Yogesh Mani Tripathi

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.

On estimating parameters of a progressively censored lognormal distribution

We consider the problem of making statistical inference on unknown parameters of a lognormal distribution under the assumption that samples are progressively censored. The maximum likelihood estimates (MLEs) are obtained by using the expectation-maximization algorithm. The observed and expected Fisher information matrices are provided as well. Approximate MLEs of unknown parameters are also obtained. Bayes and generalized estimates are derived under squared error loss function. We compute these estimates using Lindleys method as well as importance sampling method. Highest posterior density interval and asymptotic interval estimates are constructed for unknown parameters. A simulation study is conducted to compare proposed estimates. Further, a data set is analysed for illustrative purposes. Finally, optimal progressive censoring plans are discussed under different optimality criteria and results are presented.

Estimation using hybrid censored data from a two-parameter distribution with bathtub shape

The problem of estimating unknown parameters of a two-parameter distribution with bathtub shape is considered under the assumption that samples are hybrid censored. The maximum likelihood estimates are obtained using an EM algorithm. The Fisher information matrix is obtained as well and the asymptotic confidence intervals are constructed. Further, two bootstrap interval estimates are also proposed for the unknown parameters. Bayes estimates are evaluated under squared error loss function. Approximate explicit expressions for these estimates are derived using the Lindley method as well as using the Tierney and Kadane method. An importance sampling scheme is then proposed to generate Markov Chain Monte Carlo samples which have been used to compute approximate Bayes estimates and credible intervals for the unknowns. A numerical study is performed to compare the proposed estimates. Finally, two data sets are analyzed for illustrative purposes.

Bayesian Estimation and Prediction for a Hybrid Censored Lognormal Distribution

For a lognormal distribution, in Bayesian framework, we consider estimation of unknown parameters and prediction of future observable when it is known that samples are hybrid censored. We derive Bayes estimates with respect to the squared error loss function under both informative and non-informative prior situations. These estimates are then computed using Lindley method, importance sampling, and OpenBUGS software. For comparison purposes, we also obtain maximum-likelihood estimates using expectation-maximization (EM) algorithm, and compute Fisher information matrix as well. Predictive estimates of future observable are obtained under the setup of one- and two-sample prediction problems. We further compute equal-tail and highest posterior density predictive intervals, the corresponding average interval length and coverage percentage. Proposed methods of estimation and prediction are compared numerically, and comments are made based on a simulation study. Two real data sets are also analyzed for illustration purposes.

Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring

In this paper, we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution under type II progressive censored samples. Estimation of reliability and hazard functions is also considered. Maximum likelihood estimators are obtained using the Expectation–Maximization (EM) algorithm. Further, we obtain expected Fisher information matrix using the missing value principle. Bayes estimators are derived under squared error and linex loss functions. We have used Lindley, and Tiernery and Kadane methods to compute these estimates. In addition, Bayes estimators are computed using importance sampling scheme as well. Samples generated from this scheme are further utilized for constructing highest posterior density intervals for unknown parameters. For comparison purposes asymptotic intervals are also obtained. A numerical comparison is made between proposed estimators using simulations and observations are given. A real-life data set is analyzed for illustrative purposes.

Parameter and reliability estimation for an exponentiated half-logistic distribution under progressive type II censoring

In this article, we deal with a two-parameter exponentiated half-logistic distribution. We consider the estimation of unknown parameters, the associated reliability function and the hazard rate function under progressive Type II censoring. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. Bayes estimates are derived with respect to squared error, linex and entropy loss functions. Approximate explicit expressions for all Bayes estimates are obtained using the Lindley method. We also use importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo samples are further used to produce credible intervals for the unknown parameters. Asymptotic confidence intervals are constructed using the normality property of the MLEs. For comparison purposes, bootstrap-p and bootstrap-t confidence intervals are also constructed. A comprehensive numerical study is performed to compare the proposed estimates. Finally, a real-life data set is analysed to illustrate the proposed methods of estimation.

A Subset Selection Procedure for Selecting the Exponential Population Having the Longest Mean Lifetime When the Guarantee Times are the Same

ABSTRACT Consider k(≥ 2) independent exponential populations Π1, Π2, …, Π k , having the common unknown location parameter μ ∈ (−∞, ∞) (also called the guarantee time) and unknown scale parameters σ1, σ2, …σ k , respectively (also called the remaining mean lifetimes after the completion of guarantee times), σ i > 0, i = 1, 2, …, k. Assume that the correct ordering between σ1, σ2, …, σ k is not known apriori and let σ[i], i = 1, 2, …, k, denote the ith smallest of σ j s, so that σ[1] ≤ σ[2] ··· ≤ σ[k]. Then Θ i = μ + σ i is the mean lifetime of Π i , i = 1, 2, …, k. Let Θ[1] ≤ Θ[2] ··· ≤ Θ[k] denote the ranked values of the Θ j s, so that Θ[i] = μ + σ[i], i = 1, 2, …, k, and let Π(i) denote the unknown population associated with the ith smallest mean lifetime Θ[i] = μ + σ[i], i = 1, 2, …, k. Based on independent random samples from the k populations, we propose a selection procedure for the goal of selecting the population having the longest mean lifetime Θ[k] (called the “best” population), under the subset selection formulation. Tables for the implementation of the proposed selection procedure are provided. It is established that the proposed subset selection procedure is monotone for a general k (≥ 2). For k = 2, we consider the loss measured by the size of the selected subset and establish that the proposed subset selection procedure is minimax among selection procedures that satisfy a certain probability requirement (called the P*-condition) for the inclusion of the best population in the selected subset.

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

ABSTRACT We consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.

James–Stein type estimators for ordered normal means

Suppose independent observations X 1, X 2, …, X k are available from k (≥2) normal populations having means θ1, θ2, …, θ k , respectively, and common variance unity. The means are known to be ordered, that is, θ1≤θ2≤···≤θ k . Two commonly used estimators for simultaneous estimation of θ=(θ1, θ2, …, θ k ) are δ MLE, the order restricted maximum likelihood estimator (MLE), and δ p, the generalized Bayes estimator of θ with respect to the uniform prior on the restricted space Ω={θ∈R k : θ1≤θ2≤···≤θ k }, where R k denotes the k-dimensional Euclidean space. Both δ MLE and δ p improve the usual unrestricted MLE X=(X 1, X 2, …, X k ) and are minimax. But δ MLE is inadmissible for k≥2 and δ p is inadmissible for k≥3. However, no dominating estimators are yet known. Using Browns [Brown, L.D., 1979, A heuristic method for determining admissibility of estimators—with applications. Annals of Statistics, 7, 960–994] heuristic approach for proving admissibility or inadmissibility of estimators, we propose some classes of James–Stein type estimators and show, through a simulation study, that many of these estimators dominate δ p and δ MLE.

Estimation using hybrid censored data from a generalized inverted exponential distribution

ABSTRACT We consider point and interval estimation of the unknown parameters of a generalized inverted exponential distribution in the presence of hybrid censoring. The maximum likelihood estimates are obtained using EM algorithm. We then compute Fisher information matrix using the missing value principle. Bayes estimates are derived under squared error and general entropy loss functions. Furthermore, approximate Bayes estimates are obtained using Tierney and Kadane method as well as using importance sampling approach. Asymptotic and highest posterior density intervals are also constructed. Proposed estimates are compared numerically using Monte Carlo simulations and a real data set is analyzed for illustrative purposes.