Bayes Estimates Of Unknown Parameters Research Articles

Estimation of some lifetime parameter of the unit half logistic-geometry distribution under progressively type-II censored data

This research examines the estimation of the parameter for the unit-half logistic-geometry distribution (UHLGD) using progressively type-II right censored samples. The unknown parameter of UHLGD under progressively type II right censoring is estimated using maximum likelihood estimations (MLEs), maximum product space estimators, and Bayesian estimators. The confidence intervals (ACIs) are computed. Additionally, two bootstrap CIs (bootstrap-p and bootstrap-t) are recommended. For symmetric loss functions such as squared error loss (SEL), Bayesian approximations are developed. The Metropolis-Hastings samplers approach is used to get Bayes estimates of unknown parameters and associated credible intervals (CRIs) using the Markov chain Monte Carlo (MCMC) method. A set of real data that measures the tensile strength of polyester fibres is considered an application of the provided techniques.

Statistical Inference for Competing Risks Model with Adaptive Progressively Type-II Censored Gompertz Life Data Using Industrial and Medical Applications

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches.

Statistical Inference to the Parameter of the Inverse Power Ishita Distribution under Progressive Type‐II Censored Data with Application to COVID‐19 Data

The goal of the article is the inference about the parameters of the inverse power ishita distribution (IPID) using progressively type‐II censored (Prog–II–C) samples. For IPID parameters, maximum likelihood and Bayesian estimates were obtained. Two bootstrap “confidence intervals” (CIs) are also proposed in addition to “approximate confidence intervals” (ACIs). In addition, Bayesian estimates for “squared error loss” (SEL) and LINEX loss functions are provided. The Gibbs within Metropolis–Hasting samplers process is used to provide Bayes estimators of unknown parameters also “credible intervals” (CRIs) of them by using the “Markov Chain Monte Carlo” (MCMC) technique. Then, an application of the suggested approaches is considered a set of real‐life data this data set COVID‐19 data from France of 51 days recorded from 1 January to 20 February 2021 formed of mortality rate. To evaluate the quality of the proposed estimators, a simulation study is conducted.

Bayesian and Frequentist Inferences on a Type I Half-Logistic Odd Weibull Generator with Applications in Engineering.

In this article, we have proposed a new generalization of the odd Weibull-G family by consolidating two notable families of distributions. We have derived various mathematical properties of the proposed family, including quantile function, skewness, kurtosis, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, probability weighted moments, moments of (reversed) residual lifetime, entropy and order statistics. After producing the general class, two of the corresponding parametric statistical models are outlined. The hazard rate function of the sub-models can take a variety of shapes such as increasing, decreasing, unimodal, and Bathtub shaped, for different values of the parameters. Furthermore, the sub-models of the introduced family are also capable of modelling symmetric and skewed data. The parameter estimation of the special models are discussed by numerous methods, namely, the maximum likelihood, simple least squares, weighted least squares, Cramér-von Mises, and Bayesian estimation. Under the Bayesian framework, we have used informative and non-informative priors to obtain Bayes estimates of unknown parameters with the squared error and generalized entropy loss functions. An extensive Monte Carlo simulation is conducted to assess the effectiveness of these estimation techniques. The applicability of two sub-models of the proposed family is illustrated by means of two real data sets.

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

AbstractRecently, the two‐parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power‐trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.

Bayesian Estimation of Stress Strength Reliability from Inverse Chen Distribution with Application on Failure Time Data

In this article, we develop Bayesian estimation procedure for estimating the stress strength reliability R = P [X > Y ] when X (strength) and Y (stress) are the inverse Chen random variables. First, we study some statistical properties of the inverse Chen distribution such as quantiles, mode, stochastic ordering, entropy measure, order statistics and stress strength reliability. Then, we estimate the stress strength parameters and R using maximum likelihood and Bayesian estimations. A symmetric (squared error loss) and an asymmetric (entropy loss) loss functions are considered for Bayesian estimation under the assumption of gamma prior. Since, joint posterior distribution of the model parameters and R involve multiple integrations and have complex form. So, we do not get analytical solution without using any numerical techniques. Therefore, we propose to use Lindley’s approximation and Markov chain Monte Carlo techniques for Bayesian computation. A simulation study is carried out for the proposed Bayes estimators of unknown parameters and compared with the maximum likelihood estimator on the basis of mean squared error. Finally, an empirical illustration based on failure time data is presented to demonstrate the applicability of inverse Chen stress strength model.

Estimation and Prediction for a Progressively First-Failure Censored Inverted Exponentiated Rayleigh Distribution

We discuss inverted exponentiated Rayleigh distribution under progressive first-failure censoring. Maximum likelihood and Bayes estimates of unknown parameters are obtained. An expectation–maximization algorithm is used for computing maximum likelihood estimates. Asymptotic intervals are constructed from the observed Fisher information matrix. Bayes estimates of unknown parameters are obtained under the squared error loss function. We construct highest posterior density intervals based on importance sampling. Different predictors and prediction intervals of censored observations are discussed. A Monte Carlo simulations study is performed to compare different methods. Finally, three real data sets are analyzed for illustration purposes.

On progressively censored inverted exponentiated Rayleigh distribution

ABSTRACTIn this paper, we discuss a progressively censored inverted exponentiated Rayleigh distribution. Estimation of unknown parameters is considered under progressive censoring using maximum likelihood and Bayesian approaches. Bayes estimators of unknown parameters are derived with respect to different symmetric and asymmetric loss functions using gamma prior distributions. An importance sampling procedure is taken into consideration for deriving these estimates. Further highest posterior density intervals for unknown parameters are constructed and for comparison purposes bootstrap intervals are also obtained. Prediction of future observations is studied in one- and two-sample situations from classical and Bayesian viewpoint. We further establish optimum censoring schemes using Bayesian approach. Finally, we conduct a simulation study to compare the performance of proposed methods and analyse two real data sets for illustration purposes.

Robust Bayesian estimation of a bathtub-shaped distribution under progressive Type-II censoring

ABSTRACTThe bathtub-shaped failure rate function has been used for modeling the life spans of a number of electronic and mechanical products, as well as for modeling the life spans of humans, especially when some of the data are censored. This article addresses robust methods for the estimation of unknown parameters in a two-parameter distribution with a bathtub-shaped failure rate function based on progressive Type-II censored samples. Here, a class of flexible priors is considered by using the hierarchical structure of a conjugate prior distribution, and corresponding posterior distributions are obtained in a closed-form. Then, based on the square error loss function, Bayes estimators of unknown parameters are derived, which depend on hyperparameters as parameters of the conjugate prior. In order to eliminate the hyperparameters, hierarchical Bayesian estimation methods are proposed, and these proposed estimators are compared to one another based on the mean squared error, through Monte Carlo simulations for various progressively Type-II censoring schemes. Finally, a real dataset is presented for the purpose of illustration.

An Analysis of Record Statistics based on an Exponentiated Gumbel Model

This paper develops a maximum profile likelihood estimator of unknown parameters of the exponentiated Gumbel distribution based on upper record values. We propose an approximate maximum profile likelihood estimator for a scale parameter. In addition, we derive Bayes estimators of unknown parameters of the exponentiated Gumbel distribution using Lindley's approximation under symmetric and asymmetric loss functions. We assess the validity of the proposed method by using real data and compare these estimators based on estimated risk through a Monte Carlo simulation.