )}}-A_{n-i+1}^{\mu }log\left(A_{n-i+1}\right)\right]\right|}_{({\widehat{\xi }}_{ADE}=\xi )}=0,\end{array}$$

(43)

and

$${\left.-{\displaystyle \frac{\mu {\theta }^{-2}}{n}}\sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{N_i{e}^{-A_i^{\mu }}}{F(x_i;\xi )}}-N_{n-i+1}\right]\right|}_{({\widehat{\xi }}_{ADE}=\xi )}=0,$$

(44)

where $N_i=A_i^{\mu -1}\left(\alpha x_i{e}^{\alpha \theta x_i}-(e^{\alpha \theta x_i}-1)\right)e^{-A_i^{\mu }}.$

From (1) and (3), the RTAD estimators (RTADEs) for $\xi$ of the Baker–T1 distribution, denoted by ${\widehat{\xi }}_{RTADE}$, can be obtained by minimizing

$$RTAD\left(\xi \right|\mathbf{x})={\displaystyle \frac{n}{2}}-2\sum _{i=1}^n\left(F(x_i;\xi )\right)-\sum _{i=1}^n(2i-1)logR(x_{n-i+1};\xi )$$

(45)

with respect to $\alpha$, $\mu$, and $\theta$.

The RTADEs ${\widehat{\xi }}_{RTADE}$ of $\xi$ can be obtained by solving the following system of non-linear equations:

$${\left.-2\mu \sum _{i=1}^n{\epsilon }_i-\mu \sum _{i=1}^n(2i-1){\displaystyle \frac{{\epsilon }_{n-i+1}}{F(x_{n-i+1};\xi )}}\right|}_{({\widehat{\xi }}_{RTADE}=\xi )}=0,$$

(46)

$${\left.-2\sum _{i=1}^n{\kappa }_i-\sum _{i=1}^n(2i-1){\displaystyle \frac{{\kappa }_{n-i+1}}{F(x_{n-i+1};\xi )}}\right|}_{({\widehat{\xi }}_{RTADE}=\xi )}=0,$$

(47)

and

$${\left.-2\mu {\theta }^{-2}\sum _{i=1}^n{ϵ}_i{B}_i^{\mu -1}{e}^{-B_i^{\mu }}-\mu {\theta }^{-2}\sum _{i=1}^n(2i-1){\displaystyle \frac{{ϵ}_{n-i+1}{B}_{n-i+1}^{\mu -1}{e}^{-B_{n-i+1}^{\mu }}}{F(x_{n-i+1};\xi )}}\right|}_{({\widehat{\xi }}_{RTADE}=\xi )}=0,$$

(48)

respectively, where

${\epsilon }_i=x_i{B}_i^{\mu -1}{e}^{-B_i^{\mu }}{e}^{\alpha \theta x_i}$, ${\kappa }_i=B_i^{\mu }log\left(B_i\right)e^{-B_i^{\mu }}$, and ${ϵ}_i=\alpha x_i\theta e^{\alpha \theta x_i}-(e^{\alpha \theta x_i}-1)$.

5. Simulation Study

This section employs a Monte Carlo simulation study to assess the finite-sample performance of the proposed estimators under a variety of controlled settings. For each experimental configuration, 5000 independent samples are generated with sample sizes $n\in \{20,50,100,150,200\}$. The simulated data are drawn from the $\mathrm{Baker}\text{–}\mathrm{T}1\left(\xi \right)$ distribution using different combinations of the model parameters in order to represent a broad range of distributional behaviors. Specifically, two values are considered for each unknown parameter, namely $\alpha \in \{0.2,1.2\}$, $\mu \in \{0.5,1.5\}$, and $\theta \in \{0.8,1.8\}$, which are arranged into the following five parameter groups for $\xi$:

• Group-1: $(0.2,0.5,0.8)$;
• Group-2: $(1.2,1.5,1.8)$;
• Group-3: $(1.2,0.5,0.8)$;
• Group-4: $(0.2,1.5,0.8)$;
• Group-5: $(0.2,0.5,1.8)$.

The selected parameter configurations are intended to ensure a thorough assessment of estimator performance across a diverse set of plausible conditions, encompassing both symmetric and asymmetric distributions as well as light- and heavy-tailed behaviors. All numerical computations were implemented in the $\mathsf{R}$ software environment (version 4.2.2). The parameter values of $\xi$ for Group-i ($i=1,2,\dots ,5$) were chosen to represent a wide spectrum of distributional shapes associated with the Baker–T1 model. Specifically, smaller values of $\xi$ correspond to highly skewed and heavy-tailed distributions, whereas larger values lead to more symmetric and lighter-tailed forms. Consequently, the simulation design facilitates the evaluation of estimator accuracy and robustness under both extreme and moderate settings. It should be noted that the initial values of $\xi$ were set equal to their true parameter values to ensure stable convergence of the optimization algorithms while enabling a fair comparison of the intrinsic performance of the competing estimators. This choice is justified within simulation frameworks, where the true parameter values are known in advance (see Alotaibi et al. [13]), but it should not be applicable in practical applications. Alternatively, moment-based or percentage-based configuration can be used, in addition to improving network search, to assign appropriate starting points for each parameter.

Let ${\widehat{\xi }}_1,{\widehat{\xi }}_2,\dots ,{\widehat{\xi }}_N$ denote the estimates of a parameter $\xi$ obtained from N Monte Carlo replications. The estimation accuracy is evaluated using the mean squared error (MSE), mean absolute bias (MAB), and relative absolute bias (RAB), defined respectively as $\mathrm{MSE}\left(\widehat{\xi }\right)=N^{-1}{\sum }_{i=1}^N{({\widehat{\xi }}_i-\xi )}^2$, $\mathrm{MAB}\left(\widehat{\xi }\right)=N^{-1}{\sum }_{i=1}^N|{\widehat{\xi }}_i-\xi |$, and $\mathrm{RAB}\left(\widehat{\xi }\right)=N^{-1}{\sum }_{i=1}^N|({\widehat{\xi }}_i-\xi )/\xi |$. For each parameter and performance criterion, estimation methods are ranked in ascending order, with rank one indicating the best performance; this rank is referred to as the order rank (OR). The total rank (TR) of a given method is obtained by summing its ORs across all parameters (see

Table 2. Estimates (1st Col.) and ranks (2nd Col.) of $\xi$ from Group-1.

n	Metric	Par.	MLE		MPSE		LSE		WLSE		PCS		CvME		ADE		RTADE
20	MSE	$\alpha$	0.005	1	0.010	4	0.012	6	0.012	7	0.010	3	0.011	5	0.013	8	0.009	2
		$\mu$	0.008	1	0.011	3	0.010	2	0.017	6	0.024	8	0.018	7	0.013	4	0.013	5
		$\theta$	0.478	1	0.990	8	0.581	3	0.640	5	0.621	4	0.640	5	0.640	5	0.508	2
	MAB	$\alpha$	0.072	1	0.111	7	0.108	5	0.109	6	0.098	3	0.105	4	0.114	8	0.094	2
		$\mu$	0.087	1	0.105	3	0.100	2	0.132	6	0.156	8	0.133	7	0.113	4	0.116	5
		$\theta$	0.691	2	0.742	4	0.762	5	0.612	1	0.788	6	0.932	8	0.886	7	0.713	3
	RAB	$\alpha$	0.359	1	0.507	4	0.540	6	0.544	7	0.488	3	0.526	5	0.569	8	0.468	2
		$\mu$	0.174	1	0.209	3	0.200	2	0.264	6	0.312	8	0.267	7	0.225	4	0.232	5
		$\theta$	0.864	3	0.875	4	0.953	7	0.725	1	0.985	8	0.889	5	0.784	2	0.891	6
TR→				12		40		38		45		51		53		50		32
OR→				1		4		3		5		7		8		6		2
50	MSE	$\alpha$	0.004	6	0.003	3	0.005	8	0.004	4	0.003	2	0.005	7	0.002	1	0.004	5
		$\mu$	0.003	2	0.004	4	0.007	7	0.003	3	0.019	8	0.006	5	0.002	1	0.006	6
		$\theta$	0.258	3	0.234	2	0.373	7	0.300	4	0.323	5	0.453	8	0.158	1	0.359	6
	MAB	$\alpha$	0.063	6	0.055	3	0.073	8	0.059	4	0.053	2	0.072	7	0.048	1	0.060	5
		$\mu$	0.053	2	0.066	4	0.082	7	0.057	3	0.137	8	0.075	5	0.048	1	0.075	6
		$\theta$	0.508	3	0.483	2	0.611	7	0.548	4	0.568	5	0.673	8	0.397	1	0.599	6
	RAB	$\alpha$	0.316	6	0.277	3	0.367	8	0.296	4	0.267	2	0.362	7	0.239	1	0.302	5
		$\mu$	0.107	2	0.131	4	0.164	7	0.113	3	0.274	8	0.151	5	0.096	1	0.151	6
		$\theta$	0.635	3	0.604	2	0.763	7	0.685	4	0.710	5	0.841	8	0.496	1	0.749	6
TR→				33		27		66		33		45		60		9		51
OR→				3.5		2		8		3.5		5		7		1		6
100	MSE	$\alpha$	0.003	7	0.001	1	0.002	5	0.002	2	0.002	5	0.003	8	0.002	3	0.002	3
		$\mu$	0.002	2	0.003	4	0.003	5	0.001	1	0.009	8	0.003	6	0.002	3	0.004	7
		$\theta$	0.128	5	0.081	1	0.162	6	0.101	2	0.166	7	0.322	8	0.125	4	0.112	3
	MAB	$\alpha$	0.050	7	0.038	1	0.045	5	0.040	2	0.045	6	0.050	8	0.042	3	0.042	4
		$\mu$	0.045	2	0.051	4	0.052	5	0.037	1	0.096	8	0.059	6	0.048	3	0.060	7
		$\theta$	0.358	5	0.285	1	0.403	6	0.318	2	0.407	7	0.568	8	0.353	4	0.335	3
	RAB	$\alpha$	0.250	7	0.189	1	0.224	5	0.198	2	0.224	6	0.252	8	0.210	3	0.210	4
		$\mu$	0.091	2	0.102	4	0.103	5	0.073	1	0.191	8	0.118	6	0.097	3	0.120	7
		$\theta$	0.447	5	0.356	1	0.504	6	0.398	2	0.509	7	0.710	8	0.441	4	0.419	3
TR→				42		18		48		15		62		66		30		41
OR→				5		2		6		1		7		8		3		4
150	MSE	$\alpha$	0.001	1	0.002	7	0.002	5	0.002	6	0.001	2	0.002	8	0.001	4	0.001	3
		$\mu$	0.001	3	0.001	2	0.002	5	0.002	6	0.005	8	0.001	1	0.002	4	0.002	7
		$\theta$	0.059	1	0.075	2	0.164	8	0.138	7	0.102	4	0.138	6	0.103	5	0.092	3
	MAB	$\alpha$	0.027	1	0.042	7	0.039	5	0.040	6	0.032	2	0.043	8	0.033	4	0.033	3
		$\mu$	0.039	3	0.035	2	0.041	5	0.043	6	0.068	8	0.032	1	0.040	4	0.045	7
		$\theta$	0.243	1	0.273	2	0.405	8	0.372	7	0.320	4	0.371	6	0.320	5	0.303	3
	RAB	$\alpha$	0.137	1	0.210	7	0.197	5	0.199	6	0.160	2	0.216	8	0.166	4	0.165	3
		$\mu$	0.077	3	0.070	2	0.083	5	0.086	6	0.136	8	0.064	1	0.080	4	0.089	7
		$\theta$	0.304	1	0.342	2	0.506	8	0.465	7	0.399	4	0.464	6	0.400	5	0.379	3
TR→			15		33		54		57		42			45		39		39
OR→				1		2		7		8		5		6		3.5		3.5
200	MSE	$\alpha$	0.001	1	0.001	2	0.001	5	0.001	8	0.001	6	0.001	7	0.001	4	0.001	3
		$\mu$	0.001	1	0.001	2	0.001	6	0.001	4	0.005	8	0.001	3	0.001	5	0.002	7
		$\theta$	0.049	1	0.052	2	0.085	8	0.070	4	0.073	5	0.075	6	0.079	7	0.061	3
	MAB	$\alpha$	0.026	1	0.026	2	0.032	5	0.036	8	0.034	6	0.034	7	0.031	4	0.028	3
		$\mu$	0.027	1	0.031	2	0.038	6	0.032	4	0.068	8	0.031	3	0.033	5	0.042	7
		$\theta$	0.221	1	0.229	2	0.291	8	0.265	4	0.270	5	0.273	6	0.281	7	0.248	3
	RAB	$\alpha$	0.128	1	0.132	2	0.162	5	0.179	8	0.168	6	0.170	7	0.153	4	0.138	3
		$\mu$	0.054	1	0.062	2	0.077	6	0.064	4	0.136	8	0.062	3	0.065	5	0.083	7
		$\theta$	0.276	1	0.286	2	0.364	8	0.331	4	0.338	5	0.342	6	0.352	7	0.310	3
TR→				9		18		57		48		57		48		48		39
OR→				1		2		7.5		5		7.5		5		5		3

,

Table 3. Estimates (1st Col.) and ranks (2nd Col.) of $\xi$ from Group-2.

n	Metric	Par.	MLE		MPSE		LSE		WLSE		PCS		CvME		ADE		RTADE
20	MSE	$\alpha$	0.541	1	0.714	2	1.087	7	0.884	4	0.829	3	1.116	8	1.056	6	0.934	5
		$\mu$	0.210	1	0.275	3	0.275	2	0.415	8	0.277	4	0.401	7	0.294	5	0.300	6
		$\theta$	1.440	3	1.449	5	1.445	4	1.459	7	1.464	8	1.449	6	1.439	2	1.425	1
	MAB	$\alpha$	0.735	1	0.845	2	1.043	7	0.940	4	0.911	3	1.056	8	1.028	6	0.966	5
		$\mu$	0.458	1	0.525	3	0.524	2	0.644	8	0.526	4	0.634	7	0.542	5	0.548	6
		$\theta$	1.222	3	1.252	8	1.231	6	1.243	7	1.203	2	1.229	5	1.228	4	1.194	1
	RAB	$\alpha$	0.409	1	0.469	2	0.579	7	0.522	4	0.506	3	0.587	8	0.571	6	0.537	5
		$\mu$	0.305	1	0.350	3	0.349	2	0.429	8	0.351	4	0.422	7	0.362	5	0.365	6
		$\theta$	1.114	2	1.152	4	1.124	3	1.113	1	1.187	7	1.181	6	1.194	8	1.162	5
TR→				14		32		40		51		38		62		47		40
OR→				1		1		2.5		4		1		3		2		1
50	MSE	$\alpha$	0.329	2	0.274	1	0.799	7	0.425	4	0.465	5	0.846	8	0.337	3	0.533	6
		$\mu$	0.098	3	0.079	2	0.206	7	0.132	5	0.107	4	0.227	8	0.073	1	0.159	6
		$\theta$	1.055	3	0.859	1	1.440	5	1.440	5	1.159	4	1.440	5	0.985	2	1.440	5
	MAB	$\alpha$	0.574	2	0.523	1	0.894	7	0.651	4	0.682	5	0.919	8	0.580	3	0.730	6
		$\mu$	0.313	3	0.280	2	0.454	7	0.363	5	0.327	4	0.476	8	0.269	1	0.399	6
		$\theta$	1.027	3	0.927	1	1.152	5	1.225	8	1.076	4	1.185	7	0.992	2	1.176	6
	RAB	$\alpha$	0.319	2	0.291	1	0.496	7	0.362	4	0.379	5	0.511	8	0.322	3	0.406	6
		$\mu$	0.209	3	0.187	2	0.303	7	0.242	5	0.218	4	0.317	8	0.179	1	0.266	6
		$\theta$	0.856	1	0.972	5	1.076	6	0.942	4	0.897	2	1.121	8	0.927	3	1.087	7
TR→				22		16		58		44		37		68		19		54
OR→				3		1		5		3		2		3		1		1
100	MSE	$\alpha$	0.233	5	0.120	1	0.272	6	0.132	2	0.185	3	0.429	8	0.191	4	0.286	7
		$\mu$	0.071	4	0.066	3	0.105	7	0.042	1	0.090	5	0.093	6	0.053	2	0.131	8
		$\theta$	0.790	4	0.613	2	1.102	7	0.407	1	0.796	5	1.440	8	0.717	3	0.936	6
	MAB	$\alpha$	0.482	5	0.346	1	0.521	6	0.363	2	0.429	3	0.655	8	0.437	4	0.535	7
		$\mu$	0.266	4	0.257	3	0.324	7	0.205	1	0.299	5	0.306	6	0.231	2	0.361	8
		$\theta$	0.889	4	0.783	2	1.050	7	0.638	1	0.892	5	1.200	8	0.846	3	0.967	6
	RAB	$\alpha$	0.268	5	0.192	1	0.290	6	0.202	2	0.238	3	0.364	8	0.243	4	0.297	7
		$\mu$	0.177	4	0.171	3	0.216	7	0.137	1	0.200	5	0.204	6	0.154	2	0.241	8
		$\theta$	0.741	4	0.652	2	0.875	7	0.532	1	0.743	5	0.990	8	0.705	3	0.806	6
TR→				39		18		60		12		39		66		27		63
OR→				4.5		2		4		1		2		3		1		1
150	MSE	$\alpha$	0.087	1	0.107	2	0.236	7	0.189	5	0.112	3	0.305	8	0.171	4	0.190	6
		$\mu$	0.039	2	0.020	1	0.072	7	0.063	6	0.044	3	0.057	4	0.057	5	0.073	8
		$\theta$	0.405	3	0.327	1	1.131	8	0.513	4	0.337	2	0.826	7	0.530	5	0.654	6
	MAB	$\alpha$	0.295	1	0.327	2	0.486	7	0.435	5	0.334	3	0.552	8	0.414	4	0.436	6
		$\mu$	0.198	2	0.143	1	0.268	7	0.250	6	0.209	3	0.238	4	0.238	5	0.270	8
		$\theta$	0.637	3	0.572	1	1.064	8	0.716	4	0.580	2	0.909	7	0.728	5	0.809	6
	RAB	$\alpha$	0.164	1	0.182	2	0.270	7	0.242	5	0.186	3	0.307	8	0.230	4	0.242	6
		$\mu$	0.132	2	0.095	1	0.179	7	0.167	6	0.139	3	0.159	4	0.159	5	0.180	8
		$\theta$	0.530	3	0.476	1	0.886	8	0.597	4	0.484	2	0.757	7	0.607	5	0.674	6
TR→				18		12		66		45		24		57		42		60
OR→				2		1		6		3		1		2		1		1
200	MSE	$\alpha$	0.074	1	0.084	2	0.173	7	0.114	4	0.088	3	0.193	8	0.115	5	0.151	6
		$\mu$	0.022	2	0.021	1	0.050	6	0.032	4	0.028	3	0.057	7	0.035	5	0.072	8
		$\theta$	0.267	2	0.311	3	0.629	7	0.432	4	0.258	1	0.639	8	0.476	5	0.616	6
	MAB	$\alpha$	0.272	1	0.289	2	0.416	7	0.338	4	0.296	3	0.440	8	0.340	5	0.389	6
		$\mu$	0.147	2	0.146	1	0.225	6	0.178	4	0.166	3	0.240	7	0.187	5	0.267	8
		$\theta$	0.517	2	0.558	3	0.793	7	0.657	4	0.508	1	0.799	8	0.690	5	0.785	6
	RAB	$\alpha$	0.151	1	0.161	2	0.231	7	0.188	4	0.164	3	0.244	8	0.189	5	0.216	6
		$\mu$	0.098	2	0.097	1	0.150	6	0.119	4	0.111	3	0.160	7	0.125	5	0.178	8
		$\theta$	0.430	2	0.465	3	0.661	7	0.548	4	0.423	1	0.666	8	0.575	5	0.654	6
TR→				15		18		60		36		21		69		45		60
OR→				1		1		4.5		2		1		3		1		1

,

Table 4. Estimates (1st Col.) and ranks (2nd Col.) of $\xi$ from Group-3.

n	Metric	Par.	MLE		MPSE		LSE		WLSE		PCS		CvME		ADE		RTADE
20	MSE	$\alpha$	0.183	1	0.391	4	0.420	6	0.427	7	0.343	3	0.398	5	0.493	8	0.315	2
		$\mu$	0.007	1	0.011	3	0.010	2	0.017	6	0.026	8	0.019	7	0.013	4	0.013	5
		$\theta$	0.431	1	0.498	2	0.581	4	0.640	5	0.640	5	0.640	5	0.640	5	0.508	3
	MAB	$\alpha$	0.428	1	0.625	4	0.648	6	0.653	7	0.585	3	0.631	5	0.702	8	0.561	2
		$\mu$	0.084	1	0.105	3	0.100	2	0.132	6	0.160	8	0.137	7	0.116	4	0.116	5
		$\theta$	0.656	1	0.705	2	0.762	4	0.832	8	0.810	5	0.824	7	0.818	6	0.713	3
	RAB	$\alpha$	0.357	1	0.521	4	0.540	6	0.544	7	0.488	3	0.526	5	0.585	8	0.468	2
		$\mu$	0.168	1	0.209	3	0.200	2	0.264	6	0.320	8	0.274	7	0.232	4	0.232	5
		$\theta$	0.820	1	0.882	2	0.953	7	0.988	8	0.952	6	0.912	4	0.929	5	0.891	3
TR→				9		27		39		60		49		52		52		30
OR→				1		2		4		8		5		6.5		6.5		3
50	MSE	$\alpha$	0.144	6	0.117	3	0.194	8	0.131	4	0.103	2	0.189	7	0.086	1	0.131	5
		$\mu$	0.003	2	0.004	4	0.007	7	0.004	3	0.020	8	0.006	5	0.003	1	0.006	6
		$\theta$	0.247	3	0.229	2	0.373	7	0.300	4	0.323	5	0.453	8	0.166	1	0.359	6
	MAB	$\alpha$	0.379	6	0.342	3	0.440	8	0.362	4	0.321	2	0.435	7	0.293	1	0.363	5
		$\mu$	0.055	2	0.066	4	0.081	7	0.060	3	0.140	8	0.075	5	0.050	1	0.075	6
		$\theta$	0.497	3	0.479	2	0.611	7	0.548	4	0.568	5	0.673	8	0.408	1	0.599	6
	RAB	$\alpha$	0.316	6	0.285	3	0.367	8	0.301	4	0.267	2	0.362	7	0.244	1	0.302	5
		$\mu$	0.110	2	0.131	4	0.162	7	0.120	3	0.281	8	0.151	5	0.100	1	0.151	6
		$\theta$	0.622	3	0.598	2	0.763	7	0.685	4	0.710	5	0.841	8	0.510	1	0.749	6
TR→				33		27		66		33		45		60		9		51
OR→				3.5		2		8		3.5		5		7		1		6
100	MSE	$\alpha$	0.072	6	0.051	1	0.072	5	0.063	2	0.073	7	0.091	8	0.064	3	0.070	4
		$\mu$	0.002	2	0.003	4	0.003	5	0.001	1	0.010	8	0.003	6	0.002	3	0.004	7
		$\theta$	0.109	2	0.083	1	0.166	6	0.110	3	0.175	7	0.323	8	0.130	5	0.117	4
	MAB	$\alpha$	0.269	6	0.226	1	0.268	5	0.251	2	0.270	7	0.302	8	0.253	3	0.265	4
		$\mu$	0.045	2	0.051	4	0.052	5	0.037	1	0.102	8	0.059	6	0.049	3	0.061	7
		$\theta$	0.330	2	0.289	1	0.407	6	0.332	3	0.418	7	0.568	8	0.361	5	0.342	4
	RAB	$\alpha$	0.224	6	0.188	1	0.224	5	0.209	2	0.225	7	0.252	8	0.211	3	0.221	4
		$\mu$	0.089	2	0.102	4	0.105	5	0.073	1	0.204	8	0.118	6	0.099	3	0.122	7
		$\theta$	0.413	2	0.361	1	0.509	6	0.414	3	0.523	7	0.710	8	0.451	5	0.427	4
TR→				30		18		48		18		66		66		33		45
OR→				3		1.5		6		1.5		7.5		7.5		4		5
150	MSE	$\alpha$	0.027	1	0.064	6	0.056	4	0.069	8	0.037	2	0.068	7	0.056	5	0.045	3
		$\mu$	0.001	2	0.001	1	0.002	4	0.002	6	0.005	8	0.001	2	0.002	5	0.002	7
		$\theta$	0.058	1	0.075	2	0.164	8	0.163	7	0.110	4	0.147	6	0.112	5	0.105	3
	MAB	$\alpha$	0.165	1	0.254	6	0.236	4	0.262	8	0.192	2	0.261	7	0.237	5	0.211	3
		$\mu$	0.038	2	0.035	1	0.042	4	0.044	6	0.069	8	0.038	3	0.043	5	0.046	7
		$\theta$	0.241	1	0.274	2	0.405	8	0.403	7	0.332	4	0.383	6	0.334	5	0.324	3
	RAB	$\alpha$	0.137	1	0.211	6	0.197	4	0.218	8	0.160	2	0.217	7	0.197	5	0.176	3
		$\mu$	0.075	2	0.070	1	0.085	4	0.089	6	0.137	8	0.075	3	0.086	5	0.092	7
		$\theta$	0.301	1	0.343	2	0.506	8	0.504	7	0.415	4	0.479	6	0.418	5	0.405	3
TR→				12		27		48		63		42		47		45		39
OR→				1		2		7		8		4		6		5		3
200	MSE	$\alpha$	0.023	1	0.024	2	0.039	5	0.057	8	0.042	6	0.049	7	0.037	4	0.028	3
		$\mu$	0.001	1	0.001	2	0.001	6	0.001	4	0.005	8	0.001	3	0.001	5	0.002	7
		$\theta$	0.047	1	0.053	2	0.089	7	0.083	5	0.075	4	0.095	8	0.089	6	0.064	3
	MAB	$\alpha$	0.151	1	0.155	2	0.197	5	0.239	8	0.204	6	0.222	7	0.192	4	0.167	3
		$\mu$	0.027	1	0.032	2	0.038	6	0.034	4	0.068	8	0.034	3	0.035	5	0.042	7
		$\theta$	0.218	1	0.229	2	0.299	7	0.289	5	0.273	4	0.308	8	0.298	6	0.253	3
	RAB	$\alpha$	0.125	1	0.129	2	0.164	5	0.199	8	0.170	6	0.185	7	0.160	4	0.139	3
		$\mu$	0.054	1	0.064	2	0.077	6	0.068	4	0.137	8	0.067	3	0.071	5	0.084	7
		$\theta$	0.272	1	0.287	2	0.374	7	0.361	5	0.342	4	0.385	8	0.373	6	0.316	3
TR→				9		18		54		51		54		54		45		39
OR→				1		2		7		5		7		7		4		3

,

Table 5. Estimates (1st Col.) and ranks (2nd Col.) of $\xi$ from Group-4.

n	Metric	Par.	MLE		MPSE		LSE		WLSE		PCS		CvME		ADE		RTADE
20	MSE	$\alpha$	0.006	3	0.004	1	0.008	4	0.008	7	0.006	2	0.008	5	0.008	8	0.008	6
		$\mu$	0.215	4	0.172	1	0.202	3	0.351	8	0.188	2	0.320	7	0.230	5	0.247	6
		$\theta$	0.640	1	0.640	1	0.640	1	0.679	8	0.640	1	0.640	1	0.640	1	0.640	1
	MAB	$\alpha$	0.079	3	0.066	1	0.087	4	0.090	7	0.076	2	0.088	5	0.092	8	0.089	6
		$\mu$	0.463	4	0.414	1	0.450	3	0.593	8	0.433	2	0.566	7	0.479	5	0.497	6
		$\theta$	0.800	1	0.800	1	0.800	1	0.824	8	0.800	1	0.800	1	0.800	1	0.800	1
	RAB	$\alpha$	0.394	3	0.331	1	0.435	4	0.450	7	0.378	2	0.438	5	0.461	8	0.445	6
		$\mu$	0.309	4	0.276	1	0.300	3	0.395	8	0.289	2	0.377	7	0.319	5	0.331	6
		$\theta$	1.142	4	1.099	2	1.252	8	1.030	1	1.242	6	1.125	3	1.192	5	1.251	7
TR→				27		10		31		62		20		41		46		45
OR→				3		1		4		8		2		5		7		6
50	MSE	$\alpha$	0.003	1	0.003	4	0.006	7	0.003	3	0.004	5	0.007	8	0.003	2	0.005	6
		$\mu$	0.066	2	0.074	3	0.175	8	0.081	4	0.093	5	0.167	7	0.054	1	0.116	6
		$\theta$	0.476	1	0.616	4	0.640	5	0.640	5	0.606	3	0.640	5	0.500	2	0.640	5
	MAB	$\alpha$	0.051	1	0.057	4	0.076	7	0.055	3	0.064	5	0.081	8	0.054	2	0.069	6
		$\mu$	0.257	2	0.273	3	0.418	8	0.284	4	0.305	5	0.409	7	0.233	1	0.341	6
		$\theta$	0.690	1	0.785	4	0.800	5	0.800	5	0.778	3	0.800	5	0.707	2	0.800	5
	RAB	$\alpha$	0.256	1	0.283	4	0.379	7	0.275	3	0.320	5	0.403	8	0.271	2	0.344	6
		$\mu$	0.172	2	0.182	3	0.279	8	0.189	4	0.203	5	0.273	7	0.155	1	0.227	6
		$\theta$	0.862	1	0.981	4	1.025	7	0.987	5	0.973	3	1.008	6	0.884	2	1.124	8
TR→				12		33		62		36		39		61		15		54
OR→				1		3		8		4		5		7		2		6
100	MSE	$\alpha$	0.001	1	0.002	6	0.003	7	0.001	2	0.002	4	0.004	8	0.002	3	0.002	5
		$\mu$	0.062	4	0.060	3	0.088	7	0.035	1	0.068	5	0.080	6	0.039	2	0.106	8
		$\theta$	0.317	2	0.385	4	0.490	7	0.227	1	0.427	5	0.640	8	0.345	3	0.485	6
	MAB	$\alpha$	0.035	1	0.047	6	0.051	7	0.036	2	0.046	4	0.061	8	0.042	3	0.047	5
		$\mu$	0.249	4	0.246	3	0.296	7	0.188	1	0.261	5	0.283	6	0.198	2	0.326	8
		$\theta$	0.563	2	0.620	4	0.700	7	0.476	1	0.654	5	0.800	8	0.587	3	0.696	6
	RAB	$\alpha$	0.176	1	0.237	6	0.255	7	0.179	2	0.229	4	0.304	8	0.209	3	0.234	5
		$\mu$	0.166	4	0.164	3	0.197	7	0.126	1	0.174	5	0.188	6	0.132	2	0.217	8
		$\theta$	0.704	2	0.775	4	0.875	7	0.595	1	0.817	5	0.969	8	0.733	3	0.871	6
TR→				21		39		63		12		42		66		24		57
OR→				2		4		7		1		5		8		3		6
150	MSE	$\alpha$	0.001	3	0.001	1	0.002	7	0.002	4	0.001	2	0.003	8	0.002	5	0.002	6
		$\mu$	0.021	1	0.031	3	0.051	6	0.048	4	0.030	2	0.051	7	0.049	5	0.069	8
		$\theta$	0.170	2	0.216	3	0.571	8	0.243	4	0.157	1	0.445	7	0.313	5	0.372	6
	MAB	$\alpha$	0.035	3	0.029	1	0.048	7	0.040	4	0.031	2	0.052	8	0.040	5	0.041	6
		$\mu$	0.144	1	0.176	3	0.225	6	0.220	4	0.174	2	0.225	7	0.222	5	0.263	8
		$\theta$	0.412	2	0.465	3	0.755	8	0.493	4	0.396	1	0.667	7	0.559	5	0.610	6
	RAB	$\alpha$	0.174	3	0.143	1	0.239	7	0.198	4	0.157	2	0.258	8	0.201	5	0.205	6
		$\mu$	0.096	1	0.117	3	0.150	6	0.147	4	0.116	2	0.150	7	0.148	5	0.175	8
		$\theta$	0.516	2	0.581	3	0.944	8	0.617	4	0.495	1	0.834	7	0.699	5	0.762	6
TR→				18		21		63		36		15		66		45		60
OR→				2		3		7		4		1		8		5		6
200	MSE	$\alpha$	0.001	2	0.001	3	0.002	7	0.001	4	0.001	1	0.002	8	0.001	5	0.001	6
		$\mu$	0.019	1	0.019	2	0.048	6	0.021	3	0.025	4	0.048	7	0.031	5	0.055	8
		$\theta$	0.149	3	0.129	2	0.344	8	0.190	4	0.112	1	0.335	7	0.269	5	0.315	6
	MAB	$\alpha$	0.026	2	0.026	3	0.041	7	0.030	4	0.024	1	0.042	8	0.034	5	0.035	6
		$\mu$	0.136	1	0.138	2	0.220	6	0.144	3	0.158	4	0.220	7	0.176	5	0.235	8
		$\theta$	0.386	3	0.359	2	0.586	8	0.436	4	0.335	1	0.579	7	0.519	5	0.561	6
	RAB	$\alpha$	0.130	2	0.132	3	0.203	7	0.151	4	0.122	1	0.210	8	0.172	5	0.176	6
		$\mu$	0.091	1	0.092	2	0.147	6	0.096	3	0.105	4	0.147	7	0.117	5	0.156	8
		$\theta$	0.482	3	0.449	2	0.733	8	0.545	4	0.419	1	0.724	7	0.648	5	0.701	6
TR→				18		21		63		33		18		66		45		60
OR→				1.5		3		7		4		1.5		8		5		6

and

Table 6. Estimates (1st Col.) and ranks (2nd Col.) of $\xi$ from Group-5.

n	Metric	Par.	MLE		MPSE		LSE		WLSE		PCS		CvME		ADE		RTADE
20	MSE	$\alpha$	0.006	1	0.015	3	0.018	7	0.015	4	0.017	6	0.015	5	0.018	8	0.011	2
		$\mu$	0.010	1	0.014	3	0.012	2	0.021	6	0.035	8	0.021	7	0.016	4	0.019	5
		$\theta$	1.593	1	2.283	2	2.427	5	2.913	6	2.364	3	3.126	8	3.048	7	2.365	4
	MAB	$\alpha$	0.077	1	0.121	3	0.134	7	0.124	5	0.130	6	0.124	4	0.135	8	0.104	2
		$\mu$	0.097	1	0.117	3	0.109	2	0.143	6	0.186	8	0.144	7	0.126	4	0.138	5
		$\theta$	1.261	1	1.511	2	1.558	5	1.707	6	1.538	3	1.768	8	1.746	7	1.538	4
	RAB	$\alpha$	0.383	1	0.606	3	0.671	7	0.622	5	0.649	6	0.621	4	0.675	8	0.518	2
		$\mu$	0.195	1	0.235	3	0.218	2	0.286	6	0.372	8	0.288	7	0.252	4	0.275	5
		$\theta$	0.701	1	0.839	2	0.865	5	0.948	6	0.854	3	0.982	8	0.970	7	0.854	4
TR→				9		24		42		50		51		58		57		33
OR→				1		2		4		5		6		8		7		3
50	MSE	$\alpha$	0.004	4	0.004	3	0.007	7	0.004	1	0.005	5	0.008	8	0.004	2	0.006	6
		$\mu$	0.003	2	0.005	4	0.009	7	0.004	3	0.015	8	0.007	6	0.003	1	0.006	5
		$\theta$	1.038	2	1.077	3	1.690	7	1.080	4	1.574	5	1.596	6	0.721	1	1.726	8
	MAB	$\alpha$	0.065	4	0.063	3	0.084	7	0.059	1	0.073	5	0.091	8	0.061	2	0.078	6
		$\mu$	0.058	2	0.070	4	0.097	7	0.062	3	0.121	8	0.085	6	0.052	1	0.079	5
		$\theta$	1.019	2	1.038	3	1.300	7	1.039	4	1.254	5	1.263	6	0.849	1	1.314	8
	RAB	$\alpha$	0.326	4	0.315	3	0.421	7	0.296	1	0.363	5	0.456	8	0.304	2	0.388	6
		$\mu$	0.116	2	0.139	4	0.195	7	0.123	3	0.243	8	0.170	6	0.104	1	0.158	5
		$\theta$	0.566	2	0.577	3	0.722	7	0.577	4	0.697	5	0.702	6	0.472	1	0.730	8
TR→				24		30		63		24		54		60		12		57
OR→				2.5		4		8		2.5		5		7		1		6
100	MSE	$\alpha$	0.003	5	0.002	1	0.003	6	0.002	3	0.004	8	0.003	7	0.003	4	0.002	2
		$\mu$	0.002	2	0.004	5	0.003	4	0.002	1	0.010	8	0.004	6	0.003	3	0.005	7
		$\theta$	0.518	3	0.492	2	0.800	6	0.430	1	1.204	7	1.260	8	0.671	5	0.609	4
	MAB	$\alpha$	0.053	5	0.039	1	0.056	6	0.048	3	0.060	8	0.056	7	0.052	4	0.047	2
		$\mu$	0.048	2	0.059	5	0.054	4	0.043	1	0.102	8	0.064	6	0.054	3	0.070	7
		$\theta$	0.719	3	0.702	2	0.895	6	0.656	1	1.097	7	1.122	8	0.819	5	0.780	4
	RAB	$\alpha$	0.267	5	0.197	1	0.280	6	0.241	3	0.298	8	0.282	7	0.261	4	0.234	2
		$\mu$	0.097	2	0.119	5	0.109	4	0.086	1	0.204	8	0.128	6	0.108	3	0.140	7
		$\theta$	0.400	3	0.390	2	0.497	6	0.364	1	0.610	7	0.624	8	0.455	5	0.434	4
TR→				30		24		48		15		69		63		36		39
OR→				3		2		6		1		8		7		4		5
150	MSE	$\alpha$	0.001	1	0.002	5	0.002	7	0.002	6	0.002	3	0.003	8	0.002	4	0.002	2
		$\mu$	0.001	1	0.002	5	0.002	4	0.002	6	0.005	8	0.002	3	0.002	2	0.003	7
		$\theta$	0.251	1	0.438	4	0.982	8	0.471	5	0.411	2	0.661	7	0.413	3	0.616	6
	MAB	$\alpha$	0.030	1	0.045	5	0.050	7	0.047	6	0.040	3	0.051	8	0.044	4	0.040	2
		$\mu$	0.038	1	0.042	5	0.042	4	0.044	6	0.071	8	0.041	3	0.041	2	0.054	7
		$\theta$	0.501	1	0.662	4	0.991	8	0.686	5	0.641	2	0.813	7	0.642	3	0.785	6
	RAB	$\alpha$	0.152	1	0.224	5	0.249	7	0.235	6	0.201	3	0.256	8	0.218	4	0.199	2
		$\mu$	0.075	1	0.084	5	0.083	4	0.089	6	0.143	8	0.083	3	0.082	2	0.109	7
		$\theta$	0.279	1	0.368	4	0.551	8	0.381	5	0.356	2	0.452	7	0.357	3	0.436	6
TR→				9		42		57		51		39		54		27		45
OR→				1		4		8		6		3		7		2		5
200	MSE	$\alpha$	0.001	1	0.001	2	0.002	5	0.002	6	0.002	7	0.002	8	0.001	4	0.001	3
		$\mu$	0.001	1	0.001	5	0.002	6	0.001	4	0.004	8	0.001	3	0.001	2	0.003	7
		$\theta$	0.183	1	0.281	3	0.417	6	0.273	2	0.509	8	0.455	7	0.360	4	0.370	5
	MAB	$\alpha$	0.027	1	0.031	2	0.039	5	0.040	6	0.041	7	0.046	8	0.036	4	0.034	3
		$\mu$	0.029	1	0.036	5	0.042	6	0.036	4	0.062	8	0.035	3	0.034	2	0.051	7
		$\theta$	0.428	1	0.530	3	0.645	6	0.522	2	0.713	8	0.675	7	0.600	4	0.608	5
	RAB	$\alpha$	0.134	1	0.156	2	0.193	5	0.200	6	0.203	7	0.229	8	0.180	4	0.171	3
		$\mu$	0.059	1	0.072	5	0.083	6	0.071	4	0.123	8	0.071	3	0.068	2	0.102	7
		$\theta$	0.238	1	0.295	3	0.359	6	0.290	2	0.396	8	0.375	7	0.334	4	0.338	5
TR→				9		30		51		36		69		54		30		45
OR→				1		2.5		6		4		8		7		2.5		5

). To summarize performance over five different parameter settings, the mean TR (MTR) and mean OR (MOR) are computed as the averages of the corresponding TRs and ORs across all scenarios, respectively, with smaller values indicating superior overall efficiency; see

Table 7. The MTRs and MORs of $\xi$ from Group-$i,i=1,2,\dots ,5$.

Group	n	ML	MPS	LS	WLS	PC	CvM	AD	RTAD
Group-1	20	1.0	4.0	3.0	5.0	7.0	8.0	6.0	2.0
	50	3.5	2.0	8.0	3.5	5.0	7.0	1.0	6.0
	100	5.0	2.0	6.0	1.0	7.0	8.0	3.0	4.0
	150	1.0	2.0	7.0	8.0	5.0	6.0	3.5	3.5
	200	1.0	2.0	7.5	5.0	7.5	5.0	5.0	3.0
Group-2	20	1.0	1.0	2.5	4.0	1.0	3.0	2.0	1.0
	50	3.0	1.0	5.0	3.0	2.0	3.0	1.0	1.0
	100	4.5	2.0	4.0	1.0	2.0	3.0	1.0	1.0
	150	2.0	1.0	6.0	3.0	1.0	2.0	1.0	1.0
	200	1.0	1.0	4.5	2.0	1.0	3.0	1.0	1.0
Group-3	20	1.0	2.0	4.0	8.0	5.0	6.5	6.5	3.0
	50	3.5	2.0	8.0	3.5	5.0	7.0	1.0	6.0
	100	3.0	1.5	6.0	1.5	7.5	7.5	4.0	5.0
	150	1.0	2.0	7.0	8.0	4.0	6.0	5.0	3.0
	200	1.0	2.0	7.0	5.0	7.0	7.0	4.0	3.0
Group-4	20	3.0	1.0	4.0	8.0	2.0	5.0	7.0	6.0
	50	1.0	3.0	8.0	4.0	5.0	7.0	2.0	6.0
	100	2.0	4.0	7.0	1.0	5.0	8.0	3.0	6.0
	150	2.0	3.0	7.0	4.0	1.0	8.0	5.0	6.0
	200	1.5	3.0	7.0	4.0	1.5	8.0	5.0	6.0
Group-5	20	1.0	2.0	4.0	5.0	6.0	8.0	7.0	3.0
	50	2.5	4.0	8.0	2.5	5.0	7.0	1.0	6.0
	100	3.0	2.0	6.0	1.0	8.0	7.0	4.0	5.0
	150	1.0	4.0	8.0	6.0	3.0	7.0	2.0	5.0
	200	1.0	2.5	6.0	4.0	8.0	7.0	2.5	5.0
MTR→		2.02	2.24	6.02	4.04	4.46	6.16	3.34	3.90
MOR→		1	2	7	5	6	8	3	4

.

The simulation outcomes reported in Table 2, Table 3, Table 4, Table 5 and Table 6 consistently indicate that estimator accuracy improves as the sample size n increases, thereby supporting the consistency of all considered estimators. Among the competing methods, the ML and MPS procedures repeatedly yield more precise and stable estimates relative to the remaining approaches. The results listed in Table 7 reveal that the ML method exhibits superior performance in most settings, followed closely by the MPS and AD methods. Based on average ranking performance, the estimation techniques can be ordered from most to least efficient as follows: ML, MPS, AD, RTAD, WLS, PC, LS, and CvM. These conclusions are fully consistent with the detailed results presented in Table 2, Table 3, Table 4, Table 5 and Table 6. As an illustrative example, Figure 3 depicts the simulation results for Group-1, including the MSE, MAB, and RAB values associated with the parameters $\alpha$, $\mu$, and $\theta$. The fitted curves corresponding to each estimation method further corroborate the numerical findings reported in Table 2. Overall, the ML method is recommended for practical applications involving the Baker–T1 distribution, with the MPS approach representing a robust and effective alternative in situations where ML estimation is impractical or computationally demanding. It is worth noticing here that the ML and MPS methods generally achieve faster convergence with average run-times of approximately 0.35 and 0.75 s/replication and require fewer iterations compared to others. These findings highlight the reliability and robustness of likelihood-based approaches, particularly ML and MPS, in achieving efficient parameter estimation for the Baker–T1 distribution across varying sample sizes.

6. Real-World Applications

This section presents two illustrative applications—one drawn from engineering and the other from a clinical setting—to showcase the practical effectiveness of the proposed inferential procedures and to emphasize the flexibility of the Baker–T1 model in modeling diverse real-world phenomena. The considered applications are described below.

• Application 1: Reliability studies frequently rely on data describing the number of shocks experienced by a component prior to failure, as such information is critical for assessing failure dynamics under repeated loading conditions. Murthy et al. [17] emphasize that shock-count data are instrumental in modeling lifetime variability and in formulating stochastic degradation models driven by cumulative damage mechanisms. In this setting, the dataset presented in

  Table 8. Data points for petroleum reservoirs (top) and mechanical components (bottom).

  Application 1: Shocks
  0.2	0.3	0.6	0.6	0.7	0.9	0.9	1.0	1.0	1.1
  1.2	1.2	1.2	1.3	1.3	1.3	1.5	1.6	1.6	1.8
  Application 2: Leukemia
  0.315	0.496	0.616	1.145	1.208	1.263	1.414	2.025	2.036	2.162
  2.211	2.370	2.532	2.693	2.805	2.910	2.912	3.192	3.263	3.348
  3.348	3.427	3.499	3.534	3.751	3.767	3.858	3.986	4.049	4.244
  4.323	4.381	4.392	4.397	4.647	4.753	4.929	4.973	5.074	5.381

  , which contains twenty recorded shock-to-failure observations, provides a meaningful empirical foundation for analyzing the reliability behavior of systems operating in random shock environments; see Cordeiro et al. [18] and Elshahhat and Seyam [19] for a detailed description.
• Application 2: Blood cancer, commonly referred to as leukemia, affects the blood-forming tissues and can be fatal if not appropriately treated, making its investigation essential for improving clinical outcomes. Analyzing survival and time-to-event data in this context contributes to a better understanding of disease progression and supports the development of reliable statistical models for prognostic evaluation in hematologic oncology. In this application, Table 8 reports the survival times (in years) of 40 leukemia patients treated at a Ministry of Health hospital in Saudi Arabia. This dataset was provided by Imran [20] and later reanalyzed by Klakattawi [21].

Firstly, before fitting the proposed Baker–T1 model,

Table 9. Summary for the analyzed real datasets.

Application	Mean	Mode	${\mathcal{Q}}_1$	${\mathcal{Q}}_2$	${\mathcal{Q}}_3$	SD	Skew.
1	1.0650	1.2000	0.8500	1.1500	1.3000	0.4283	−0.3589
2	3.1407	3.3480	2.1988	3.3480	4.2638	1.3589	−0.4167

provides a detailed descriptive summary of the datasets considered in Applications 1 and 2, including the mean, mode, quartiles (${\mathcal{Q}}_i,i=1,2,3$), standard deviation (SD), and skewness. The reported statistics indicate that:

• Application 1: The data show low central tendency with small variability, indicating consistently lower values and a slightly left-skewed distribution, suggesting most observations cluster around the mean with few lower-end extremes.
• Application 2: The data exhibit a higher mean and greater dispersion, reflecting more variability in outcomes, while the slight negative skewness suggests a longer tail toward lower values despite generally higher measurements.

The slight negative skewness observed in both applications is visually confirmed by the violin plots (see Figure 4a), which show greater density toward higher values with thinner lower tails, consistent with the summarized skewness measures in Table 9. For clarification, in Application 1, the Baker–T1 distribution reflects an IFR pattern as evidenced by the TTT plot (see Figure 4b), indicating aging behavior where the risk of failure grows over time and aligns with systems showing gradual wear-out rather than early instability. As shown in Figure 4b, for Application 2, the TTT plot likewise confirms an IFR structure, and the fitted Baker–T1 parameters suggest heavier tails, highlighting increasing risk with time alongside non-negligible probabilities of extreme performance outcomes relevant to reliability margin assessment. These combined graphical and experimental results confirm the suitability of the Baker–T1 distribution for modeling data with increasing failure rates and slight asymmetry particularly.

To assess the adequacy of the proposed Baker–T1 distribution for the complete datasets analyzed in Applications 1 and 2, its performance was benchmarked against nine competing lifetime models possessing flexible and unbounded hazard rate forms (see Table 10). Model evaluation was carried out using eight commonly adopted goodness-of-fit and information-based criteria: (i) the negative log-likelihood ($\mathsf{LL}$), (ii) Akaike Information Criterion ($\mathsf{AI}$), (iii) Bayesian Information Criterion ($\mathsf{BI}$), (iv) Consistent Akaike Information Criterion ($\mathsf{CAI}$), (v) Hannan–Quinn Information Criterion ($\mathsf{HQI}$), (vi) Anderson–Darling statistic ($\mathsf{AD}$), (vii) Cramér–von Mises statistic ($\mathsf{CvM}$), and (viii) the Kolmogorov–Smirnov statistic ($\mathsf{KS}$) together with its associated $p$-value.

The ML estimates of the model parameters $(\alpha ,\mu ,\theta )$ along with their corresponding standard errors (SEs) alongside the corresponding numerical values of criteria (i)–(viii) were obtained using the AdequacyModel package v2.0.0 (Marinho et al. [22]) and are reported in

Table 11. Summary fit for the Baker–T1 and its competitor models.

Model	$\mathit{\alpha }$		$\mathit{\mu }$		$\mathit{\theta }$		$\mathsf{LL}$	$\mathsf{AI}$	$\mathsf{BI}$	$\mathsf{CAI}$	$\mathsf{HQI}$	$\mathsf{AD}$	$\mathsf{CvM}$	$\mathsf{KS}$
	Est.	SE	Est.	SE	Est.	SE								Distance	p-Value
Application 1
Baker–T1	0.2106	0.3550	1.2568	0.9197	8.6726	23.726	10.070	26.140	27.640	29.127	26.723	0.1970	0.0313	0.1141	0.9570
NEW	0.2864	0.4086	3.1794	0.7805	0.4381	0.2466	10.916	27.831	29.331	30.818	28.414	0.3050	0.0486	0.1296	0.8903
APW	10.938	2.2253	2.0125	0.1688	0.0105	0.0044	11.267	28.534	30.034	31.521	29.117	0.3252	0.0517	0.1495	0.7624
PGW	0.0335	0.0704	2.2280	0.4392	13.438	27.587	10.195	26.390	27.890	29.378	26.973	0.2401	0.0387	0.1452	0.9562
EW	0.6353	0.0857	0.2495	0.2487	7.6254	6.1088	10.104	26.255	27.767	29.266	26.843	0.2263	0.0370	0.1245	0.9157
WE	0.0646	0.1014	1.2432	0.9129	1.8571	1.9408	10.150	26.348	27.860	29.359	26.935	0.1976	0.0318	0.1148	0.9547
HEE	0.3125	0.3617	9.0580	8.6048	83.956	80.344	10.330	26.660	28.160	29.647	27.243	0.2027	0.0315	0.1157	0.9546
VFW	-	-	2.5133	0.8296	0.4616	0.2108	10.334	26.888	27.657	29.159	27.312	0.3475	0.0574	0.1245	0.9160
APE	-	-	55.598	47.848	1.8998	0.2906	14.260	32.520	33.226	34.511	32.909	0.6191	0.0997	0.2240	0.2683
BS	-	-	0.5745	0.0907	0.9108	0.1122	15.568	35.136	35.842	37.128	35.525	1.2598	0.2068	0.2417	0.1930
NH	-	-	24.242	27.106	0.0285	0.0324	16.339	36.678	37.384	38.670	37.067	0.4491	0.0717	0.3210	0.0324
GE	-	-	4.8403	1.8220	2.0706	0.4128	13.875	31.750	32.456	33.742	32.139	0.9060	0.1466	0.1922	0.4507
G	-	-	4.5091	1.3764	4.2339	1.3671	13.015	30.030	30.736	32.021	30.419	0.7588	0.1220	0.1755	0.5691
W	-	-	2.8129	0.5240	0.6093	0.1672	11.108	26.216	27.999	29.336	27.669	0.3841	0.0612	0.1386	0.8373
Application 2
Baker–T1	0.0399	0.0254	0.9513	0.2346	20.705	18.837	65.73	137.46	138.12	142.52	139.29	0.1491	0.0185	0.0627	0.9975
NEW	0.4678	0.4112	2.7429	0.4156	0.0272	0.0182	69.11	144.22	144.88	149.28	146.05	0.6275	0.0941	0.1074	0.7450
APW	28.254	4.3583	1.7120	0.3236	0.2330	0.1113	66.74	139.49	139.20	144.42	139.65	0.6596	0.0998	0.1011	0.8077
PGW	0.0062	0.0048	2.0547	0.2937	8.4054	7.0650	66.87	139.74	140.41	144.81	141.57	0.3609	0.0521	0.0909	0.8953
EW	0.2025	0.0074	0.1461	0.0244	11.166	0.0440	66.04	137.54	138.21	142.56	139.29	0.1511	0.0198	0.0655	0.9954
WE	0.0514	0.0358	0.8896	0.4663	0.9035	0.5799	66.16	138.81	139.48	143.92	140.66	0.1535	0.0192	0.0642	0.9967
HEE	0.3687	0.1568	2.3197	0.8280	49.084	30.117	67.09	140.19	140.86	145.26	142.02	0.3193	0.0441	0.0758	0.9755
VFW	-	-	3.4723	1.0945	0.0068	0.0113	67.68	139.36	139.68	142.74	140.58	0.3698	0.0605	0.2898	0.0024
APE	-	-	76.626	57.319	0.6544	0.0729	73.35	150.70	151.02	154.08	151.92	1.1779	0.1876	0.1538	0.3005
BS	-	-	0.7055	0.0788	2.4833	0.2598	80.58	165.17	165.49	168.55	166.39	2.6380	0.4517	0.2208	0.0405
NH	-	-	28.570	25.656	0.0081	0.0073	76.43	156.85	157.18	160.23	158.07	0.8141	0.1257	0.2705	0.0057
GE	-	-	3.5190	0.8757	0.6142	0.0910	74.96	153.92	154.25	157.30	155.15	1.7001	0.2799	0.1612	0.2495
G	-	-	3.4648	0.7405	1.1032	0.2537	73.55	151.10	151.42	154.48	152.32	1.4850	0.2414	0.1582	0.2693
W	-	-	2.4984	0.3352	0.0432	0.0213	69.56	143.12	143.44	146.49	144.34	0.7734	0.1188	0.1185	0.6284

. All parameters were initialized using admissible starting values consistent with their theoretical domains; see, for example, Mariel et al. [23]. According to the adopted selection rules, models yielding smaller values of $\mathsf{LL}$, $\mathsf{AI}$, $\mathsf{BI}$, $\mathsf{CAI}$, $\mathsf{HQI}$, $\mathsf{AD}$, $\mathsf{CvM}$, and $\mathsf{KS}$, together with larger $p$-values, are preferred. The results summarized in Table 11 clearly indicate that the corresponding fitted values for the criteria (i)–(viii) of the Baker–T1 model are the lowest compared to other competitors. Subsequently, the Baker–T1 model provides the best overall fit among all competing distributions listed in Table 10.

Table 10. Thirteen competitive models of the Baker–T1 distribution.

Model	Symbol	Author (s)
New Extended Weibull	NEW $(\alpha ,\mu ,\theta )$	Peng and Yan [24]
Alpha Power Weibull	APW $(\alpha ,\mu ,\theta )$	Nassar et al. [25]
Power Generalized Weibull	PGW $(\alpha ,\mu ,\theta )$	Bagdonavicius and Nikulin [26]
Exponentiated Weibull	EW $(\alpha ,\mu ,\theta )$	Mudholkar and Srivastava [27]
Weibull Exponential	WE $(\alpha ,\mu ,\theta )$	Oguntunde et al. [28]
Harris Extended Exponential	HEE $(\alpha ,\mu ,\theta )$	Pinho et al. [29]
Very Flexible Weibull	VFW $(\mu ,\theta )$	Ahmad and Hussain [30]
Alpha Power Exponential	APE $(\mu ,\theta )$	Mahdavi and Kundu [31]
Birnbaum Saunders	BS $(\mu ,\theta )$	Birnbaum and Saunders [32]
Nadarajah Haghighi	NH $(\mu ,\theta )$	Nadarajah and Haghighi [33]
Generalized Exponential	GE $(\mu ,\theta )$	Gupta and Kundu [34]
Gamma	G $(\mu ,\theta )$	Johnson et al. [35]
Weibull	W $(\mu ,\theta )$	Weibull [36]

Table 10. Thirteen competitive models of the Baker–T1 distribution.

Model	Symbol	Author (s)
New Extended Weibull	NEW $(\alpha ,\mu ,\theta )$	Peng and Yan [24]
Alpha Power Weibull	APW $(\alpha ,\mu ,\theta )$	Nassar et al. [25]
Power Generalized Weibull	PGW $(\alpha ,\mu ,\theta )$	Bagdonavicius and Nikulin [26]
Exponentiated Weibull	EW $(\alpha ,\mu ,\theta )$	Mudholkar and Srivastava [27]
Weibull Exponential	WE $(\alpha ,\mu ,\theta )$	Oguntunde et al. [28]
Harris Extended Exponential	HEE $(\alpha ,\mu ,\theta )$	Pinho et al. [29]
Very Flexible Weibull	VFW $(\mu ,\theta )$	Ahmad and Hussain [30]
Alpha Power Exponential	APE $(\mu ,\theta )$	Mahdavi and Kundu [31]
Birnbaum Saunders	BS $(\mu ,\theta )$	Birnbaum and Saunders [32]
Nadarajah Haghighi	NH $(\mu ,\theta )$	Nadarajah and Haghighi [33]
Generalized Exponential	GE $(\mu ,\theta )$	Gupta and Kundu [34]
Gamma	G $(\mu ,\theta )$	Johnson et al. [35]
Weibull	W $(\mu ,\theta )$	Weibull [36]

To facilitate the numerical optimization of the Baker–T1 model parameters $\alpha$, $\mu$, and $\theta$ for the two empirical datasets considered in Applications 1 and 2, the contour plots of the log-likelihood function are illustrated in Figure 5. These plots show that the selected initial values, indicated by the red points, lie in close proximity to the corresponding maximum likelihood estimates reported in Table 11. Furthermore, the contours provide evidence that the MLEs of $\alpha$, $\mu$, and $\theta$ not only exist but are also unique.

Figure 6 presents a comprehensive graphical comparison between the Baker–T1 model and its competing distributions for the datasets examined in Applications 1 and 2, using four complementary diagnostic panels: (i) probability–probability (PP) plots, (ii) quantile–quantile (QQ) plots, (iii) fitted reliability functions (RFs), and (iv) fitted probability density functions (PDFs). The graphical evidence in Figure 6a–c is consistent with the numerical findings reported in Table 11, demonstrating that the Baker–T1 model yields fitted values that closely track the empirical data in both applications. Moreover, Figure 6d reveals that the estimated Baker–T1 densities exhibit right-skewed and strictly right-skewed shapes for Applications 1 and 2, respectively.

In summary, although several competing models listed in Table 11 provide satisfactory fits to the proposed two real-world datasets, the Baker–T1 distribution demonstrates clear overall superiority when assessed using multiple complementary criteria. Specifically, it consistently yields the smallest values for all information-based measures while simultaneously producing the largest $p$-value, reflecting an optimal trade-off between model fit and parsimony. In addition, the graphical diagnostics presented in Figure 6 indicate a closer concordance between the empirical distributions and their theoretical counterparts under the Baker–T1 model. Beyond these empirical results, the Baker–T1 distribution offers a notable theoretical advantage: unlike many unit distributions, it is capable of accommodating decreasing, increasing, bathtub-shaped, and inverted bathtub-shaped hazard rate functions with considerable flexibility.

Overall, the empirical findings from both application domains provide compelling evidence that the Baker–T1 distribution is not merely an alternative, but a strictly superior modeling choice relative to the 13 competing lifetime distributions considered. Its dominance across two fundamentally different real-world datasets confirms its robustness, adaptability, and broad applicability, thereby establishing the Baker–T1 model as a powerful and reliable tool for lifetime and reliability analysis in both biomedical and engineering contexts.

7. Conclusions

This paper develops a comprehensive and rigorous treatment of the Baker–T1 distribution, a recently proposed transformation-based extension of the Weibull model derived via Manly’s exponential transformation. By developing its full probabilistic structure, the work elevates the Baker–T1 model from a constructional concept to a fully operational statistical tool for lifetime and reliability analysis. Fundamental distributional properties—including reliability measures, hazard and reversed-hazard functions, quantiles, moments, dispersion indices, skewness, kurtosis, and order statistics—were derived and examined in detail, revealing a high degree of analytical tractability alongside substantial modeling flexibility.

A key finding of this study is the ability of the Baker–T1 distribution to accommodate a wide range of density and hazard rate shapes, including monotone, bathtub-shaped, and unimodal forms, within a single parametric framework. This flexibility enables the model to capture complex failure mechanisms and heterogeneous aging behaviors that are commonly observed in real-world data but are not adequately represented by classical Weibull or many of its existing generalizations.

From an inferential perspective, an extensive estimation framework was developed by implementing eight classical parametric estimation methods. A large-scale Monte Carlo simulation study demonstrated that all estimators exhibit consistency, with estimation accuracy improving as the sample size increases. Among the competing approaches, maximum likelihood and maximum product of spacings estimators consistently provided superior finite-sample performance, confirming their suitability for practical implementation. The empirical analyses based on genuine biomedical and engineering datasets further validated the practical relevance of the Baker–T1 model, where it consistently outperformed thirteen competing lifetime distributions according to likelihood-based and information-theoretic criteria. These results collectively establish the Baker–T1 distribution as a robust, flexible, and empirically superior alternative for modeling complex lifetime data.

7.1. Practical Recommendations

Based on the theoretical developments, simulation evidence, and real-data applications presented in this study, several practical recommendations can be made. First, the Baker–T1 distribution is strongly recommended for applications involving non-monotone hazard structures, heavy-tailed behavior, or pronounced skewness, particularly in biomedical survival analysis and engineering reliability studies. Second, for parameter estimation, the maximum likelihood method should be regarded as the primary choice due to its overall efficiency and stability, with the maximum product of spacings method serving as a reliable alternative in situations involving small samples or numerical challenges. Third, the availability of a closed-form quantile function makes the Baker–T1 model especially attractive for simulation-based studies, percentile estimation, and risk assessment applications. Finally, practitioners are encouraged to consider the Baker–T1 distribution as a competitive benchmark when conducting model comparison studies involving Weibull-type or transformation-based lifetime models.

7.2. Future Directions

Although this study provides a comprehensive foundation for the Baker–T1 distribution, several promising avenues for future research remain open. One natural extension is the development of Bayesian inference procedures for the Baker–T1 model, including prior sensitivity analysis and hierarchical formulations for complex data structures. Another important direction involves extending the Baker–T1 framework to accommodate censored, truncated, or progressively censored data, which frequently arise in reliability testing and medical follow-up studies.

In addition, while both real datasets analyzed exhibit mild to moderate skewness, future work may extend the analysis to more heavily skewed data to further assess the flexibility of the proposed Baker–T1 model.

Further research may also explore regression and accelerated failure-time versions of the Baker–T1 model, allowing covariate information to be incorporated into the scale or shape parameters. Multivariate extensions and dependence modeling using copula-based constructions represent another fruitful area for investigation. Finally, comparative studies involving additional real datasets and alternative goodness-of-fit measures could further strengthen the empirical evidence supporting the Baker–T1 model and broaden its adoption across diverse applied domains.