Some Estimation Methods for the Shape Parameter and Reliability Function of Burr Type XII Distribution / Comparison Study

Burr type XII distribution plays an important role in reliability modeling, risk analyzing and process capability estimation. The choice of the best estimation method is one of the goals in estimating parameters of the distribution. The main aim of this paper is to obtain and compare the classical "maximum likelihood and uniformly minimum variance unbiased" estimators and the Bayesian estimators of the shape parameter, ???? and reliability function based on a complete sample when the other shape parameter, ? known. The Bayes estimators are obtained under non-informative priors "Jeffrey’s prior, modified and extension of Jeffrey’s prior" as well as under informative gamma prior based on different symmetric and asymmetric loss functions "squared error, quadratic, LINEX, precautionary and entropy". The Monte Carlo experiment was performed under a wide range of cases and sample size. The estimates of the unknown shape parameter were compared by employing the mean square errors and the estimates of reliability function were compared by employing the integrated mean squared error.   Keywords: Burr type XII distribution; Maximum likelihood estimator; Uniformly Minimum Variance Unbiased estimator; Bayes estimators; non-informative Prior; informative Prior; Squared error loss function; quadratic loss function; LINEX loss function; Precautionary loss function; Entropy Loss function; Mean squared error; integrated mean squared error

Non-Bayes, Bayes and Empirical Bayes Estimators for the Shape Parameter of Lomax Distribution

Point estimation is one of the core topics in mathematical statistics. In this paper we consider the most common methods of point estimation: non-Bayes, Bayes and empirical Bayes methods to estimate the shape parameter of Lomax distribution based on complete data. The maximum likelihood, moment and uniformly minimum variance unbiased estimators are obtained as non-Bayes estimators. Bayes and empirical Bayes estimators are obtained corresponding to three informative priors "gamma, chi-square and inverted Levy" based on symmetric "squared error" and asymmetric "LINEX and general entropy" loss functions. The estimates of the shape parameter were compared empirically via Monte Carlo simulation study based upon the mean squared error. Among the set of conclusions that have been reached, it is observed that, for all sample sizes and different cases, the performance of uniformly minimum variance unbiased estimator is better than other non-Bayes estimators. Further that, Monte Carlo simulation results indicate that the performance of Bayes and empirical Bayes estimator in some cases are better than non-Bayes for some appropriate of prior distribution, loss function, values of parameters and sample size. Keywords: Lomax distribution; maximum likelihood estimator; moment estimator; uniformly minimum variance unbiased estimator; Bayes estimator; empirical Bayes estimator; informative prior; squared error loss function; LINEX  loss  function;  general  entropy  loss  function;  mean  squared  error

Statistical inferences of Rs;k = Pr(Xk-s+1:k \u3e Y ) for general class of exponentiated inverted exponential distribution with progressively type-II censored samples with uniformly distributed random removal

The problem of statistical inference of the reliability parameter Pr(Xk-s+1:k \u3e Y ) of an s-out-of-k : G system with strength components X1,X2,…,Xk subjected to a common stress Y when X and Y are independent two-parameter general class of exponentiated inverted exponential (GCEIE) progressively type-II right censored data with uniformly random removal random variables, are discussed. We use p-value as a basis for hypothesis testing. There are no exact or approximate inferential procedures for reliability of a multicomponent stress-strength model from the GCEIE based on the progressively type-II right censored data with random or fixed removals available in the literature. Simulation studies and real-world data analyses are given to illustrate the proposed procedures. The size of the test, adjusted and unadjusted power of the test, coverage probability and expected confidence lengths of the confidence interval, and biases of the estimator are also discussed

Analysis of Generalized Inverted Exponential Distribution under Adaptive Type-I Progressive Hybrid Censored Competing Risks Data

The estimation of the unknown parameters of generalized inverted exponential distribution under adaptive type-I progressive hybrid censored scheme (AT-I PHCS) with competing risks data will be discussed. The reason why AT-I PHCS has exceeded other failure censored types; Time censored types enable analysts to accomplish their trials and experiments in a shorter time and with higher efficiency. In this regards, we obtain the maximum likelihood estimation of the parameters and the asymptotic confidence intervals for the unknown parameters. Further, Bayes estimates of the parameters which obtained based on squared error and LINEX loss functions under the assumptions of independent gamma priors of the scale parameters. For Bayesian estimation, we take advantage of Markov Chain Monte Carlo techniques to derive Bayesian estimators and the credible intervals. Finally, two data sets with Monte Carlo simulation study and a real data set are analyzed for illustrative purposes

Bayesian Inference of δ = P(X < Y) for Burr Type XII distribution based on progressively first failure-censored samples

[[abstract]]Let X and Y follow two independent Burr type XII distributions and δ=P(X<Y) . If X is the stress that is applied to a certain component and Y is the strength to sustain the stress, then δ is called the stress–strength parameter. In this study, The Bayes estimator of δ is investigated based on a progressively first failure-censored sample. Because of computation complexity and no closed form for the estimator as well as posterior distributions, the Markov Chain Monte Carlo procedure using the Metropolis–Hastings algorithm via Gibbs sampling is built to collect a random sample of δ via the joint distribution of the progressively first failure-censored sample and random parameters and the empirical distribution of this collected sample is used to estimate the posterior distribution of δ . Then, the Bayes estimates of δ using the square error, absolute error, and linear exponential error loss functions are obtained and the credible interval of δ is constructed using the empirical distribution. An intensive simulation study is conducted to investigate the performance of these three types of Bayes estimates and the coverage probabilities and average lengths of the credible interval of δ . Moreover, the performance of the Bayes estimates is compared with the maximum likelihood estimates. The Internet of Things and a numerical example about the miles-to-failure of vehicle components for reliability evaluation are provided for application purposes.[[notice]]補正完

On estimating the reliability in a multicomponent system based on progressively-censored data from Chen distribution

This research deals with classical, Bayesian, and generalized estimation of stress-strength reliability parameter, Rs;k = Pr(at least s of (X1;X2; :::;Xk) exceed Y) = Pr(Xks+1:k \u3eY) of an s-out-of-k : G multicomponent system, based on progressively type-II right-censored samples with random removals when stress and strength are two independent Chen random variables. Under squared-error and LINEX loss functions, Bayes estimates are developed by using Lindley’s approximation and Markov Chain Monte Carlo method. Generalized estimates are developed using generalized variable method while classical estimates - the maximum likelihood estimators, their asymptotic distributions, asymptotic confidence intervals, bootstrap-based confidence intervals - are also developed. A simulation study and a real-world data analysis are provided to illustrate the proposed procedures. The size of the test, adjusted and unadjusted power of the test, coverage probability and expected lengths of the confidence intervals, and biases of the estimators are also computed, compared and contrasted

Shrinkage Estimation and Prediction for Joint Type-II Censored Data from Two Burr-XII Populations

The main objective of this paper is to apply linear and pretest shrinkage estimation techniques to estimating the parameters of two 2-parameter Burr-XII distributions. Further more, predictions for future observations are made using both classical and Bayesian methods within a joint type-II censoring scheme. The efficiency of shrinkage estimates is compared to maximum likelihood and Bayesian estimates obtained through the expectation-maximization algorithm and importance sampling method, as developed by Akbari Bargoshadi et al. (2023) in "Statistical inference under joint type-II censoring data from two Burr-XII populations" published in Communications in Statistics-Simulation and Computation". For Bayesian estimations, both informative and non-informative prior distributions are considered. Additionally, various loss functions including squared error, linear-exponential, and generalized entropy are taken into account. Approximate confidence, credible, and highest probability density intervals are calculated. To evaluate the performance of the estimation methods, a Monte Carlo simulation study is conducted. Additionally, two real datasets are utilized to illustrate the proposed methods.Comment: 33 pages and 33 table

Analysis of Two Generalized Exponential Populations Under Joint Type-I Progressive Hybrid Censoring Scheme

This paper discussed inference for two generalized exponential using the joint type-I progressively hybrid censoring (JPHC-I) scheme. It assumed that the lifetime distribution of the items from the two populations follow generalized exponential distribution. Based on the JPHC-I scheme, we first consider the maximum likelihood estimators of the unknown parameters along with thier asymptotic confidence intervals. Next, we provide the Bayesian inferences of the unknown parameters under the assumptions of independent gamma priors on the scale parameters using squared error (SE) and linear-exponential (LINEX) loss functions. Markov Chain Monte Carlo (MCMC) techniques is applied to carry out the Bayesian estimation procedure and in turn calculate the credible intervals. To evaluate the performance of the estimators, numerical example is carried out

Bayesian Inference for Concomitants based on Weibull Subfamily of Morgenstern Family Under Generalized Order Statistics

In this paper, for Weibull subfamily of Morgenstern family, the joint density of the concomitants of generalized order statistics (GOS's) is used to obtain the maximum likelihood estimates (MLE) and Bayes estimates for the distribution parameters. Applications of these results for concomitants of order statistics are presented