Estimating Parameters of Certain Lifetime Statistical Models Using Censored Samples

Abstract

In Chapter 1, we discuss the literature review on the related problems and give some motivation for studying the problem under consideration. Also, the summary of the results obtained in the thesis has been discussed there. In Chapter 2, we discuss some basic results from decision theoretic as well as classical points of view on estimation theory. In Chapter 3, the problem of estimating parameters of the generalized inverse Lindley (GIL) distribution using type-I hybrid and general progressive type-II censored samples has been considered. As the maximum likelihood estimators (MLEs) do not have closed-form expressions, they are obtained by numerically solving a system of nonlinear equations using certain approximation methods. The two approximate Bayes estimators using Tierney and Kadane’s method and Gibbs sampling method under general entropy and LINEX loss functions are derived. For deriving Bayes estimators gamma prior for the parameters is considered. Several confidence intervals are also proposed, such as the asymptotic confidence intervals (ACIs), bootstrap confidence intervals (Boot-p and Boot-t), and the highest posterior density (HPD) intervals. An extensive simulation study has been conducted to evaluate all the estimators’ performances numerically. The point estimators are compared through their biases and mean squared errors (MSEs). The performances of confidence intervals are evaluated using coverage probability (CP), average length (AL), and probability coverage density (PCD). Some real-life datasets have been considered for illustrative purposes. In Chapter 4, the problems of point and interval estimation for the associated parameters, along with the reliability function (r(x)) and the hazard rate function (h(x)) of the inverse Gompertz distribution, are considered using a unified hybrid censoring scheme. In the case of point estimation, some classical estimators, such as the MLE and the maximum product spacing estimator (MPSE), are derived. Further, Bayes estimators are obtained with respect to a suitable prior for the associated parameters under the balanced loss functions (balanced LINEX loss function and balanced general entropy loss function). The ACIs using the MLEs and MPSEs of the parameters are derived as the interval estimators. Further, the equal tailed and the HPD intervals are derived using the posterior samples. The MSEs are used to compare the point estimators, whereas the interval estimators are compared through their CP and ALs. A real dataset is considered for application purposes. In Chapter 5, the problems of point and interval estimation of the parameters of the inverse Gaussian distribution is considered using the sample observed in the presence of a unified type-II progressive hybrid censoring scheme with binomial removals. In the case of point estimation, the MLEs are derived using Newton’s method, whereas the Bayes estimators are derived using Tierney-Kadane’s approximation and Gibbs sampling procedure. The Bayes estimators are derived assuming that the parameters follow two independent gamma priors, where the loss function is taken as the general entropy loss function. Further, the ACIs, Boot-p confidence intervals, Boot-t confidence intervals, and HPD intervals are constructed as the interval estimators. An extensive simulation study has been carried out to compare the performances of all the point and interval estimators using the Monte Carlo simulation procedure. The point estimators are compared in terms of their biases and MSEs, whereas the interval estimators are compared by their CPs and ALs. Moreover, the one- and two sample Bayesian prediction procedure has been utilized to predict censored units from the real data under the considered censoring scheme and also to predict the order statistics from a future sample. A real-life example is considered for illustration purposes. In Chapter 6, the problem of estimating scale parameters and their reciprocals (known as hazard rates) from two exponential populations with ordered scale parameters has been considered under the progressive type-II censoring scheme. The MLEs and the uniformly minimum variance unbiased estimators (UMVUEs) are derived. Sufficient conditions are derived for improving the estimators in certain classes using a decision-theoretic approach. Consequently, some improved estimators over the MLEs and the UMVUEs are derived. A numerical comparison among all the proposed estimators is made, and conclusions are drawn regarding their performances with respect to the certain bowl-shaped loss function. In Chapter 7, the problems of point and interval estimation for the parameters of two gamma populations with common shape or rate parameter have been considered in the presence of the progressive type-II censoring scheme. The MLEs and MPSEs are derived for the common shape/rate parameter and the other two nuisance parameters after numerically solving a system of non-linear equations. The ACIs using the MLEs and MPSEs are derived for the associated model parameters. Moreover, the approximate Bayes estimators are derived using the Markov chain Monte Carlo (MCMC) procedure, known as the Gibbs sampling. Further, the HPD intervals for the model parameters are also derived using the posterior samples obtained from the Gibbs sampling. An extensive simulation study has been carried out to compare all the proposed estimators numerically. Some real datasets have been considered to demonstrate the potential application of our model problem.