Different methods to estimate reliability for Exponentiated Mukherjee--Islam distribution under type II censoring

Abstract

In the field of reliability and survival times, we note that many data are naturally limited above. The Exponentiated Mukherjee--Islam distribution (EMID) is considered one of the most important finite-range survival time distributions. It has two shape parameters and a scale parameter. Among its properties is that it can track the skewness and behavior of the hazard function while maintaining support at $(0, \theta)$. In this paper, more than one method was used to estimate both parameters and reliability of the EMID under type~II censoring. The model estimators were derived using maximum likelihood (ML) and maximum product spacing (MPS) methods. To demonstrate the efficiency of the estimators obtained in this paper, we presented an extended Monte Carlo simulation study in which each estimator was compared based on the value of the root mean square error (RMSE) using \texttt{R~Studio}. The simulation study used eight groups of parameters, sample sizes $n = 30, 50, 100, 300$, and censoring ratios $C = 0.0, 0.1, 0.2, 0.3$. The results show a decrease in RMSE with larger~$n$. High censoring ratios lead to an amplified RMSE, especially for the scaling parameter~$\theta$. Maximum likelihood estimates (MLE) are better in small samples with low censoring ratios, whereas MPS estimators tend to provide more robust and efficient results.