Statistical Inference Based on Progressive type-II Censored Samples from Lifetime Distributions

Abstract

The problems of estimation and prediction for statistical models based on progressively type-II censored sample play a crucial role in various areas of research such as reliability theory, survival analysis and statistics. In this thesis, statistical inferences for five distributions are considered under progressively type-II censored sample. For generalized Rayleigh, gamma-mixed Rayleigh, log-logistic and generalized Fréchet distributions, the estimators have been obtained for the unknown parameters, reliability and hazard rate functions. For the case of exponentiated Gumbel type-II distribution, the estimands are proposed for the model parameters. Various estimates are proposed in this thesis. The maximum likelihood estimates are obtained. These are not of closed-form. Thus, Newton-Raphson method, expectation-maximization and stochastic expectation-maximization algorithms are used to compute the maximum likelihood estimates. Sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are obtained for the exponentiated Gumbel type-II distribution. For each problem, Bayes estimates are derived with respect to various symmetric and asymmetric loss functions. The priors are considered as independent gamma distributions for the purpose of Bayesian estimation. It is observed that the Bayes estimates are not of closed-form. So, approximation techniques such as Lindley's method, importance sampling method and Metropolis-Hastings algorithm are employed. The approximate confidence intervals are constructed using the normal approximation of the maximum likelihood estimates, normal approximation of the log-transformed maximum likelihood estimates and two bootstrap procedures. Highest posterior density credible intervals are introduced. In addition, the problem of Bayesian prediction and interval estimation is studied for gamma-mixed Rayleigh and generalized Fréchet distributions. In this purpose, one- and two-sample prediction problems are studied. To observe the performance of the proposed estimates, a detailed simulation is conducted using R software. The performance of the maximum likelihood and Bayes estimates is observed based on the average values and mean squared errors. The interval estimates are compared with respect to the average lengths and coverage probabilities. For every problem, real datasets are considered and analysed for illustrative purposes.