Objective Posterior Analysis of kth Record Statistics in Gompertz Model

Abstract

The Gompertz distribution has proven highly valuable in modeling human mortality rates and assessing the impacts of catastrophic events, such as plagues, financial crashes, and famines. Record data, which capture extreme values and critical trends, are particularly relevant for analyzing such phenomena. In this study, we propose an objective Bayesian framework for estimating the parameters of the Gompertz distribution using record data. We analyze the performance of several objective priors, including the reference prior, Jeffreys’ prior, the maximal data information (MDI) prior, and probability matching priors. The suitability and properties of the resulting posterior distributions are systematically examined for each prior. A detailed simulation study is performed to assess the effectiveness of various estimators based on the performance criteria. To demonstrate the practical application of the methodology, it is applied to a real-world dataset. This study contributes to the field by providing a thorough comparative evaluation of objective priors and showcasing their impact and applicability in parameter estimation for Gompertz distribution based on record values.

1. Introduction and Motivation

Initially developed to model mortality rates, the Gompertz distribution has demonstrated remarkable versatility across various domains. Its primary appeal lies in its ability to capture growth patterns and survival probabilities in contexts where failure rates or hazard functions increase exponentially over time.

Its probability density function (pdf) and cumulative distribution function (cdf) are given by

$$\begin{array}{cc}\hfill f(x;\alpha ,\beta )& =\beta e^{\alpha x}{e}^{-{\displaystyle \frac{\beta }{\alpha }}(e^{\alpha x}-1)},x>0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill F(x;\alpha ,\beta )& =1-e^{-{\displaystyle \frac{\beta }{\alpha }}(e^{\alpha x}-1)},x>0,\hfill \end{array}$$

(2)

where $\alpha >0$ is the shape parameter, and $\beta >0$ is the scale parameter.

For instance, in biology, the Gompertz distribution has been used to model abalone growth increments [1], reflecting the biological growth process where the rate of growth decelerates over time. Similarly, it has been employed in actuarial science and biomedical studies to model human and animal survival rates, tumor growth dynamics, and cell proliferation patterns.

In economics, the Gompertz distribution has found applications in modeling income distribution [2] by capturing the skewed nature of income variations in populations. It has also been used in reliability engineering and risk assessment, particularly in insurance and finance, where it helps model claim severity, lifetime of financial instruments, and default probabilities.

In demography, the Gompertz model is widely applied in fertility rate modeling [3] and population predictions [4,5], as it effectively captures age-dependent mortality and birth rates. It has also been employed in urban planning to forecast population growth and assess the sustainability of resources in growing metropolitan areas.

Beyond these fields, the Gompertz distribution has seen extensive use in engineering, where it helps model system reliability and failure rates in mechanical and electronic components. In climate science, it has been applied to estimate extreme weather event occurrences, such as hurricanes or drought durations. Additionally, in marketing and business analytics, the distribution has been used to model consumer behavior, product adoption curves, and technology diffusion, as it aligns well with the idea of early adopters followed by slower market saturation.

The utility of the Gompertz distribution has inspired numerous extensions and generalizations, broadening its applicability to even more complex datasets. Notable extensions include the following:

• Generalized Gompertz distribution [6], which provides greater flexibility in modeling hazard functions with non-monotonic characteristics.
• Beta-Gompertz distribution [7], which incorporates additional shape parameters to better model diverse life phenomena.
• Gamma-Gompertz distribution [8], which extends the original model by incorporating a gamma-distributed heterogeneity component, making it useful in survival analysis.
• McDonald–Gompertz distribution [9], which introduces additional shape parameters for greater adaptability in real-world applications.

Given its versatility, the Gompertz distribution continues to be a valuable tool in statistical modeling, offering a robust framework for analyzing time-to-event data and dynamic growth processes across multiple disciplines.

For instance, during an epidemic, high death rates are perceived as dominant, and particular attention is paid to them because their behavior is crucial for determining the process of the plague. Similar circumstances could arise in the stock market, when investors are curious about whether a stock would outperform its predecessors. Additionally, cutting down on the number of records to observe from the entire set of data under consideration could save time and money during the memorization process. This suggests that record values in these cases include the majority of the information as well; see [10]. This is one of the explanations for the ongoing interest in record event modeling and observation.

Record values are one of the most useful sample schemes in reliability and life testing designs. They were introduced in [11]. In this scheme, the interest relies on the units that exceed all previous ones during the investigation. Note that there are two methods of extracting record values from a sample. Random sampling is used as the term for the first one. Within this approach, the number of records is seen as a random variable, and the sample size is specified in advance. The second approach implies that the sample size is seen as a random variable by choosing the number of records. This technique is known as inverse sampling (see [12]), and it is commonly used in practice. Under this setting, m units are placed on a life-testing experiment and only $n(n\le m)$ units of records are observed. This is the occurrence in a vast number of experiments, where it seems easier to collect and memorize only record values instead of the whole sample. Examples may be found in breaking wooden beams [13], extreme weather conditions [14], biology [15], etc. Readers who find this interesting may refer to [16,17] for the fundamentals within this theory.

This paper deals with so-called kth record values. Let $\{X_i,i\ge 1\}$ be the sequence of independent and identically distributed (iid) random variables that serves us as the scheme where the extraction procedure of records is applied. We can select the random variables $T_{1,k}=k$ and $R_{1\left(k\right)}=X_{1:k}$ as the first kth record time of occurrence and kth record value. Immediately following the first time of occurrence $T_{1,k}$, $T_{2,k}$ is defined as the minimum of the set $\{j:j>k,X_j>X_{1:k}\}$, i.e., $T_{2,k}=min\{j:j>k,X_j>X_{1:k}\}$. Then, following the second kth record time $T_{2,k}$, $T_{3,k}$ denotes the minimum of the set $\{j:j>T_{2,k},X_j>X_{T_{2,k}-k+1:k}\}$, i.e., $T_{3,k}=min\{j:j>T_{2,k},X_j>X_{T_{2,k}-k+1:k}\}$. The process continues until all remaining kth times are collected. Clearly, the upper kth record values are defined recursively as $\{R_{n\left(k\right)}=X_{T_{n-1,k}-k+1:T_{n-1,k}},n\ge 1\}$ providing a sample scheme of records ${\mathbf{R}}_{\left(\mathbf{k}\right)}=(R_{1\left(k\right)},R_{2\left(k\right)},\dots ,R_{n\left(k\right)})$ useful for inferential purposes and denoting the observed record values as ${\mathbf{r}}_{\left(\mathbf{k}\right)}=(r_{1\left(k\right)},r_{2\left(k\right)},\dots ,r_{n\left(k\right)})$. Directly, the conventional record values are specific cases of kth record values by considering the k value to be one. For more details on kth record times and values, we refer to [16,18,19]. For recent inference under the kth record sample scheme, see, for example [20,21,22,23,24,25,26,27,28].

Jaheen [29] used sampling-based computational methods to estimate the parameters of the Gompertz model under the classical record framework. This framework represents a specific case of kth records when $k=1$. It was revealed that there is no tractable form for the maximum likelihood estimator (MLE) of parameter $\alpha$ in the case of classical record values. As a result, a numerical iterative process is required for its evaluation.

In this paper, a similar situation arises as we demonstrate that there is no tractable form for the MLEs of parameter $\alpha$ in the case of kth upper record values. Therefore, numerical computation methods based on iterations must be applied. However, several challenges can emerge during such iterations, including divergence, convergence to multiple solutions, and dependency on initial values. The nature of these potential issues is discussed in Appendix A.1.

In general, drawing valid conclusions about the parameters becomes difficult when using small samples. These samples exhibit a high degree of skewness and display noticeable asymmetry in their behavior, similar to record values. To address these challenges, Bayesian methods are employed to provide clearer and more reliable parameter estimates as indicated in [30].

Moreover, the Gompertz distribution plays a crucial role in reliability analysis, aligning closely with advancements in statistical inference and degradation modeling. Recent developments propose multivariate models for analyzing dependent tail-weighted degradation data, emphasizing the importance of robust statistical methods in reliability applications [31].

Additionally, statistical inference techniques for k-out-of-n: G systems with switching failures under Poisson shocks highlight the necessity for flexible and accurate modeling in reliability and life testing [32]. These contributions support the current study by emphasizing the significance of advanced statistical approaches, such as those based on record values and Bayesian inference, in addressing reliability challenges.

By integrating these methodologies, this study bridges theoretical advancements with practical needs in reliability modeling.

Here, we consider Bayesian analysis for the Gompertz distribution when record values are present. As is known in the literature, the analysis uses the so-called subjective priors as an illustration of already acquired knowledge that the researcher possesses. Such knowledge is based on past experience, background motivation within the problem, or even some evidence already collected. Let us consider the Bayesian analysis of the model parameters $(\alpha ,\beta )$ under the subjective fuzzy prior

$$\begin{array}{c}\hfill \pi (\alpha ,\beta )\propto {\displaystyle \frac{1}{\beta }}{\alpha }^{\tau -1}{e}^{-\alpha },\end{array}$$

(3)

where $\tau >0$ is the hyperparameter.

In Appendix A, we see that under the prior (3), the posterior distributions $\pi (\alpha ,\beta \mid {\mathbf{R}}_{\left(\mathbf{k}\right)})$ are proper. For the purpose of illustrating the influence of the prior distribution’s shape on the posterior distributions, we use true values of parameters $(\alpha ,\beta )$, while the hyperparameter $\tau$ is assigned values of 1, 5, and 9 with performance metrics such as RMSE and CP, defined in Section 4. The RMSE is used to quantify the estimation accuracy, focusing on $\alpha$ since the results for $\beta$ are similar and thus omitted for brevity. The estimates $\widehat{\alpha }$ and $\widehat{\beta }$ are obtained through Bayesian methods as described in Section 4. Specifically, the posterior distributions $\pi (\alpha ,\beta \mid {\mathbf{r}}_{\left(\mathbf{k}\right)})$ are derived under the prior specified in (3), and the parameter estimates are taken as the posterior means of $\alpha$ and $\beta$. This ensures that the estimates reflect the characteristics of the posterior distributions while accounting for the influence of the prior. The record sample sizes considered in this study are $n=3,4,7,9,$ and 10, and the results are presented graphically in Figure 1, providing a detailed visualization of how $\tau$ affects posterior analysis. These findings emphasize the challenge of selecting an appropriate value for $\tau$ that balances prior assumptions with data-driven inference.

When the underlying data are unknown or when we want to highlight the information found in the data alone, we might employ what are known as objective priors. These priors are based on adequate scientific principles that address the interest of the problem at hand. Several authors have utilized the objective priors for model parameters based on iid samples and discussed their respective properties and influence on the posteriors. Examples can be found in [33,34,35,36,37,38]. For situations in which the iid structure is compromised, such analysis becomes more complex but at the same time appealing. This is the reason why there are limited papers that deal with this topic in literature. The case of record values is one such instance. As a result, this paper’s practical value is mandated by the Bayesian inference under record values and objective priors, which uses the global popularity of records and can find several applications in diverse spheres of life. For example, Refs. [39,40] contain such applications.

This paper aims to provide a detailed analysis of objective Bayesian methods for the Gompertz model based on kth record values. We demonstrate that a vague prior exists on the entire positive axis and that the properness of the posterior distributions is satisfied under this prior. Similar conclusions are drawn for objective priors. Bayesian and objective Bayesian analyses for the Gompertz model using independent and identically distributed (iid) samples have been discussed in [41,42,43]. The importance of Bayesian inference for Gompertz model parameters was highlighted in the context of an age mimicry problem in [44]. Objective Bayesian analysis for the Gumbel distribution under a record framework was presented in [40]. Furthermore, as noted in [23], when both parameters of the Gompertz model are unknown, finding a joint prior with support on the positive real axis can lead to computational complexities. Using extensive computer simulations, we compare the performance of Bayesian estimators under various objective priors. Finally, we compute credible intervals and evaluated their correspondence with mean square errors (MSEs).

Overall, this paper contributes to the growing body of work on Bayesian inference for the Gompertz distribution by focusing on record data. The specific contributions are as follows:

• We develop a comprehensive objective Bayesian framework for estimating the parameters of the Gompertz distribution using kth record values. This includes deriving and evaluating various objective priors such as Jeffreys’ prior, reference prior, maximal data information (MDI) prior, and probability matching priors.
• We rigorously establish the properness of the posterior distributions under these priors, ensuring the validity of the proposed approach.
• A detailed comparative analysis of the objective priors is performed through an extensive simulation study, highlighting their influence on Bayesian estimators in terms of mean squared error (MSE) and coverage probabilities (CPs).
• We address the computational challenges associated with maximum likelihood estimation (MLE) for kth record values and demonstrate the advantages of Bayesian methods, particularly under small sample settings, where MLE methods often struggle.
• The proposed methodology is applied to real-world record data, showcasing its practical relevance and robustness in modeling and inference.

Compared to existing works, our study is distinguished by its focus on kth record values and the integration of multiple objective priors in the Bayesian framework. While ref. [29] explored Bayesian inference for Gompertz distribution parameters using classical record values ($k=1$), this paper generalizes the framework to kth record values and extends the analysis to include objective priors. Furthermore, we address computational issues related to iterative MLE methods, as noted in [29], and demonstrate how Bayesian methods circumvent these challenges. Prior works on objective Bayesian inference, such as [39], primarily focused on iid data or simpler distributions; our approach broadens the scope to include record-based inference for the Gompertz model, which has not been extensively studied in the literature.

The remainder of the paper is structured as follows: Section 2 introduces the derivation and properties of objective priors for the Gompertz distribution under the record framework. Section 3 presents the results of a simulation study comparing the performance of Bayesian estimators under different priors. In Section 4, the proposed methodology is applied to real-world datasets, followed by conclusions and discussions in Section 5 and Section 6. Detailed proofs and additional results are provided in Appendix A.

2. Non-Informative Priors and Their Properties

In this part, we introduce deterministically generated priors for the Gompertz distribution parameters, which highlight and retain as much as possible the information found in a record sample. The next section includes a number of significant non-informative priors for parameters $(\alpha ,\beta )$, including Jeffrey’s, the maximal data information, reference, and probability matching priors.

2.1. Probability Matching Priors

In statistical analysis, the comparison between Bayesian inference and conventional frequentist methodologies remains a critical area of study. A key motivation for these comparisons is the desire to align the behaviors of the two approaches as closely as possible. This is where probability matching priors become particularly valuable. These priors are constructed to meet a fundamental practical requirement: ensuring that the posterior coverage probability of a Bayesian credible interval closely aligns with the frequentist coverage probability. For further details on this approach, refer to [41]. However, within record sampling schemes, selecting an appropriate prior becomes especially challenging due to the inherent skewness of record value distributions.

To formalize this concept, consider a prior $\pi \left(·\right)$ for the parameters $(\varphi ,\xi )$, where $\varphi$ is the parameter of interest. Let ${\varphi }^{(1-\gamma )}(\pi \left(·\right),\mathbf{X})$ denote the $(1-\gamma )$th percentile of the marginal posterior distribution of $\varphi$. The prior $\pi \left(·\right)$ is termed a second-order probability matching prior if it satisfies

$$P\left(\varphi \le {\varphi }^{(1-\gamma )}(\pi \left(·\right),\mathbf{X})\right)=1-\gamma +o\left(n^{-1}\right),$$

for all $\gamma \in (0,1)$. For more detailed insights, see [42].

Expressions like (1) and (2) are instrumental in defining second-order probability matching priors for the parameters $(\alpha ,\beta )$ of the Gompertz distribution. These priors ensure that the frequentist coverage probabilities of Bayesian credible intervals align closely with their nominal levels, a desirable property in statistical inference. Their construction carefully balances the interplay between the parameter of interest and the nuisance parameter, enabling a tailored approach to inference. Moreover, the properness of the posterior under these priors ensures their applicability for Bayesian analysis, mitigating computational challenges often associated with such distributions. Consequently, these expressions serve as a foundation for constructing objective priors for the Gompertz distribution within the record framework.

Theorem 1. 

(a) When α is the parameter of interest and β is the nuisance parameter, the second-order probability matching prior has the approximated form

$$\begin{array}{c}\hfill {\pi }_{M_1}(\alpha ,\beta )\propto F_1\left(\alpha \right)·G_1\left(\beta \right),\end{array}$$

(4)

where $F_1\left(\alpha \right)=e^{-\int {\displaystyle \frac{c_1+2}{\alpha }}d\alpha }$ and $G_1\left(\beta \right)={\beta }^{c_1}$, and $c_1$ as an arbitrary constant.

(b) When β is the parameter of interest and α is the nuisance parameter, the second-order probability matching prior has the approximated form

$$\begin{array}{c}\hfill {\pi }_{M_2}(\beta ,\alpha )\propto F_2\left(\alpha \right)·G_2\left(\beta \right),\end{array}$$

(5)

where $F_2\left(\alpha \right)=e^{-\int {\displaystyle \frac{2{\alpha }^2(c_2+1)-1}{2\alpha }}d\alpha }$ and $G_2\left(\beta \right)={\beta }^{c_2}$, and $c_2$ as an arbitrary constant. (c) For the case when $c_1=c_2=-1$, the posterior distribution under ${\pi }_{M_1}$ or ${\pi }_{M_2}$ is proper.

2.2. Maximal Data Information Priors

Lindley [43] developed an information-theoretical study of the Bayesian modeling structure using Shannon entropy. His work aimed to formalize the principles behind Bayesian inference by leveraging entropy-based measures to assess the informativeness of prior distributions. The fundamental idea is to construct a prior distribution $\pi$ that maximizes the information obtained from the data themselves while effectively incorporating prior knowledge about the parameters. This ensures that the prior is not overly restrictive or uninformative but is instead optimally balanced to make full use of the available information.

Building upon this concept, Zellner [44] introduced a systematic approach to deriving informative priors based on maximizing the information content in the likelihood function of the unknown parameters. He proposed a prior distribution that optimally reflects the structure of the likelihood while maintaining a coherent Bayesian updating framework. This prior, known as the maximum data information (MDI) prior, is formulated to enhance the inferential process by ensuring that the prior contributes meaningfully to parameter estimation without overshadowing the evidence provided by the data. The MDI prior is particularly useful in situations where subjective prior information is limited or unreliable, allowing for a more objective and data-driven approach to Bayesian analysis.

The next theorem provides the MDI prior for parameters $\alpha$ and $\beta$ of the Gompertz model (1).

Theorem 2. 

(a) 

The MDI prior for the parameters ($\alpha ,\beta )$ is selected as

$$\begin{array}{c}\hfill {\pi }_{MDI}(\alpha ,\beta )\propto \beta e^{\alpha }.\end{array}$$

(b) 

The posterior distribution under ${\pi }_{MDI}(\alpha ,\beta )$ is proper.

2.3. Reference Priors

In order to maximize the missing information on the parameters, the reference analysis use information-theoretic ideas to properly specify the objective prior, which should be maximally dominated by the data [45]. In [46], reference priors were first formulated in an essentially informal manner. More detailed explanations of the sequential reference process in continuous multiparameter issues were provided in [47]. Furthermore, in [47], a formal, rigorous definition of reference priors for a single block of parameters was provided. Reference priors divide the parameters into distinct ordering groups of interest as is well known. One can find more details in [39].

The following theorem represents the reference priors under different ordering groups for parameters $\alpha$ and $\beta$ from model (1).

Theorem 3. 

(a) If α is the parameter of interest and β is the nuisance parameter, the reference prior holds the form

$$\begin{array}{c}\hfill {\pi }_{R1}(\alpha ,\beta )\propto {\displaystyle \frac{1}{\alpha \beta }}.\end{array}$$

and if β is the parameter of interest and α is the nuisance parameter, the reference prior for ($\beta ,\alpha )$ is

$$\begin{array}{c}\hfill {\pi }_{R2}(\beta ,\alpha )\propto {\displaystyle \frac{1}{\alpha \sqrt{\beta }}}.\end{array}$$

(b) The posterior distribution under the prior ${\pi }_{R1}$ and ${\pi }_{R2}$ is proper.

2.4. Jeffrey’s Prior

Jeffery [48] suggested an objective prior to represent a scenario with weak information about the unknown model parameters. The Fisher information (FI) matrix I, provided in (A18), is the source of this prior, indicated by

$$\begin{array}{c}\hfill {\pi }_J(\alpha ,\beta )\propto \sqrt{detI(\alpha ,\beta )}\end{array}$$

(6)

where

$$\begin{array}{c}\hfill I(\alpha ,\beta )\approx \left(\begin{array}{cc}{\displaystyle \frac{2}{\alpha }}-{\displaystyle \frac{2}{\alpha \beta }}+{\displaystyle \frac{1}{{\beta }^2}}& {\displaystyle \frac{1}{{\beta }^2}}-{\displaystyle \frac{1}{\alpha \beta }}\\ {\displaystyle \frac{1}{{\beta }^2}}-{\displaystyle \frac{1}{\alpha \beta }}& {\displaystyle \frac{1}{{\beta }^2}}\end{array}\right),\end{array}$$

(7)

Although there is continuous debate on whether the multivariate form prior is acceptable, Jeffrey’s prior is commonly utilized because of its invariance property under one-to-one transformations of parameters. Thus, from (6) and (7), the Jeffrey’s prior for $(\alpha ,\beta )$ is stated in following theorem. We gain a simplified version by approximation.

Theorem 4. 

(a) The Jeffrey’s prior for $(\alpha ,\beta )$ is

$$\begin{array}{c}\hfill {\pi }_J(\alpha ,\beta )\propto {\displaystyle \frac{1}{\alpha \beta }}.\end{array}$$

(b) The posterior distribution under ${\pi }_J(\alpha ,\beta )$ is proper.

The proof of Theorems 1–4 is postponed to Appendix A. Theorems 1–4 imply that posterior inferences are possible with respect to all objective priors discussed above. On the other hand, while ${\pi }_{MDI},{\pi }_{R2}$ is not a second-order probability matching prior, ${\pi }_{R1}$ and ${\pi }_J$ are, and they coincide. In this regard, prospective users are advised to use ${\pi }_J$. Indeed, the following numerical investigations corroborate this.

While this study evaluates multiple priors, the choice of a prior should be context dependent. For instance, Jeffreys’ prior is recommended in small sample settings due to its second-order probability matching property. On the other hand, the maximal data information (MDI) prior may be more suitable for exploratory analyses where the focus is on leveraging the data’s inherent structure. Practitioners should also consider computational trade-offs; priors requiring complex numerical integration may not be ideal for real-time applications. Guidelines for selecting priors in specific domains, such as meteorology or reliability testing, are provided in Section 4.

3. Implementing the MCMC Algorithm

As is standard in Bayesian analysis, we explicitly state the likelihood function and derive the joint posterior distribution for the parameters $(\alpha ,\beta )$ of the Gompertz distribution. The likelihood function for the parameters $(\alpha ,\beta )$ given the observed record data ${\mathbf{r}}_{\left(\mathbf{k}\right)}=(r_{1\left(k\right)},r_{2\left(k\right)},\cdots ,r_{n\left(k\right)})$ is defined as (A2) with a prior distribution $\pi (\alpha ,\beta )$, and the joint posterior distribution is obtained as (A12) with the form

$$\pi (\alpha ,\beta \mid {\mathbf{r}}_{\left(\mathbf{k}\right)})\propto k^n{\beta }^n\pi (\alpha ,\beta )exp\left(\alpha \sum _{i=1}^n{r}_{i\left(k\right)}-{\displaystyle \frac{\beta k}{\alpha }}\left(e^{\alpha r_{n\left(k\right)}}-1\right)\right).$$

The joint posterior distribution lacks a closed-form solution, making direct sampling of $(\alpha ,\beta )$ infeasible; hence, we employ the Markov Chain Monte Carlo (MCMC) algorithm as the pivotal in Bayesian statistical analysis, particularly when dealing with posterior distributions that lack a closed-form solution. In this study, we employ the random-walk Metropolis–Hastings (MH) algorithm [49] due to its flexibility and robustness in handling complex posterior landscapes into practice within the R software (version 4.4.0), which is a widely used MCMC method. This approach is particularly advantageous in scenarios where posterior distributions are challenging to sample from directly due to their intricate structures, conditional posteriors lack standard forms that preclude the use of Gibbs sampling (see Appendix A.7), and the parameter space involves strong dependencies between parameters, which the MH algorithm effectively navigates by adjusting the proposal distribution’s step size.

The MH algorithm demonstrates robust performance in estimating parameters for the Gompertz distribution under the kth record framework. The choice of an appropriate proposal distribution and tuning parameters is critical for achieving high acceptance rates and efficient exploration of the parameter space.

In this study, we employ truncated normal distributions at vectors of zeros as candidate-generating densities for the MH algorithm. The variance matrix of the proposal distribution is constructed to be directly proportional to the Fisher information matrix. This choice is motivated by the Fisher information’s ability to reflect the local geometry of the parameter space, thereby guiding the proposal distribution to regions of higher posterior density.

Specifically, let the Fisher information matrix for the parameters $\theta =(\alpha ,\beta )$ be denoted by $I\left(\theta \right)$. The variance matrix for the proposal distribution is set as

$$\mathrm{\Sigma }=c·I{\left(\theta \right)}^{-1},$$

where $c>0$ is a scaling factor adjusted during preliminary tuning to achieve an acceptance rate in the range of 10–40%. The truncation ensures that all proposed values remain within the parameter space’s feasible bounds, particularly for parameters like $\alpha$ and $\beta$, which are constrained to be positive.

Using this approach offers two main advantages: it provides efficient exploration by aligning the variance matrix with the curvature of the posterior distribution, allowing the algorithm to propose moves that better match the underlying geometry and improve convergence; and it ensures robustness through the Fisher information-based variance matrix, which adapts to the data’s properties and maintains flexibility across different scenarios.

This implementation is further validated through diagnostic checks, including trace plots and autocorrelation analysis, ensuring effective mixing of the Markov chain and convergence to the posterior distribution.

The steps of the random-walk Metropolis algorithm are as follows:

• Initialize the parameters ${\alpha }_0$ and ${\beta }_0$.
• Propose new candidate values ${\alpha }^{\prime }$ and ${\beta }^{\prime }$ from the truncated normal distribution centered at the current values with variance matrix $\mathrm{\Sigma }$.
• Compute the acceptance probability

  $$A=min\left(1,{\displaystyle \frac{\pi ({\alpha }^{\prime },{\beta }^{\prime }|{\mathbf{r}}_{\left(\mathbf{k}\right)})}{\pi (\alpha ,\beta |{\mathbf{r}}_{\left(\mathbf{k}\right)})}}\right).$$

  (8)
• Accept the candidate with probability A. If rejected, retain the current values.
• Repeat steps 2–4 for a pre-specified number of iterations.

The reliance on iterative numerical procedures for parameter estimation, such as the MCMC algorithm, introduces computational challenges. These include potential divergence, multiple solutions, and dependency on initial conditions. These issues are exacerbated in small sample sizes or high-dimensional parameter spaces. The study addresses these challenges by adopting robust Bayesian frameworks, as detailed in the next section, ensuring convergence and computational feasibility. Practitioners are encouraged to adopt hybrid strategies, combining analytic approximations with iterative refinement, to mitigate these issues in practical applications.

4. Simulation Study

In order to evaluate the impact of the various prior distributions on the posterior, a Monte Carlo simulation analysis is carried out in this section. A total of 100,000 random variates are sampled; the first 40,000 are discarded as a burn-in sample, and each process is repeated 500 times. By targeting the 10–40% acceptance rate, the algorithm balances between sufficient exploration of the state space and a reasonable number of accepted moves as proposed in [50]. The samples are then reduced by a factor of 15 to produce a low mutual autocorrelation within chains and used to estimate the posterior density with the remaining observations. In order to provide a reliable performance, the median is chosen as a Bayesian estimator. This intuitively makes sense since the median provides more optimal and robust estimation compared to the sample mean. We select the highest posterior density intervals (HPDs) as the appropriate estimate of Bayesian credible intervals (CIs) due to the non-symmetrical form of the marginal posteriors of parameters $\alpha$ and $\beta$ based on records as recognized in [51,52].

All estimates are evaluated based on their computed values and MSEs. For example, the MSE for $\alpha$ is calculated using the formula

$$\mathrm{MSE}\left(\alpha \right)={\displaystyle \frac{1}{M}}\sum _{i=1}^M{\left({\widehat{\alpha }}_i-\alpha \right)}^2,$$

where ${\widehat{\alpha }}_i$ denotes the estimate of $\alpha$ obtained in the ith simulation, $\alpha$ is the true parameter value, and M is the total number of simulations. Also, the root mean square error (RMSE) is defined as the square root of MSE, i.e., $\sqrt{MSE}$. Further, the true parameter values are assigned as $(\alpha ,\beta )=\left\{\right(3,1),(2,1.5),(1.5,2),(1.5,5),(3,10\left)\right\}$. Value k is chosen to be 1 and 2. These values of k are intuitively chosen because, for $k=1$, they represent ordinary records that are encountered frequently in practice, indicating a pragmatic viewpoint. In the meantime, $k=2$ emphasizes other aspects of records, including the convergence of their moments, which can be more profound in real situations (see, for example, [53]). This consideration becomes particularly significant in machine learning techniques, as the selection of k values can profoundly impact the dimensionality reduction of the problem. For instance, in techniques like Principal Component Analysis (PCA), choosing different values of k alters the number of principal components retained, thus affecting the representation and compression of the data. Therefore, the thoughtful selection of k values is crucial for reducing dimensionality and retaining knowledge in machine learning tasks in an efficient manner (see [54,55,56,57]). The sample sizes are selected at $n=5,10,15$, and 20 in order to illustrate the performance of Bayesian estimators with respect to different objective priors.

From the reported values found in

Table A1. Empirical MSEs (CPs in parentheses) of Bayesian estimators based on priors ${\pi }_J$, ${\pi }_{R2}$, ${\pi }_{M2}$, and ${\pi }_{MDI}$ for $k=1$.

Prior	Parameter	$\mathit{n}=5$	$\mathit{n}=10$	$\mathit{n}=15$	$\mathit{n}=20$
$\alpha =3,\beta =1$
${\pi }_J$	$\alpha$	1.7362 (0.932)	1.2102 (0.932)	1.0539 (0.902)	0.7827 (0.924)
	$\beta$	1.6424 (0.978)	1.9788 (0.974)	1.925 (0.948)	1.6002 (0.972)
${\pi }_{MDI}$	$\alpha$	1.7592 (0.97)	0.9793 (0.932)	0.8322 (0.92)	0.7636 (0.904)
	$\beta$	1.1298 (0.998)	1.7745 (0.974)	1.8069 (0.978)	1.6755 (0.98)
${\pi }_{R2}$	$\alpha$	1.8439 (0.842)	1.4707 (0.79)	1.0848 (0.85)	0.8439 (0.878)
	$\beta$	2.377 (0.964)	2.785 (0.924)	2.4906 (0.952)	1.9889 (0.96)
${\pi }_{M2}$	$\alpha$	1.5546 (0.978)	1.2402 (0.924)	0.8938 (0.952)	0.7753 (0.966)
	$\beta$	1.0959 (0.988)	1.4218 (0.97)	1.1956 (0.968)	1.0505 (0.974)
$\alpha =2,\beta =1.5$
${\pi }_J$	$\alpha$	1.4957 (0.942)	1.048 (0.886)	0.8575 (0.884)	0.7146 (0.894)
	$\beta$	1.3724 (0.978)	2.146 (0.938)	2.1382 (0.94)	2.1277 (0.93)
${\pi }_{MDI}$	$\alpha$	1.3398 (0.984)	0.7242 (0.922)	0.6621 (0.9)	0.5555 (0.94)
	$\beta$	1.1552 (1)	1.9265 (0.986)	2.1219 (0.958)	1.9719 (0.968)
${\pi }_{R2}$	$\alpha$	1.3055 (0.88)	1.0662 (0.806)	0.8812 (0.808)	0.7455 (0.846)
	$\beta$	2.2305 (0.96)	2.5247 (0.944)	2.6387 (0.912)	2.6454 (0.92)
${\pi }_{M2}$	$\alpha$	1.5645 (0.982)	1.0097 (0.948)	0.7429 (0.938)	0.6122 (0.948)
	$\beta$	1.0612 (0.978)	1.4921 (0.962)	1.5654 (0.95)	1.469 (0.946)
$\alpha =1.5,\beta =2$
${\pi }_J$	$\alpha$	0.9934 (0.972)	0.8581 (0.9)	0.6904 (0.892)	0.5961 (0.91)
	$\beta$	1.5894 (0.972)	1.9405 (0.958)	2.1222 (0.948)	2.1179 (0.93)
${\pi }_{MDI}$	$\alpha$	1.8438 (0.984)	0.5918 (0.956)	0.5644 (0.882)	0.4974 (0.89)
	$\beta$	1.0913 (0.996)	1.8687 (0.996)	2.3238 (0.952)	2.1677 (0.96)
${\pi }_{R2}$	$\alpha$	0.8766 (0.944)	0.844 (0.842)	0.7265 (0.81)	0.6344 (0.806)
	$\beta$	1.9598 (0.97)	2.3987 (0.956)	2.6637 (0.938)	2.7873 (0.908)
${\pi }_{M2}$	$\alpha$	1.3672 (0.99)	0.8012 (0.974)	0.6592 (0.93)	0.5548 (0.93)
	$\beta$	1.1989 (0.982)	1.434 (0.968)	1.7181 (0.94)	1.6663 (0.934)
$\alpha =1.5,\beta =5$
${\pi }_J$	$\alpha$	2.4513 (0.962)	1.049 (0.946)	0.8486 (0.91)	0.7696 (0.868)
	$\beta$	3.805 (0.94)	2.8276 (0.956)	3.6008 (0.934)	3.7584 (0.918)
${\pi }_{MDI}$	$\alpha$	12.7472 (0.706)	1.4345 (0.968)	0.6719 (0.962)	0.5508 (0.958)
	$\beta$	3.2466 (0.754)	2.276 (0.98)	2.6135 (0.986)	3.117 (0.97)
${\pi }_{R2}$	$\alpha$	1.192 (0.962)	0.9704 (0.894)	0.8767 (0.834)	0.7855 (0.808)
	$\beta$	4.1568 (0.962)	3.4359 (0.946)	3.8024 (0.928)	4.4419 (0.89)
${\pi }_{M2}$	$\alpha$	2.8096 (0.982)	1.1815 (0.982)	0.9151 (0.928)	0.6964 (0.912)
	$\beta$	2.5463 (0.954)	2.4604 (0.962)	2.8278 (0.912)	3.1741 (0.936)
$\alpha =3,\beta =10$
${\pi }_J$	$\alpha$	4.3602 (0.966)	2.4604 (0.914)	1.8525 (0.88)	1.6201 (0.848)
	$\beta$	6.3537 (0.954)	6.8124 (0.942)	7.7799 (0.906)	8.325 (0.916)
${\pi }_{MDI}$	$\alpha$	44.3649 (0.126)	6.8346 (0.656)	2.2841 (0.898)	1.4796 (0.926)
	$\beta$	9.6124 (0.058)	6.4391 (0.69)	4.9064 (0.886)	4.8237 (0.92)
${\pi }_{R2}$	$\alpha$	2.5706 (0.952)	1.9613 (0.89)	1.7866 (0.826)	1.6454 (0.79)
	$\beta$	7.178 (0.946)	7.3819 (0.942)	8.6658 (0.896)	9.5319 (0.872)
${\pi }_{M2}$	$\alpha$	6.2331 (0.97)	2.5782 (0.966)	1.7945 (0.94)	1.4292 (0.922)
	$\beta$	5.7429 (0.944)	5.2623 (0.932)	5.7694 (0.944)	6.4812 (0.926)

and

Table A2. Empirical MSEs (CPs in parentheses) of Bayesian estimators based on priors ${\pi }_J$, ${\pi }_{R2}$, ${\pi }_{M2}$, and ${\pi }_{MDI}$ for $k=2$.

Prior	Parameter	$\mathit{n}=5$	$\mathit{n}=10$	$\mathit{n}=15$	$\mathit{n}=20$
$\alpha =3,\beta =1$
${\pi }_J$	$\alpha$	2.1334 (0.906)	1.455 (0.904)	1.1254 (0.912)	0.9594 (0.91)
	$\beta$	1.2199 (0.96)	1.5031 (0.978)	1.5222 (0.956)	1.3702 (0.964)
${\pi }_{MDI}$	$\alpha$	3.0018 (0.944)	1.0543 (0.958)	0.8895 (0.936)	0.7713 (0.942)
	$\beta$	0.6066 (0.988)	1.0881 (0.994)	1.1939 (0.978)	1.2017 (0.98)
${\pi }_{R2}$	$\alpha$	1.9617 (0.858)	1.6118 (0.816)	1.2976 (0.844)	1.0139 (0.87)
	$\beta$	1.6104 (0.966)	1.8605 (0.946)	1.9594 (0.94)	1.6754 (0.954)
${\pi }_{M2}$	$\alpha$	2.6099 (0.974)	1.3474 (0.936)	1.0111 (0.946)	0.8385 (0.942)
	$\beta$	0.7729 (0.99)	1.0413 (0.966)	1.1612 (0.962)	1.0505 (0.952)
$\alpha =2,\beta =1.5$
${\pi }_J$	$\alpha$	1.4819 (0.97)	1.1575 (0.9)	1.0355 (0.862)	0.8459 (0.88)
	$\beta$	1.1532 (0.976)	1.4445 (0.948)	1.7347 (0.898)	1.6963 (0.922)
${\pi }_{MDI}$	$\alpha$	6.8213 (0.892)	0.9073 (0.972)	0.7113 (0.94)	0.6326 (0.914)
	$\beta$	0.7371 (0.962)	1.183 (0.994)	1.3923 (0.98)	1.4029 (0.972)
${\pi }_{R2}$	$\alpha$	1.2328 (0.928)	1.1683 (0.842)	1.0485 (0.812)	0.8664 (0.83)
	$\beta$	1.6365 (0.95)	1.7396 (0.952)	2.0795 (0.912)	1.9476 (0.934)
${\pi }_{M2}$	$\alpha$	2.5595 (0.986)	1.0307 (0.968)	0.8997 (0.922)	0.7498 (0.914)
	$\beta$	1.1424 (0.974)	1.1226 (0.98)	1.2761 (0.952)	1.2239 (0.946)
$\alpha =1.5,\beta =2$
${\pi }_J$	$\alpha$	1.6862 (0.964)	0.9628 (0.934)	0.8384 (0.862)	0.7328 (0.886)
	$\beta$	1.1864 (0.956)	1.3059 (0.956)	1.7547 (0.936)	1.7597 (0.94)
${\pi }_{MDI}$	$\alpha$	6.7843 (0.852)	0.8998 (0.982)	0.624 (0.958)	0.5602 (0.928)
	$\beta$	1.0036 (0.946)	1.1376 (0.988)	1.3229 (0.994)	1.5534 (0.962)
${\pi }_{R2}$	$\alpha$	0.869 (0.954)	0.9311 (0.896)	0.8535 (0.82)	0.7681 (0.784)
	$\beta$	1.7139 (0.96)	1.5577 (0.962)	1.8474 (0.942)	2.1392 (0.892)
${\pi }_{M2}$	$\alpha$	2.388 (0.988)	0.9196 (0.984)	0.8015 (0.95)	0.6599 (0.922)
	$\beta$	1.0435 (0.972)	1.0425 (0.976)	1.3135 (0.952)	1.3494 (0.946)
$\alpha =1.5,\beta =5$
${\pi }_J$	$\alpha$	2.9618 (0.97)	1.3554 (0.952)	0.9985 (0.936)	0.9052 (0.906)
	$\beta$	2.8079 (0.93)	2.3359 (0.94)	2.2688 (0.97)	2.7031 (0.94)
${\pi }_{MDI}$	$\alpha$	33.0919 (0.262)	3.2629 (0.816)	1.2834 (0.954)	0.7814 (0.97)
	$\beta$	4.4792 (0.214)	2.3849 (0.878)	1.9536 (0.952)	2.0055 (0.968)
${\pi }_{R2}$	$\alpha$	1.614 (0.974)	1.0475 (0.954)	0.9707 (0.89)	0.873 (0.878)
	$\beta$	3.6881 (0.97)	2.5887 (0.958)	2.5859 (0.948)	3.0233 (0.932)
${\pi }_{M2}$	$\alpha$	4.3126 (0.994)	1.5492 (0.986)	1.0858 (0.986)	0.8529 (0.956)
	$\beta$	2.611 (0.908)	2.1368 (0.958)	2.0915 (0.956)	2.1985 (0.956)
$\alpha =3,\beta =10$
${\pi }_J$	$\alpha$	7.3097 (0.966)	2.8326 (0.96)	2.2558 (0.9)	1.8764 (0.86)
	$\beta$	6.8628 (0.922)	4.4389 (0.946)	5.1001 (0.944)	5.6842 (0.924)
${\pi }_{MDI}$	$\alpha$	84.4238 (0.004)	20.9314 (0.27)	5.8657 (0.634)	2.7898 (0.824)
	$\beta$	9.9741 (0)	8.4092 (0.216)	5.9635 (0.634)	4.6369 (0.8)
${\pi }_{R2}$	$\alpha$	4.1755 (0.972)	2.2731 (0.94)	1.9344 (0.908)	1.8518 (0.84)
	$\beta$	6.1125 (0.96)	5.4758 (0.966)	5.3633 (0.944)	6.3066 (0.934)
${\pi }_{M2}$	$\alpha$	10.8352 (0.976)	3.218 (0.988)	2.1488 (0.964)	1.7812 (0.946)
	$\beta$	5.7046 (0.88)	4.1613 (0.952)	4.3392 (0.952)	4.5652 (0.94)

, we can derive the following conclusions:

• The simulations reveal that larger sample sizes consistently reduced the MSE for all priors and improved the CPs of HPD intervals, with values converging to the nominal level of 0.95. This is readily apparent for instances $(\alpha ,\beta )=(1.5,5)$ and $(\alpha ,\beta )=(3,10)$. In contrast, small sample sizes ($n\le 10$) exhibit higher variability and less reliable coverage probabilities, underscoring the challenges of record-based inference with limited data. For $k=2$, the MSEs show similar trends to those observed with $k=1$, highlighting the flexibility of Bayesian estimation under different record settings. In general, inference based on record values suffers from the low precision of estimators. This limitation is well recognized in the literature, where similar issues have been observed; see, for example, [19]. This only confirms the necessity of such an analysis.
• For the case when true values $(\alpha ,\beta )=(3,10)$, the deviations have the highest values, while the CPs holds stable values, except for the case of the ${\pi }_{MDI}$ prior.
• The performance of the Bayesian estimators under the Jeffrey’s prior ${\pi }_J$ is generally superior to that under any other priors as indicated by both the MSEs and CPs. This is reasonable since ${\pi }_J$ is a second-order probability matching prior, in contrast to ${\pi }_{MDI}$ and ${\pi }_{R2}$. See Figure A2 and Figure A3 for a graphical presentation.

The selection of an appropriate estimator involves balancing its performance with computational cost, especially in scenarios with limited resources or stringent time constraints. High-performing estimators, such as those derived from Bayesian methods using Jeffreys’ prior, often necessitate iterative numerical procedures that can be computationally intensive. In contrast, simpler methods like maximum likelihood estimation (MLE) might be computationally cheaper but exhibit higher bias or variance, particularly in small sample settings.

Moreover, iterative methods introduce potential challenges such as non-convergence or sensitivity to initial conditions, as illustrated in Appendix A.1. Practitioners should weigh these trade-offs based on the application context. For instance, during time-critical decision-making scenarios like disaster response, methods with faster convergence may be prioritized over those yielding marginally better statistical accuracy.

5. Data Analysis

In this section, we discuss the analyses on the performance of Bayesian estimators based on kth upper record values with Gompertz distribution. We apply the proposed methods to real datasets of ordinary ($k=1$) records extracted from a sequence of 30-year periods of the amount of annual rainfall (in inches) from the Los Angeles Civic Center (Dataset II) found in [22] and a simulated record data (Dataset I) found in [29]. These datasets are shown in

Table 1. Record datasets.

Dataset I	0.12528	0.21211	0.22784	0.26063	0.65258
	0.66056	0.68255	0.79385	0.83778	0.92206
Dataset II	4.85	18.79	20.44	22.00	27.47
	33.44

, while in

Table 2. Parameters $\alpha$ and $\beta$ of the fitted Gompertz model.

	$\mathit{\alpha }$	$\mathit{\beta }$
Dataset I	1.483	5.572
Dataset II	0.0049	0.2659

, the values of the fitted parameters $\alpha$ and $\beta$ of the Gompertz model are presented.

The Bayesian techniques for estimating the parameters $\alpha$ and $\beta$ based on the ${\pi }_{MDI}$, ${\pi }_{M2}$, ${\pi }_{R2}$ and ${\pi }_J$ priors are provided here for the sake of comparison. Medians are selected as appropriate Bayes estimates and computed using MH methods. Using the MH algorithm, we initially sample 300,000 values for the MCMC chains. We then use the first 30,000 samples as a burn-in sample and filter the rest of them by selecting every 15th observation for analysis for reducing any potential autocorrelation within chains. We confirm the convergence of MH samples through graphical examination. Figure A2 and Figure A3 show the trace plots for the MH sequence of values for $\alpha$ and $\beta$, varying in relation to the implemented priors ${\pi }_J$ and ${\pi }_{MDI}$ as well as the used dataset of records. The trace plots derived from these figures demonstrate that, for both parameters, the values of $\alpha$ and $\beta$ are randomly distributed around the average. Furthermore, it is evident from the MH sequence histograms for $\alpha$ and $\beta$ in Figure A2 and Figure A3 that selecting the truncated normal distribution for the proposal distribution is a reasonable choice. The same conclusion follows with the usage of ${\pi }_{R2}$ and ${\pi }_{M2}$ priors.

Table 3. Summary of the Bayesian estimates for parameters $(\alpha .\beta )$.

	Prior	Parameter	Median	SD	95% HDI
Dataset I
	${\pi }_J$	$\alpha$	0.6191	0.9874	(0, 2.2369)
		$\beta$	7.2644	4.1427	(1.3615, 14.5149)
	${\pi }_{MDI}$	$\alpha$	2.3136	1.6505	(0, 5.1734)
		$\beta$	3.0662	3.5271	(0.0473, 9.9768)
	${\pi }_{R2}$	$\alpha$	0.4777	1.0322	(0, 1.8967)
		$\beta$	8.2162	4.7862	(2.1967, 15.9144)
	${\pi }_{M2}$	$\alpha$	0.9855	0.9589	(0.0002, 2.9002)
		$\beta$	6.1871	3.7258	(0.627, 13.4511)
Dataset II
	${\pi }_J$	$\alpha$	0.0235	0.0368	(0, 0.0823)
		$\beta$	0.1041	0.1652	(0.0062, 0.2482)
	${\pi }_{MDI}$	$\alpha$	0.0348	0.0565	(0, 0.1319)
		$\beta$	0.0905	0.1838	(0.0001, 0.3362)
	${\pi }_{R2}$	$\alpha$	0.1967	0.0318	(0, 0.0731)
		$\beta$	0.1193	0.1537	(0.0124, 0.2693)
	${\pi }_{M2}$	$\alpha$	0.0369	0.0518	(0, 0.1097)
		$\beta$	0.0829	0.1835	(0.0009, 0.2171)

lists the Bayesian estimators for $\alpha$ and $\beta$ together with the appropriate standard deviation (SD) and 95% highest density posterior (HDI) coverage interval [58]. Based on the SD and the width of the HDIs, the result shows that the Bayesian estimates of $\alpha$ are slightly more precise than in the case of parameter $\beta$ for all priors. In the case of ${\pi }_{M2}$, Bayesian estimates produce the most accurate results. In addition, the performance results of the four Bayesian estimates are near to one other, although the prior ${\pi }_J$ performs slightly better than others in terms of the width of coverage intervals. This is as expected since ${\pi }_J$ is also a probability matching prior. Overall, we may choose the ${\pi }_J$ prior as the optimal choice.

The results obtained in this study bridge the gap between theoretical development and practical implementation. For instance, the use of Jeffreys’ prior consistently demonstrates superior performance in small sample settings, a common scenario in fields like epidemiology and finance. The ability to generate credible intervals and high posterior density regions underpins applications where uncertainty quantification is critical. Such outcomes reinforce the utility of Bayesian methods for operational risk assessment, weather forecasting, and resource optimization in engineering and public health. The MDI prior may be suitable for exploratory analyses where the primary goal is to leverage inherent data structure. Reference priors are advantageous in computationally constrained settings, offering simplicity and robustness. Practitioners should consider computational feasibility and application-specific requirements, such as the need for rapid decision-making or detailed uncertainty quantification.

6. Conclusions

This paper addresses key challenges in Bayesian inference for the Gompertz model under kth record data, contributing to both theoretical and practical aspects. By developing objective priors, establishing the propriety of posteriors, and employing computationally efficient methods like the MH algorithm, it strengthens the Bayesian framework while ensuring practical applicability. Extensive simulations validate the proposed approach, demonstrating robustness in terms of MSEs and CPs.

A significant contribution of this study is extending the application of objective priors to record-based data, a relatively unexplored area. The finding that neither the MDI prior nor the reference prior ${\pi }_{R2}$ belongs to the class of probability-matching priors provides valuable insights into prior selection. These results have practical implications, particularly in decision-making contexts involving kth record values.

Another key aspect addressed in this study is the trade-off between the estimator performance and computational cost. While Bayesian methods utilizing priors such as Jeffreys’ prior offer high accuracy, they can also be computationally demanding, especially in resource-limited settings. Practitioners must balance these factors based on specific application needs, such as emergency response or resource allocation during crises.

Furthermore, this research highlights the relevance of Bayesian methods in policy development under extreme conditions, such as pandemics or financial disruptions. The ability to obtain robust estimates from limited and skewed data enables informed decision-making in such high-stakes scenarios. Future work could explore refinements to sampling techniques or the development of alternative priors tailored to specific datasets, further advancing Bayesian inference for complex real-world challenges.

Despite its contributions, this study faces several limitations, primarily due to small sample sizes and the inherent properties of record values.

One major challenge is the reliance on small sample sizes. Since record values represent extreme observations, the dataset is naturally reduced, leading to higher variability in parameter estimates. This limitation affects precision, making it difficult to generalize findings or draw robust inferences, as estimates may be biased with wider credible intervals.

Another limitation arises from the high degree of skewness and asymmetry commonly observed in record value distributions. Such skewness complicates inference, making it challenging to apply traditional statistical techniques. Even Bayesian methods may struggle with convergence issues or sensitivity to prior selection when faced with extreme skewness.

Additionally, the computational complexity of the proposed framework can be a constraint, especially with small and highly skewed samples. Iterative algorithms, such as the MCMC methods employed in this study, may encounter difficulties like non-convergence or dependence on initial conditions, further complicating estimation.

Lastly, the accuracy of posterior estimates depends on the choice of prior distributions. While objective priors provide a theoretically sound basis, their suitability for highly skewed and asymmetric datasets remains an open question. Conducting sensitivity analyses is crucial to ensuring the robustness of conclusions drawn from the Bayesian approach.

While this study advances Bayesian inference for the Gompertz model under kth records, several avenues for future research remain.

One promising direction is extending the Bayesian framework to accommodate multivariate or dependent record data. This could be particularly useful in reliability analysis and epidemiological modeling, where relationships between multiple variables play a crucial role.

Another area for exploration is the integration of alternative prior distributions, especially those informed by domain-specific knowledge. The use of hierarchical or empirical priors may enhance flexibility and improve the accuracy of Bayesian estimates, making them more adaptable to diverse datasets.

Developing more computationally efficient algorithms is also a key area for future research. Tailoring Bayesian inference methods to specific priors or data structures could improve scalability, particularly for applications involving large datasets or real-time decision-making.

Finally, there is potential to apply these Bayesian methods in emerging fields such as machine learning and artificial intelligence. Optimizing predictive models that leverage record-based datasets could enhance decision-making processes across various domains.

By addressing these challenges and opportunities, future research can build upon the findings of this study, further refining Bayesian methodologies and expanding their real-world applications.