Inference for Simple Step Stress Accelerated Life Test Model Under Progressively Censored Gompertz Data

This study introduces a robust Bayesian framework for simple-step stress accelerated life testing. For modeling lifetime characteristics, this study adopts the Xgamma distribution as the principal life model. Bayesian parameter estimation is conducted using the Markov Chain Monte Carlo method under three distinct loss functions: the squared error loss function, generalized entropy loss function, and linear exponential loss function. The resulting Bayesian estimators are analyzed and compared with those obtained through the classical maximum likelihood estimation method. Finally, the proposed methods are illustrated with the analysis of two real data sets.

Reliability analysis is the key to evaluate software’s quality. Since the early 1970s, the Power Law Process, among others, has been used to assess the rate of change of software reliability as time-varying function by using its intensity function. The Bayesian analysis applicability to the Power Law Process is justified using real software failure times. The choice of a loss function is an important entity of the Bayesian settings. The analytical estimate of likelihood-based Bayesian reliability estimates of the Power Law Process under the squared error and Higgins-Tsokos loss functions were obtained for different prior knowledge of its key parameter. As a result of a simulation analysis and using real data, the Bayesian reliability estimate under the Higgins-Tsokos loss function not only is robust as the Bayesian reliability estimate under the squared error loss function but also performed better, where both are superior to the maximum likelihood reliability estimate. A sensitivity analysis resulted in the Bayesian estimate of the reliability function being sensitive to the prior, whether parametric or non-parametric, and to the loss function. An interactive user interface application was additionally developed using Wolfram language to compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given data.

This article investigates the estimation of the parameters for power hazard function distribution and some lifetime indices such as reliability function, hazard rate function, and coefficient of variation based on adaptive Type-II progressive censoring. From the perspective of frequentism, we derive the point estimations through the method of maximum likelihood estimation. Besides, delta method is implemented to construct the variances of the reliability characteristics. Markov chain Monte Carlo techniques are proposed to construct the Bayes estimates. To this end, the results of the Bayes estimates are obtained under squared error and linear exponential loss functions. Also, the corresponding credible intervals are constructed. A simulation study is utilized to assay the performance of the proposed methods. Finally, a real data set of COVID-19 mortality rate is analyzed to validate the introduced inference methods.

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

In this paper, we make a statistical inference for the Gompertz distribution based on multiply Type-I censoreddata, where we determine a censoring time point for each unit in the life time test. Estimation of the parameters is obtainedusing maximum likelihood method and Bayesian method, also the asymptotic con dence intervals for the parameters areconstructed based on the approximate of the Fisher information matrix. The necessary condition for existence and uniquenessof the maximum likelihood estimators is discussed. The Bayesian estimates are obtained depending on the squared error lossfunction, linear exponential loss function and the generalized entropy loss function. The One-sample Bayesian predictionintervals are constructed for the unobserved lifetimes in the same sample. A real data example is presented to illustrate themethods of inference developed here. Finally, the simulation study is executed to compare the performance of the proposedmethods.

Accelerated life testing (ALT) is commonly used to obtain reliability information about a product in a timely manner. Several stress loading designs have been proposed and recent research interests have emerged concerning the development of equivalent ALT plans. Step‐stress ALT (SSALT) is one of the most commonly used stress loadings because it usually shortens the test duration and reduces the number of required test units. This article considers two fundamental questions when designing a SSALT and provides formal proofs in answer to each. Namely:(1)can a simple SSALT be designed so that it is equivalent to other stress loading designs?(2)when optimizing a multilevel SSALT, does it degenerate to a simple SSALT plan? The answers to both queries, under certain reasonable model assumptions, are shown to be a qualified YES. In addition, we provide an argument to support the rationale of a common practice in designing a SSALT, that is, setting the higher stress level as high as possible in a SSALT plan. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

In data-processing standpoint, an efficient algorithm for identifying the minimum value among a set of measurements are record statistics. From a sequence of n independent identically distributed continuous random variables only about log(n) records are expected, so we expect to have little data, hence any prior information is welcome (Houchens, Record value theory and inference, Ph.D. thesis, University of California, Riverside, 1984). In this paper, non-Bayesian and Bayesian estimates are derived for the two parameters of the Exponential distribution based on record statistics with respect to the squared error and Linear-Exponential loss functions and then compared with together. The admissibility of some estimators is discussed.

The Bayesian approach offers a flexible, interpretable and powerful framework for statistical analysis, making it a valuable tool to help in making optimal decisions under uncertainty. It incorporates prior knowledge or beliefs about the parameters, which can lead to more accurate and informative results. Also, it offers credible intervals as a measure of uncertainty, which are often more interpretable than confidence intervals. Hence, the Bayesian approach is utilized to estimate the parameters, reliability function, hazard rate function and reversed hazard rate function of a new competing risks model. A squared error loss function as a symmetric loss function and a linear exponential loss function as an asymmetric loss function are employed to derive the Bayesian estimators. Credible intervals of the parameters, reliability function, hazard rate function and reversed hazard rate function are obtained. Predicting future observations is important in many fields, from finance and weather forecasting to healthcare and engineering. Thus, two-sample prediction (as a special case of the multi-sample prediction) for future observation is considered. An adaptive Metropolis algorithm is applied to conduct a simulation study to evaluate the performance of the Bayes estimates and predictors. Moreover, two applications of medical and engineering data sets are used to test and validate the theoretical results, ensuring that they are accurate, applicable to real-world scenarios and contribute to the understanding of the world and inform decision-making.

In this article, the concept of asymptotic pointwise optimality in a single continuous-time process provided by Hwang (2001) is extended to more than one continuous-time process. We propose an asymptotically pointwise optimal (APO) rule, which consists of a sequential allocation procedure and a stopping time, and derive some properties of asymptotic optimality of the rule. In particular, some APO rules are proposed in the Bayes sequential estimation problem for several Poisson processes under both squared error loss and linear exponential loss functions. They are shown to be asymptotically optimal for arbitrary priors and asymptotically nondeficient for conjugate priors.

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.

In this research, Log logistic distribution under Bayesian paradigm is studied. Posterior distribution has been derived using Uniform and Jeffrey’s’ priors. The three loss functions taken up are squared error loss function (SELF), weighted balanced loss function (WBLF) and precautionary loss function (PLF). The performance of an estimator is assessed on the basis of its relative posterior risk. An extensive simulation was carried out to illustrate the applicability of this research and to compare the performance of different estimators. The study indicated that for estimation of the shape parameter using Bayesian estimation technique, the precautionary loss function can effectively be employed. Keywords : Bayesian estimation, Log logistic distribution, Uniform prior, Jeffrey’s prior, loss functions.

Stress-strength models are of special importance in reliability literature and engineering applications. This paper deals with the estimation problem of a stress-strength model incorporating multi-component system. The system is regarded as alive only if at least S out of k ( S < k ) strength components exceed the stress. The reliability of such system is obtained when strength and stress variables have Weibull distributions. Maximum likelihood estimator of R S , k and asymptotic confidence intervals are obtained based on upper record values. Bayesian estimator under squared error and linear exponential loss functions using gamma prior distributions and the corresponding credible intervals are obtained. Due to the lack of explicit forms for the Bayes estimates, the Markov Chain Monte Carlo (MCMC) method is employed. A simulation study is implemented to assess the performance of estimates. A real-life example is presented to show how the proposed model may be utilized in breaking strength of jute fibre data.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.