Parameters and Reliability Estimation of Left Truncated Gumbel Distribution under Progressive Type II Censored with Repairable Mechanical Equipment Data
The estimation of two parameters of the left truncated Gumbel distribution using the progressive type II censoring scheme is discussed. We first derived the maximum likelihood estimators of the unknown parameters. The approximate asymptotic variance-covariance matrix and approximate confidence intervals based on the asymptotic normality of the classical estimators are calculated. Also, the survival and hazard functions are derived. Further, the delta method is used to construct approximate confidence intervals for survival and hazard functions. Using the left truncated normal prior for the location parameter and an inverted gamma prior for the scale parameter, several Bayes estimates based on squared error and general entropy loss functions are computed. Bayes estimators of the unknown parameters cannot be calculated in closed forms. Markov chain Monte Carlo method, namely Metropolis-Hastings algorithm, has been used to derive the approximate Bayes estimates. Also, the credible intervals are constructed by using Markov chain Monte Carlo samples. Finally, The Monte Carlo simulation study compares the performances among various estimates in terms of their root mean squared errors, mean absolute biased, average confidence lengths, and coverage probabilities under different sets of values of sample sizes, number of failures and censoring schemes. Moreover, a numerical example with a real data set and Markov chain Monte Carlo data sets are tackled to highlight the importance of the proposed methods. Bayes Markov chain Monte Carlo estimates have performed better than those obtained based on the likelihood function
