A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

Selecting the best one of several meifull populations

Consider k(k ≥ 2) two-parameter Meibull populations. Using type II censored data we want to select a best population. Me have proposed procedures which can be used for maximum likelihood estimators or simplified linear estimators of Che unknown parameters. The ranking of the populations is done by comparing their reliabilities at certain fixed time or by comparing their 2-eh quantiles. In selected cases, the constants needed for the procedures are tabulated using Monte Carlo methods.     

On testing hypotheses with divergence statistics

Using divergence measures based on entropy functions, a procedure to test statistical hypotheses is proposed. Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained. Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators are considered, so they can be used to construct confidence intervals and to test statistical hypotheses based on one or more samples. These results can also be applied to multinomial populations. Tests of goodness of fit and tests of homogeneity can be constructed.     

Classifying weibull populations with respect to a control

This paper is concerned with classifying k Weibull populations by their reliabilities with respect to a control population. He have proposed procedures which can be used for maximum likelihood estimators or simplified linear estimators of the parameters. Estimators are based on type II censored data. The cases considered include unknown shape parameters being equal or unequal. In selected cases, the constants needed for the procedures are tabulated using Monte Carlo methods.     

Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case

In this article, we propose some procedures to get confidence intervals for the reliability in stress-strength models. The confidence intervals are obtained either through a parametric bootstrap procedure or using asymptotic results, and are applied to the particular context of two independent normal random variables. The performance of these estimators and other known approximate estimators are empirically checked through a simulation study which considers several scenarios.     

Hypothesis testing for quantiles of two-parameter binary models

A procedure is developed to test the equality of the quantiles from k populations, assuming the responses follow a two-parameter binary model.The method utilizes the asymptotic distribution of the maximum likelihood estimators.The exact distribution of the test statistic is discussed in general.This exact distribution is generated for the logit model in order to investgate the convergence properties of the asymptotic procedure.     

Generalized inverted exponential distribution under progressive first-failure censoring

This article deals with progressive first-failure censoring, which is a generalization of progressive censoring. We derive maximum likelihood estimators of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using progressive first-failure censored samples. The asymptotic confidence intervals and coverage probabilities for the parameters are obtained based on the observed Fisher's information matrix. Bayes estimators of the parameters and reliability characteristics under squared error loss function are obtained using the Lindley approximation and importance sampling methods. Also, highest posterior density credible intervals for the parameters are computed using importance sampling procedure. A Monte Carlo simulation study is conducted to analyse the performance of the estimators derived in the article. A real data set is discussed for illustration purposes. Finally, an optimal censoring scheme has been suggested using different optimality criteria.     

Inference on the Kumaraswamy distribution

Many lifetime distribution models have successfully served as population models for risk analysis and reliability mechanisms. The Kumaraswamy distribution is one of these distributions which is particularly useful to many natural phenomena whose outcomes have lower and upper bounds or bounded outcomes in the biomedical and epidemiological research. This article studies point estimation and interval estimation for the Kumaraswamy distribution. The inverse estimators (IEs) for the parameters of the Kumaraswamy distribution are derived. Numerical comparisons with maximum likelihood estimation and biased-corrected methods clearly indicate the proposed IEs are promising. Confidence intervals for the parameters and reliability characteristics of interest are constructed using pivotal or generalized pivotal quantities. Then, the results are extended to the stress–strength model involving two Kumaraswamy populations with different parameter values. Construction of confidence intervals for the stress–strength reliability is derived. Extensive simulations are used to demonstrate the performance of confidence intervals constructed using generalized pivotal quantities.     

Statistical inference for two exponential populations under joint progressive type-I censored scheme

In this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.     

Estimation of generalized exponential distribution based on an adaptive progressively type-II censored sample

In this paper, based on an adaptive Type-II progressively censored sample from the generalized exponential distribution, the maximum likelihood and Bayesian estimators are derived for the unknown parameters as well as the reliability and hazard functions. Also, the approximate confidence intervals of the unknown parameters, and the reliability and hazard functions are calculated. Markov chain Monte Carlo method is applied to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. Moreover, results from simulation studies assessing the performance of our proposed method are included. Finally, an illustrative example using real data set is presented for illustrating all the inferential procedures developed here.     

Reliability estimation in multicomponent stress–strength model for Topp-Leone distribution

In this paper, we consider the estimation reliability in multicomponent stress-strength (MSS) model when both the stress and strengths are drawn from Topp-Leone (TL) distribution. The maximum likelihood (ML) and Bayesian methods are used in the estimation procedure. Bayesian estimates are obtained by using Lindley’s approximation and Gibbs sampling methods, since they cannot be obtained in explicit form in the context of TL. The asymptotic confidence intervals are constructed based on the ML estimators. The Bayesian credible intervals are also constructed using Gibbs sampling. The reliability estimates are compared via an extensive Monte-Carlo simulation study. Finally, a real data set is analysed for illustrative purposes.     

Reliability of stress–strength model for exponentiated Pareto distributions

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical exponentiated Pareto distributed random variables. Different point estimations and interval estimations are proposed. The point estimators obtained are maximum likelihood, uniformly minimum variance unbiased and Bayesian estimators. The interval estimations obtained are approximate, exact, bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval. Different methods and the corresponding confidence intervals are compared using Monte-carlo simulations.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Exact Likelihood Inference for Two Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of two competing products with regard to their reliability. In this article, we consider two exponential populations and when joint progressive Type-II censoring is implemented on the two samples. We then derive the moment generating functions and the exact distributions of the maximum likelihood estimators (MLEs) of the mean lifetimes of the two exponential populations under such a joint progressive Type-II censoring. We then discuss the exact lower confidence bounds, exact confidence intervals, and simultaneous confidence regions. Next, we discuss the corresponding approximate results based on the asymptotic normality of the MLEs as well as those based on the Bayesian method. All these confidence intervals and regions are then compared by means of Monte Carlo simulations with those obtained from bootstrap methods. Finally, an illustrative example is presented in order to illustrate all the methods of inference discussed here.     

Inference for Weibull distribution based on progressively Type-II hybrid censored data

Progressive Type-II hybrid censoring is a mixture of progressive Type-II and hybrid censoring schemes. In this paper, we discuss the statistical inference on Weibull parameters when the observed data are progressively Type-II hybrid censored. We derive the maximum likelihood estimators (MLEs) and the approximate maximum likelihood estimators (AMLEs) of the Weibull parameters. We then use the asymptotic distributions of the maximum likelihood estimators to construct approximate confidence intervals. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are obtained under suitable priors on the unknown parameters and also by using the Gibbs sampling procedure. Monte Carlo simulations are then performed for comparing the confidence intervals based on all those different methods. Finally, one data set is analyzed for illustrative purposes.     

Stress-strength reliability of exponentiated Fréchet distributions based on Type-II censored data

In this paper, we consider inference of the stress-strength parameter, R, based on two independent Type-II censored samples from exponentiated Fréchet populations with different index parameters. The maximum likelihood and uniformly minimum variance unbiased estimators, exact and asymptotic confidence intervals and hypotheses testing for R are obtained. We conduct a Monte Carlo simulation study to evaluate the performance of these estimators and confidence intervals. Finally, two real data sets are analysed for illustrative purposes.     

Conditional Maximum Likelihood and Interval Estimation for Two Weibull Populations under Joint Type-II Progressive Censoring

Comparative lifetime experiments are important when the object of a study is to determine the relative merits of two competing duration of life products. This study considers the interval estimation for two Weibull populations when joint Type-II progressive censoring is implemented. We obtain the conditional maximum likelihood estimators of the two Weibull parameters under this scheme. Moreover, simultaneous approximate confidence region based on the asymptotic normality of the maximum likelihood estimators are also discussed and compared with two Bootstrap confidence regions. We consider the behavior of probability of failure structure with different schemes. A simulation study is performed and an illustrative example is also given.     

Interval estimators for reliability: the bivariate normal case

This paper proposes procedures to provide confidence intervals (CIs) for reliability in stress–strength models, considering the particular case of a bivariate normal set-up. The suggested CIs are obtained by employing either asymptotic variances of maximum-likelihood estimators or a bootstrap procedure. The coverage and the accuracy of these intervals are empirically checked through a simulation study and compared with those of another proposal in the literature. An application to real data is provided.     

ASYMPTOTIC PROPERTIES FOR MAXIMUM LIKELIHOOD ESTIMATORS FOR RELIABILITY AND FAILURE RATES OF MARKOV CHAINS

ABSTRACT

In this paper, we shall study a homogeneous ergodic, finite state, Markov chain with unknown transition probability matrix. Starting from the well known maximum likelihood estimator of transition probability matrix, we define estimators of reliability and its measurements. Our aim is to show that these estimators are uniformly strongly consistent and converge in distribution to normal random variables. The construction of the confidence intervals for availability, reliability, and failure rates are also given. Finally we shall give a numerical example for illustration and comparing our results with the usual empirical estimator results.     

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.