Modeling Fatigue Data of Complex Metallic Alloys Using a Generalized Student’s t-Birnbaum–Saunders Family of Lifetime Models: A Comparative Study with Applications

Abstract

The mechanical reliability of metallic alloys under cyclic loading is crucial for optimizing their microstructure–property relationships. Understanding the statistical behavior of fatigue failure data is essential for designing alloys that endure extreme environmental conditions. This study introduces a generalization of the Student’s t-Birnbaum–Saunders distribution to improve the modeling of fatigue life data, which often exhibit heavy tails and are common in advanced alloy systems. Seven different estimation methods are employed to estimate and compare the parameters of the proposed distribution, providing a comprehensive statistical framework for fatigue failure analysis. The goodness-of-fit of the proposed model and its sub-models, along with the joint relative efficiency of parameter estimates, is assessed using real fatigue data within the maximum likelihood framework. Additionally, the robustness of estimation methods is examined through Monte Carlo simulations across various sample sizes and parameter configurations. The results highlight the effectiveness of the generalized Student’s t-Birnbaum–Saunders distribution in capturing the stochastic nature of fatigue failure in metallic alloys, offering valuable insights for materials design and predictive reliability modeling. These findings align with advancements in computational modeling and simulation, contributing to developing alloys with tailored mechanical properties.

1. Introduction

The mechanical reliability of metals and alloys under repeated stress (e.g., cyclic loading) is of paramount importance in structural engineering, aerospace, and materials science. Fatigue failure, defined as progressive structural damage resulting from the combined action of one or more stress factors (e.g., loading), plays a crucial role in determining the lifespan of metallic components. As industrial researchers continue to develop advanced metallic alloys with tailored properties, there is a pressing need for statistical models that accurately capture the stochastic nature of fatigue life data. In practice, well-known lifetime models (e.g., exponential and Weibull distributions) often fail to accommodate the heavy-tailed behavior observed in such data, particularly for complex or novel alloys.

In recent years, various studies have addressed the modeling of fatigue failure. For example, ref. [1] proposed a fatigue crack growth model by integrating finite element modeling with an approximate Bayesian computation framework to predict rail defects, specifically transverse defects, in data from a U.S. Class I railroad. Ref. [2] introduced a new empirical method for estimating the cumulative distribution function of fatigue life under given loads by statistically transforming fatigue data across different stress or strain levels, thereby improving distribution estimation. Ref. [3] proposed a probability-based basis for predicting the fatigue life of notched components, accounting for size effects. The underlying life distribution of pultruded fiber-reinforced compounds was studied by [4], who explored its statistical characteristics. A comprehensive review on fatigue modeling using neural networks was conducted by [5], summarizing past progress and outlining future research directions. Ref. [6] examined the extreme value distribution of maximum fatigue indicators in large-scale engineering components using an upscaling method extrapolated from simulations involving statistical volume elements, each containing approximately 264 grains. Lastly, ref. [7] fitted a set of 950 NASA-conducted gear surface fatigue test results to a three-parameter Weibull distribution, while ref. [8] recently discussed advances in probabilistic modeling of low-cycle fatigue.

In practice, objects made of materials and alloys are subjected to various loading parameters, such as cyclic loading, which causes fatigue in the long run. The latter phenomenon describes the beginning and spread of cracks in materials and alloys due to loading parameters. This phenomenon is considered in the design of structural components in many engineering disciplines. Fatigue modeling and prediction are essential to ensure the reliability and safety of elements considered in cyclic loading conditions, especially in applications like civil infrastructure and automotive. From a statistical perspective, fatigue can be considered as a random variable from a lifetime model, which is typically a probability distribution with non-negative support and at least one parameter that can be linked to a stress factor. Various lifetime probability distributions and statistical inferential methods have been applied to model the statistical variability in fatigue data. For example, the applicability of the normal distribution versus the three-parameter Weibull distribution was investigated and compared by [9] according to the results of six series of 18 to 30 similar fatigue tests on three specimens. A random fatigue-limit model was considered for data of specimens in four-point out-of-plane bending tests of carbon eight-harness satin/epoxy laminate by [10], while ref. [11] analyzed nickel base superalloy fatigue data with runouts via a model with nonconstant standard deviation and a fatigue limit parameter. In this connection, an article by [12] investigated some issues when applying Markov chain Monte Carlo and Laplacian approximations estimators to the experimental design problem in the random fatigue-limit model with application to laminate panel data. In [13], a competitive Bayesian treatment of stress-life data was drawn from a collection of records of fatigue experiments that were performed on 75S-T6 aluminum alloys, assuming at least one model was provided. Recently, ref. [14] proposed the Bayesian model averaging method, which offers a general framework for developing probabilistic fatigue models with superior robustness and precision in their predictions. A direct application of statistical distribution theory in fatigue analysis was a probabilistic model called the Birnbaum–Saunders (BS) fatigue life distribution [15,16]. The latter distribution was proposed to model fatigue failure under cyclic loading based on a physical mechanism in which failure results from the progressive growth of initial micro-cracks. Specifically, crack growth is assumed to be normally distributed, arising from the accumulation of minor damage due to repeated stress cycles, with failure occurring once a critical crack length is reached. On the one hand, the BS model offers a probabilistic framework based on a stochastic representation of damage accumulation. On the other hand, physics-based fatigue models (e.g., the Coffin–Manson model) rely on material-specific parameters such as strain energy amplitude. Overall, the BS model provides a data-driven approach for lifetime prediction under repetitive stress, particularly when detailed material or loading information is unavailable.

Assume that X is a continuous random variable denoting the time-to-failure due to fatigue. Hence, X is considered to follow the unimodal two-parameter BS distribution if its cumulative distribution function (CDF) is defined as:

$$F(x;\alpha ,\beta )=\mathrm{\Phi }\left({\alpha }^{-1}\left[\sqrt{{\displaystyle \frac{x}{\beta }}}-\sqrt{{\displaystyle \frac{\beta }{x}}}\right]\right),x>0,$$

(1)

such that $\mathrm{\Phi }(·)$ is the CDF of the standard normal distribution, while $\alpha >0$ is a shape parameter, and $\beta >0$ is a scale parameter. The study by [17] offered a broader derivation of the BS distribution based on a biological framework, reinforcing the physical justification for its application by easing the initial assumptions made in [15,16]. Since its introduction, the BS distribution has received considerable attention from many researchers due to its close relation to the normal distribution and other desirable properties. According to a thorough review by [18], over two hundred research articles and one dedicated monograph have been published detailing the properties, advancements, and various extensions and generalizations of this distribution. Recently, ref. [19] discussed frequentist and Bayesian estimation methods for the two-parameter conventional BS distribution and the prediction of missing data (i.e., failure times) under the Type-II censoring plan. The latter lifetime distribution is used to analyze the time to fatigue of a sample of a specific type of aluminum coupon. The coupons are cut in a corresponding direction to the rolling, and they were oscillated at specific levels of cycles per second and maximum stress per cycle. The study by [20] considered a goodness-of-fit test for the Birnbaum–Saunders distribution based on the probability plot and provided an application to the strength of glass fiber of length 15 cm. The BS distribution does not accurately model the heavy-tailed pattern that may occur in fatigue data of complex alloys. Researchers have considered modifying or generalizing the BS distribution to compensate for this limitation. For instance, refs. [21,22] developed the generalized Birnbaum–Saunders distribution (GBS) by substituting the normal kernel with elliptically symmetric kernels (e.g., Student’s t, Cauchy, etc.) to enhance the model’s flexibility. Afterward, ref. [23] considered Student’s t BS survival regression models with heavy-tailed errors, presented some inferential results, and performed diagnostics analysis, while ref. [24] introduced the BS distributions based on scale mixtures of normal models and illustrated the results by analyzing fatigue failure data of aluminum specimens of type 6061-T6. Recently, another approach considered by [25] to generalize this model is by changing the square root in the CDF by another shape parameter $\theta >0$. Recent research about the BS distribution and its generalizations has been considered by researchers, including, but not limited to, the mentioned contributions. A study by [26] introduced a new type of BS model as an alternative to the conventional model to fit fatigue data, while ref. [27] used a new bivariate BS distribution to establish a regression of generalized linear models. One of the desirable properties of the BS distribution is that its scale parameter represents the median lifetime, unlike the scale parameter of the exponential distribution, which represents the mean lifetime. Consequently, some researchers focused on studying the median of the BS distribution. For instance, ref. [28] developed a test to assess the impact of two interacting factors on the median for BS distributed response via the integrated likelihood ratio test framework. To address a serious challenge in lifetime data analysis, ref. [29] considered the model misspecification problem between the log-normal and BS distributions. A study by [30] assumed a varying-stress accelerated life test for a generalized BS model to establish an extension for this model. The study outlined the aspects of this highly flexible distribution, obtained the classical maximum likelihood estimators (MLEs), developed a novel goodness-of-fit procedure, and proposed a new inference approach using Bayesian theory. Under the latter framework, ref. [31] considered estimating the parameters of the BS distribution in the presence of right-censored data. A study by [32] recently employed the BS distribution to model metallic materials fatigue life under cyclic loading. The study compared this model to the normal distribution. The considered models were used to analyze three sets of data of unnotched specimens of 75S-T6 aluminum alloys and carbon coating with several types of loading. Recently, ref. [33] investigated the asymptotic aspects of the method of moments estimators for a newly parametrized BS distribution.

Estimating the model parameters is a topic of significant interest to researchers and has been extensively studied in the statistical literature. Since there are many estimation methods, researchers conducted Monte Carlo comparative simulation studies to compare their performance in terms of various statistical and computational aspects. Such studies are imperative for evaluating under different settings, providing robust comparisons between estimation accuracy and efficiency. Several studies demonstrate this, notably those listed in [34,35,36,37,38,39,40,41,42,43], among others.

This study proposes a robust statistical framework based on the generalized Student’s t-Birnbaum–Saunders (GTBS) distribution as a suitable model for fatigue failure. The aim of this study is twofold. First, a flexible generalization for the BS distribution is proposed to model fatigue failure. Second, since estimating the distributional properties of a probability distribution (e.g., survivability, hazard rate, etc.) necessitates obtaining appropriate estimations for the model parameters using appropriate estimation procedures, the model parameters are being estimated via seven frequentist parametric estimation procedures. For this study, the considered frequentist parametric estimation methods are the maximum likelihood estimation (MLE), the least-squares estimation (LSE), the weighted least-squares estimation (WLSE), the maximum product of spacings estimation (MPSE), the Cramér–von Mises estimation (CVME), the Anderson–Darling estimation (ADE), and the right-tailed Anderson–Darling estimation (RADE) methods. It is important to mention that it is assumed that the obtained estimators from these methods exist and are unique. It is essential to note that the GTBS model and its sub-models do not directly incorporate microstructural variables (e.g., grain size or phase distribution). However, its formulation inherently captures the statistical characteristics of fatigue life data, often resulting from such underlying heterogeneity. In complex metallic alloys, variations in microstructural features frequently lead to substantial scatter in fatigue life, particularly in the presence of extreme values or heavy tails. The GTBS distribution extends the classical two-parameter BS model by introducing a shape parameter and incorporating heavy-tailed behavior through the Student’s t kernel via a degrees-of-freedom parameter, allowing for a possible modeling flexibility. As a result, the GTBS model offers a statistically rigorous and computationally tractable framework for representing the stochastic effects of material structure, particularly when direct physical measurements are unavailable. It is therefore especially well-suited for analyzing fatigue life data in advanced alloys, where conventional models such as the Weibull, gamma, lognormal, or BS distributions may be insufficient.

The remainder of this study is arranged as follows. Section 2 briefly overviews the proposed generalization for the BS distribution and its sub-models. Section 3 defines the previously named estimation approaches of the study. Section 4 reports the statistical analysis of two real datasets. Section 5 summarizes Monte Carlo simulation outcomes from which estimation efficiency is numerically examined and assessed. The study is concluded in Section 6 with a summary and future research directions.

2. A Generalized Student’s t-Birnbaum–Saunders Distribution

A non-negative continuous random variable X is said to follow the generalized Student’s t BS (GTBS) distribution with location parameter $\mu >0$, scale parameter $\sigma >0$, shape parameter $\theta >0$, and degrees of freedom parameter $\nu >0$ if the associated cumulative distribution function (CDF) is expressed as:

$$F(x;\mathit{\theta })={\mathrm{\Phi }}_{\nu }\left({\left[{\displaystyle \frac{x-\mu }{\sigma }}\right]}^{\theta }-{\left[{\displaystyle \frac{\sigma }{x-\mu }}\right]}^{\theta }\right),x>\mu ,$$

(2)

such that ${\mathrm{\Phi }}_{\nu }(·)$ is the CDF of the standard Student’s t distribution with degrees of freedom parameter $\nu >0$, and $\mathit{\theta }=(\mu ,\sigma ,\theta ,\nu )$ is the vector of model parameters defined on a parameter space ${\mathbb{R}}_{+}^4$. The probability density function (PDF) of the GTBS distribution is obtained by differentiating (2) with respect to x, i.e.,

$$f(x;\mathit{\theta })={\displaystyle \frac{\theta }{(x-\mu )}}\left({\left[{\displaystyle \frac{x-\mu }{\sigma }}\right]}^{\theta }+{\left[{\displaystyle \frac{\sigma }{x-\mu }}\right]}^{\theta }\right){\varphi }_{\nu }\left({\left[{\displaystyle \frac{x-\mu }{\sigma }}\right]}^{\theta }-{\left[{\displaystyle \frac{\sigma }{x-\mu }}\right]}^{\theta }\right),x>\mu ,$$

(3)

where ${\varphi }_{\nu }(·)$ is the PDF of the standard Student’s t distribution with degrees of freedom parameter $\nu >0$. For more details about Student’s t distribution and its properties, see, for example, [44,45]. It is important to mention that most conducted studies about the BS distribution and its extensions and generalizations do not assume the existence of a location parameter; instead, it is typically assumed that the BS distribution has a shape parameter $\alpha$ and scale parameter $\beta$ with CDF (1).

Table 1. The sub-models of the GTBS lifetime distribution with model parameters vector $\mathit{\theta }=(\mu ,\sigma ,\theta ,\nu )\in {\mathbb{R}}_{+}^4$.

Model	$\mathit{\theta }$
Location-scale Cauchy BS distribution (CBS)	$(\mu ,\sigma ,{\displaystyle \frac{1}{2}},1)$
Location-scale BS distribution (BS)	$(\mu ,\sigma ,{\displaystyle \frac{1}{2}},\infty )$
Location-scale Student’s t BS distribution (TBS)	$(\mu ,\sigma ,{\displaystyle \frac{1}{2}},\nu )$
Generalized Cauchy BS distribution (GCBS)	$(\mu ,\sigma ,\theta ,1)$
Generalized BS distribution (GBS)	$(\mu ,\sigma ,\theta ,\infty )$

shows the sub-models of the GTBS distribution, while Figure 1 illustrates the shape of the PDF of the GTBS distribution for various values of $\theta$ and $\nu$ assuming the location parameter $\mu =0$ and the scale parameter $\sigma =1$ without loss of any generality. It is essential to note that some sub-models have counterintuitive (i.e., undefined) statistical aspects (e.g., the mean and variance). An example of a sub-model is the Cauchy BS (CBS) distribution.

3. Estimation Procedures

The estimation approaches of interest for the parameters of the GTBS distribution mentioned in the preceding introductory section are considered in this part of the study. It is essential to mention that this study considers the degrees of freedom parameter to be a natural number, i.e., $\nu =1,2,\dots$ without loss of generality. Moreover, all estimation methods are obtained using the following profiling algorithm when estimating the model parameters:

• For ${\nu }_1=1$ to ${\nu }_N$ by 1:

  (a)

  Determine the estimators of the model parameters $\mu ,\sigma$ and $\theta$ by optimizing an objective function proposed in this section using suitable initial values;

  (b)

  Compete the value of the optimized objective function;

• Using a convergence criterion, choose the value of $\nu$ that maximizes the objective function $g\left(\mathit{\theta }\right|\mathbf{x})$ if the estimators are obtained by solving a maximization problem; otherwise, choose the value of $\nu$ that minimizes the objective function $g\left(\mathit{\theta }\right|\mathbf{x})$ if the estimators are obtained by solving a minimization problem. For this study, the considered convergence criterion is given by:

  $$\left|1-{\displaystyle \frac{g\left({\mathit{\theta }}_k\right|\mathbf{x})}{g\left({\mathit{\theta }}_{k+1}\right|\mathbf{x})}}\right|<ϵ,$$

  where $ϵ>0$ is relatively small and $k=1,\dots ,N$.

A similar approach is considered in the study by [24]. Furthermore, all mathematical derivatives with respect to the model parameters required by the upcoming optimization problems are provided in Appendix A for the sake of conciseness.

Much research is still required. For example, Bayesian estimation methods could be explored and compared with the frequentist approaches considered in this study. Furthermore, estimation efficiency and model fit could be assessed under censored data structures, such as progressive or hybrid censoring schemes.

3.1. Maximum Likelihood Estimation

The estimation theory of maximum likelihood is an extensively used statistical framework in which the parameters of lifetime models are estimated by maximizing a likelihood-based objective function; namely, the log-likelihood function. This type of estimation is versatile and asymptotically efficient; thus, the estimators obtained from this approach have become a fundamental tool in numerous scientific disciplines. Suppose $\mathbf{x}=[x_1,x_2,\dots ,x_n]$ is an observed random sample of size n from the GTBS distribution. The MLEs are obtained by solving the following maximization problem:

$$\begin{array}{cc}\hfill \mathrm{maximize}& \ell \left(\mathit{\theta }\right|\mathbf{x})=nlog\theta -\sigma \sum _{i=1}^{n}log{z}_i+\sum _{i=1}^{n}log\left(z_i^{\theta }+z_i^{-\theta }\right)+\sum _{i=1}^{n}log\left[{\varphi }_{\nu }\left(z_i^{\theta }-z_i^{-\theta }\right)\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4,\hfill \end{array}$$

(4)

where $z_i={\sigma }^{-1}(x_i-\mu )$.

3.2. Least-Squares Estimations

The least-squares estimators (LSEs) and the weighted least-squares estimators (WLSEs) are obtained using an approach similar to [46] in which the beta distribution parameters are estimated. Recall that $\mathbf{x}=[x_1,x_2,\dots ,x_n]$ is an observed random sample, and let $x_{1:n}<x_{2:n}<\cdots <x_{n:n}$ be the corresponding observed order statistics. Thus, $F(x_{1:n};\mathit{\theta }),\dots ,F(x_{n:n};\mathit{\theta })$ are the order statistics form a standard uniform distribution with expected values and variances given by:

$$\mathrm{E}\left[F(x_{i:n};\mathit{\theta })\right]={\displaystyle \frac{i}{n+1}}\mathrm{and}\mathrm{V}\left[F(x_{i:n};\mathit{\theta })\right]={\displaystyle \frac{i(n-i+1)}{{(n+1)}^2(n+1)}},$$

respectively, such that $i=1,\dots ,n$ [47]. The LSEs of $\mathit{\theta }$ of the GTBS distribution are obtained by solving the following minimization problem:

$$\begin{array}{cc}\hfill \mathrm{minimize}& \sum _{i=1}^n{w}_i{\left[{\mathrm{\Phi }}_{\nu }\left(z_{i:n}^{\theta }-z_{i:n}^{-\theta }\right)-{\displaystyle \frac{i}{n+1}}\right]}^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4\hfill \end{array}$$

(5)

such that $z_{i:n}={\sigma }^{-1}(x_{i:n}-\mu )$ and $w_i=1,\forall i$. However, if $w_i^{-1}=\mathrm{V}\left[F(x_{i:n};\mathit{\theta })\right]$ in the above minimization problem, then the solution for the minimization problem is the WLSEs for $\mathit{\theta }$.

3.3. Maximum Produce of Spacings Estimation

The maximum product of spacings estimation framework is another approach that requires solving a maximization problem to obtain maximum product of spacings estimators (MPSEs). The latter estimators rival those obtained under the maximum likelihood framework in terms of estimation effectiveness and asymptotic aspects [48,49,50,51]. An important note is that the calculation of the MPSEs becomes complicated if ties are present, as the standard method cannot be applied directly. Instead, a generalized maximum product of spacings approach proposed by [52] must be used. Recall that $x_{1:n}<x_{2:n}<\cdots <x_{n:n}$ represent the observed order statistics from a random sample of size n. The MPSEs for $\mathit{\theta }$ are acquired numerically by solving the following maximization problem:

$$\begin{array}{cc}\hfill \mathrm{maximize}& {\displaystyle \frac{1}{n+1}}\sum _{i=1}^{n+1}log\left({\mathsf{\Delta }}_i\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4,\hfill \end{array}$$

(6)

where

$${\mathsf{\Delta }}_i=\left\{\begin{array}{cc}{\mathrm{\Phi }}_{\nu }\left(z_{1:n}^{\theta }-z_{1:n}^{-\theta }\right)\hfill & \mathrm{if}i=1\hfill \\ {\mathrm{\Phi }}_{\nu }\left(z_{i:n}^{\theta }-z_{i:n}^{-\theta }\right)-{\mathrm{\Phi }}_{\nu }\left(z_{i-1:n}^{\theta }-z_{i-1:n}^{-\theta }\right)\hfill & \mathrm{if}1<i\le n\hfill \\ 1-{\mathrm{\Phi }}_{\nu }\left(z_{n:n}^{\theta }-z_{n:n}^{-\theta }\right)\hfill & \mathrm{if}i=n+1\hfill \end{array}\right..$$

3.4. Minimum Distance Estimations

The remaining three estimators, the CVMEs, ADEs, and RADEs, are obtained from the CVME, ADE, and RADE methods, respectively. These methods minimize goodness-of-fit statistics based on the theoretical CDF of the underlying model. The CVMEs of $\mathit{\theta }$ are obtained by evaluating the following minimization problem recalling that $x_{1:n}<x_{2:n}<\cdots <x_{n:n}$ are the corresponding observed order statistics of a random sample of size n:

$$\begin{array}{cc}\hfill \mathrm{minimize}& {\displaystyle \frac{1}{12n}}+\sum _{i=1}^n{\left[{\mathrm{\Phi }}_{\nu }\left(z_{i:n}^{\theta }-z_{i:n}^{-\theta }\right)-{\displaystyle \frac{2i-1}{2n}}\right]}^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4\hfill \end{array}$$

(7)

Alternatively, the ADEs and RADEs of $\mathit{\theta }$ are obtained by solving:

$$\begin{array}{cc}\hfill \mathrm{minimize}& -n-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)log{\mathrm{\Phi }}_{\nu }\left(z_{i:n}^{\theta }-z_{i:n}^{-\theta }\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)log\left[1-{\mathrm{\Phi }}_{\nu }\left(z_{n+1-i:n}^{\theta }-z_{n+1-i:n}^{-\theta }\right)\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill \mathrm{minimize}& {\displaystyle \frac{n}{2}}-2\sum _{i=1}^n{\mathrm{\Phi }}_{\nu }\left(z_{i:n}^{\theta }-z_{i:n}^{-\theta }\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)log\left[1-{\mathrm{\Phi }}_{\nu }\left(z_{n+1-i:n}^{\theta }-z_{n+1-i:n}^{-\theta }\right)\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mu ,\sigma ,\theta ,\nu \in {\mathbb{R}}_{+}^4,\hfill \end{array}$$

(9)

respectively.

4. Applications

This section is bipartite. The first part of this section illustrates the applicability of the proposed model by analyzing real data and evaluating the goodness-of-fit compared to the sub-models under the maximum likelihood framework. In the second part of this section, the methods in the preceding section are used to estimate the main model parameters, and the acquired estimators are compared per some statistical aspects obtained using parametric bootstrapping. As a simulation technique, bootstrapping requires the ability to simulate random variates from the model of interest, which can be performed using the quantile function (QF). The latter function for the GTBS distribution is given by:

$$Q(u,\mathit{\theta })=\mu +\sigma {\left\{{\displaystyle \frac{{\mathrm{\Phi }}_{\nu }^{-1}\left(u\right)+\sqrt{{\left[{\mathrm{\Phi }}_{\nu }^{-1}\left(u\right)\right]}^2+4}}{2}}\right\}}^{{\displaystyle \frac{1}{\theta }}},0<u<1,$$

(10)

such that ${\mathrm{\Phi }}_{\nu }^{-1}(·)$ is the inverse of the CDF of the standard Student’s t distribution with degrees of freedom parameter $\nu >0$.

The first dataset considered in this study was obtained from [53,54], containing stress-life (SN) data for welded specimens made of steel S690QL, assuming 144 megapascal (MPa). SN data are used to evaluate the relationship between stress and the number of cycles to failure. The considered data (in 100 cycles) are: 2826.80, 3095.70, 3551.00, 3983.19, 4560.20 4564.80, 4966.52, 5232.00, 5547.26, 6477.26, 6615.50, 6832.64, 7394.15, 8274.73, 8856.00, 8971.83, 9890.00, and 12,006.40.

The second dataset to be analyzed was acquired from [55,56], containing strain-life (eN) data for commercial pure titanium (CP-Ti), assuming a modulus of elasticity of 108.21 GPa and tensile strength (TS) of 418.23 MPa. The base material plates were made by rolling and the fatigue specimens were cut in the rolling direction. All specimens were ground and then polished using metallographic abrasive paper and a grinding rod to exclude the effect of outer and inner surface roughness. These treatments were applied before the fatigue tests. The considered data (in log cycles) are: 4.329682569, 4.230959556, 3.923658422, 3.798995734, 3.720076573, 3.77815125, 3.431685345, 3.343999069, 3.362859303, 3.418963831, 3.211921084, 3.121231455, 3.087071206, 3.077004327, 2.953276337, and 2.851258349.

All numerical analyses and results presented in this part of the study and the following one were performed using R, an environment for statistical computing [57]. To account for potential outliers in the data, robust summary statistics were computed for each dataset, including the minimum, first quartile (${\mathrm{Q}}_1$), median (${\mathrm{Q}}_2$), third quartile (${\mathrm{Q}}_3$), maximum, and the median absolute deviation (MAD). Moreover, the objective functions in the previous section were optimized using a limited-memory BFGS method [58]. Although it has a built-in implementation within R via the optim function, the lbfgs function from the nloptr is considered instead. The latter package solves optimization problems using an R interface to NLopt, a free/open-source library for nonlinear optimization. It has different free optimization routines available online as well as original implementations of various other algorithms [59]. The reason for choosing the limited-memory efficient optimization approach is that the optimization problems of interest have box constraints.

Before applying the limited-memory BFGS method to solve the optimization problems, it is essential to note that the location parameter $\mu$ is part of the domain of the random variable. This situation disrupts one of the standard regularity conditions required for maximum likelihood estimation. These conditions are crucial for ensuring the consistency, asymptotic normality, existence, and efficiency of MLEs; see [60,61] for further discussion. Consequently, a re-parameterization of the location parameter is given by $\mu =x_{1:n}-\delta$, where $0<\delta <x_{1:n}$. This re-parameterization resolves the underlying issue.

4.1. Goodness-of-Fit Evaluation

The parameters of the GTBS distribution and its sub-models per Table 1 are estimated under the maximum likelihood framework as part of this study. The Kolmogorov–Smirnov (KS) [62,63], Anderson–Darling (AD) [64], and Cramér–von Mises (CVM) [65,66] statistics are used to check the goodness-of-fit of the fitted models. A parametric bootstrap approach with $B=1000$ resamples is considered for calculating the p-values for these statistics. The latter statistical approach is implemented as follows. First, the GTBS model parameters $\mu ,\sigma$ and $\theta$ are estimated from the observed data, producing $\widehat{\mu },\widehat{\sigma }$ and $\widehat{\theta }$, while the estimator of $\widehat{\nu }$ is obtained via profiling. Afterward, for the sub-model, part of the model parameters is estimated since some of the parameters become known; for example, in the case of the CBS distribution, $\theta =0.5$ and $\nu =1$, while $\theta =0.5$ and $\nu \to \infty$ in the case of the BS distribution. Consequently, the obtained estimators are then utilized to acquire B bootstrap samples from the underlying model. The goodness-of-fit statistics are computed for each bootstrap sample, and this procedure is repeated B times to approximate the distribution of test statistics. The bootstrap-based p-value is calculated as the proportion of bootstrap statistics exceeding the observed test statistic. For additional details and information about bootstrapping, readers may refer to [67,68,69].

Table 2. Summary statistics for SN and eN datasets. The distances between ${\mathrm{Q}}_2$, ${\mathrm{Q}}_3$, and the maximum value are noticeably large, which may suggest that the data follow a heavy-tail probabilistic model.

Data	Minimum	${\mathit{Q}}_1$	Median	${\mathit{Q}}_3$	Maximum	MAD
SN	2826.8	4561.35	6012.26	8054.585	12,006.4	2580.562
eN	2.8512583	3.1126914	3.3909116	3.7833624	4.3296826	0.4767095

reports the SN and eN data summary statistics. These summary statistics reveal that the considered datasets seem to be from heavy-tailed populations since the gap between the median and ${\mathrm{Q}}_3$ and the maximum is noticeably not small.

Table 3. SN data analysis and goodness-of-fit. The GBS model achieved the smallest values for all goodness-of-fit statistics. Since the corresponding parametric bootstrap p-value exceeds the standard significance level, there is insufficient evidence to reject the hypothesis that the GTBS, GCBS, and GBS models describe the data better than their sub-models.

Model	$\widehat{\mathit{\mu }}$	$\widehat{\mathit{\sigma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\nu }}$	KS	p-Value	AD	p-Value	CVM	p-Value
GTBS	203.11	5595.92	1.2534	6	0.1084	0.9371	0.1687	0.9800	0.0251	0.9650
GCBS	2156.48	3927.38	1.2208	1 *	0.1206	0.8412	0.4357	0.6803	0.0605	0.6933
GBS	203.11	5595.92	1.1335	∞ *	0.1026	0.9600	0.1502	0.9960	0.0213	0.9930
TBS	2705.21	2802.40	$0.5$ *	3	0.1464	0.4805	0.5549	0.3477	0.0749	0.3776
CBS	2826.80	3138.20	$0.5$ *	1 *	0.2312	0.0260	1.5333	0.0949	0.1584	0.0519
BS	2395.51	2775.06	$0.5$ *	∞ *	0.1794	0.2537	0.5999	0.2637	0.1104	0.2118

* The values are fixed, not estimated.

shows the data analysis for the SN dataset alongside the considered goodness-of-fit metrics, while

Table 4. The analysis of the eN data and the corresponding goodness-of-fit. The GBS model achieved the smallest values for all goodness-of-fit statistics. Once again, the corresponding parametric bootstrap p-value was above the standard significance level, i.e., there is not enough evidence to reject the claim that the GTBS, GBS and BS distributions properly fit the data.

Model	$\widehat{\mathit{\mu }}$	$\widehat{\mathit{\sigma }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\nu }}$	KS	p-Value	AD	p-Value	CVM	p-Value
GTBS	2.3208	1.0758	1.2843	20	0.1254	0.6194	0.2014	0.7882	0.0313	0.7353
GCBS	2.8513	0.5525	0.5051	1 *	0.2444	0.0310	1.2857	0.0170	0.1484	0.0420
GBS	2.2840	1.1144	1.2921	∞ *	0.1226	0.7143	0.1982	0.8392	0.0306	0.7982
TBS	2.7853	0.5089	$0.5$ *	16	0.1654	0.3996	0.5644	0.2478	0.0913	0.2398
CBS	2.8513	0.5521	$0.5$ *	1 *	0.2460	0.0300	1.2960	0.0999	0.1515	0.0549
BS	2.7745	0.5085	$0.5$ *	∞ *	0.1748	0.4006	0.5960	0.2857	0.1014	0.2657

* The values are fixed, not estimated.

provides similar information but for the eN dataset. Both tables indicate that the sub-model of the GTBS distribution, i.e., the GBS distribution, fits the data better than its counterparts, followed by its generalization. The GBS and GTBS are very close in terms of goodness-of-fit. To strengthen the latter conclusion, Figure 2 and Figure 3 visually support the claim that the GTBS model is suitable for both data sets.

4.2. Estimation Efficiency Assessment

In the previous part of the study, goodness-of-fit analysis has been conducted under the maximum likelihood framework; however, is it the best approach to estimate the considered model parameters? The joint relative efficiency (JRE) is calculated using bootstrap samples to answer this question. According to [70], one can define the joint relative efficiency (JRE) as:

$$\mathrm{JRE}={\displaystyle \frac{{\mathrm{RMSE}}^{\ast }\left(\tilde{\mu }\right)+{\mathrm{RMSE}}^{\ast }\left(\tilde{\sigma }\right)+{\mathrm{RMSE}}^{\ast }\left(\tilde{\theta }\right)}{{\mathrm{RMSE}}^{\ast }\left(\widehat{\mu }\right)+{\mathrm{RMSE}}^{\ast }\left(\widehat{\sigma }\right)+{\mathrm{RMSE}}^{\ast }\left(\widehat{\theta }\right)}}\times 100\%,$$

such that ${\mathrm{RMSE}}^{\ast }\left(\widehat{\vartheta }\right)$ and ${\mathrm{RMSE}}^{\ast }\left(\tilde{\vartheta }\right)$ are the bootstrap root mean square errors (RMSEs) of the MLEs and the other estimators under consideration for $\vartheta$, where $\vartheta$ corresponds to $\mu$, $\sigma$ or $\theta$. Note that the bootstrap estimator is acquired using an approach similar to the one mentioned previously. If the value of the JRE is larger than 100%, then the MLEs are jointly more efficient than the other estimators.

Table 5. Estimation efficiency assessment of seven estimation methods in terms of JREs based on SN data. Note that ${\mathrm{RMSE}}^{\ast }\left(\widehat{\mu }\right)=2185.23$, ${\mathrm{RMSE}}^{\ast }\left(\widehat{\sigma }\right)=521.18$, and ${\mathrm{RMSE}}^{\ast }\left(\widehat{\theta }\right)=0.6324$.

Model	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\mu }}\right)$	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\sigma }}\right)$	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\theta }}\right)$	JRE (%)
LSE	2268.99	308.50	0.6619	95.2388
WLSE	2079.67	529.08	0.5303	96.3884
MPSE	1634.82	641.45	0.5106	84.1059
CVME	2418.33	281.60	0.7406	99.7649
ADE	2488.87	174.21	0.7881	98.4053
RADE	2550.32	412.58	0.7846	109.4804

reveals that the MLEs did not outperform the considered estimation methods except for the RADEs in terms of JRE. In contrast,

Table 6. Estimation efficiency evaluation of seven estimation methods in terms of JREs based on eN data. Note that ${\mathrm{RMSE}}^{\ast }\left(\widehat{\mu }\right)=1.0309$, ${\mathrm{RMSE}}^{\ast }\left(\widehat{\sigma }\right)=1.0758$, and ${\mathrm{RMSE}}^{\ast }\left(\widehat{\theta }\right)=1.6558$.

Model	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\mu }}\right)$	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\sigma }}\right)$	${\mathbf{RMSE}}^{\ast }\left(\tilde{\mathit{\theta }}\right)$	JRE (%)
LSE	1.1746	1.2281	1.3998	101.06
WLSE	1.1207	1.1789	1.3209	96.22
MPSE	1.0806	1.1184	1.4709	97.54
CVME	1.2483	1.3001	1.8118	115.88
ADE	1.0598	1.1130	1.4767	97.00
RADE	1.1976	1.3149	1.7335	112.85

shows that the MLEs are jointly more efficient than the LSEs, CVMEs, and RADEs. This might be due to the small sample sizes of the considered datasets.

5. Monte Carlo Simulation Outcomes

Simulation studies have a vital role in evaluating the performance of estimation procedures across different settings, offering a reliable framework for assessing their effectiveness and precision. Researchers frequently employ Monte Carlo simulation studies to analyze and compare estimation efficiency when multiple estimators are considered for the parameters of a given model in practical applications. Thus, numerical insights from both statistical and sometimes computational viewpoints are provided.

This part of the study reports the Monte Carlo simulation outcomes. These results reflect the performance of the considered estimation methods, assuming different sample sizes and parameters with different values. That is, the simulation study results help identify the most effective and consistent estimation method for the GTBS model parameters. All numerical outcomes are based on $N=$ 1000 random samples generated from the GTBS model. The options for the sample size and the model parameters are $n=10,25,50,100,250$, $\mu =0,\sigma =1,\theta =0.25,1,2,4$, and $\nu =5,10,15,30$ without loss of generality. The comparison and evaluation metric of choice for the estimation efficiency is the RMSE, and it is defined similarly to what was mentioned in the preceding application section. For instance, the RMSE for the estimator of $\mu$ is given by:

$$\mathrm{RMSE}\left(\widehat{\mu }\right)=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left({\widehat{\mu }}_i-\mu \right)}^2},$$

where $N=1000$ and ${\widehat{\mu }}_i$ is an estimator of the model parameter $\mu$ based on simulation run i. As shown in the previous section, goodness-of-fit analysis for the fitted model parameters alongside estimation efficiency provides additional insights on performance. The mean absolute difference between the actual and estimated CDFs ($D_{\mathrm{abs}}$) and the maximum absolute difference between the actual and estimated CDFs ($D_{\max }$) are two goodness-of-fit metrics that are given by:

$$D_{\mathrm{abs}}={\displaystyle \frac{1}{n\times N}}\sum _{i=1}^N\sum _{j=1}^n\left|F(x_j;\mathit{\theta })-F(x_j;{\widehat{\mathit{\theta }}}_i)\right|,$$

and

$$D_{\max }={\displaystyle \frac{1}{N}}\sum _{i=1}^N\underset{j=1,\dots ,n}{max}\left|F(x_j;\mathit{\theta })-F(x_j;{\widehat{\mathit{\theta }}}_i)\right|,$$

respectively, where $F(x;\mathit{\theta })$ is given by (2), and ${\widehat{\mathit{\theta }}}_i$ is an estimator of the vector of model parameters $\mathit{\theta }$ based on simulation run i. For computational convenience, min-max normalization is considered for all metrics to help in data visualization and interpretation and to reduce the impact of outliers (if any); see, for example, ref. [71] for further details.

When the normalized RMSEs approach zero as the sample size increases, the estimators are computationally efficient, regardless of the underlying probability distribution. Moreover, estimators are deemed effective if the goodness-of-fit metrics $D_{\mathrm{abs}}$ and $D_{\max }$ are minimized and tend to zero as the sample size increases and remains robust to the other factors, such as the heaviness of the distribution’s tail or the existence of outliers, for example. Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 reveal the following observations:

• The estimators of $\mu$ behave similarly for small values of $\theta$. As the value of $\theta$ starts to increase, both MLE and MPSE of $\mu$ perform better than the remaining estimation procedures; nevertheless, when the value of $\theta$ surges, some of the other estimation methods provide better estimates for $\mu$.
• The MLEs of $\sigma$ and $\theta$ are less efficient than the other estimators for small values of $\theta$; however, once the value of $\theta$ increases, the methods behave similarly, especially when the sample size is large enough. MLEs, MPSEs, and ADEs of $\sigma$ and $\theta$ surpass the other estimates for larger values of $\theta$ in terms of RMSEs.
• For the goodness-of-fit measurements $D_{\mathrm{abs}}$ and $D_{\max }$, MLEs performed well compared to the remaining estimators for small values of $\theta$, while MLEs, MPSEs, and ADEs slightly did not perform well for large values of $\theta$ unless the sample size is large enough.
• Overall, all methods behave similarly when the sample size is large enough as expected in terms of estimation efficiency and goodness-of-fit.

6. Conclusions

This paper addressed one potential approach to modeling fatigue data using a statistical model called the GTBS family of lifetime distributions. The main contributions and findings of this study can be summarized as follows:

• A generalization of the conventional two-parameter BS distribution was considered. The model of interest, known as the GTBS distribution, extends the classical BS family by incorporating a shape parameter and heavy-tailed behavior via a Student’s t kernel.
• The model parameters were estimated using seven frequentist methods, including maximum likelihood, least squares, and spacing- and distance-based approaches.
• The validity of the model and its sub-models was verified using real fatigue data. Specifically, two real fatigue datasets were analyzed, and the GTBS model provided superior goodness-of-fit compared to its sub-models, as evidenced by bootstrapped KS, AD, and CVM statistics. Nevertheless, the GTBS model is expected to perform even better when the sample size is sufficiently large.
• In addition to the practical applications, estimation efficiency was demonstrated through Monte Carlo simulations. The results showed that the GTBS model, particularly under the MLE and MPSE methods, performs well in terms of RMSE and goodness-of-fit metrics across a range of parameter settings and sample sizes.