Parameter Estimation Based on the Frèchet Progressive Type II Censored Data with Binomial Removals

This paper considers the estimation problem for the Frèchet distribution under progressive Type II censoring with random removals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood method to obtain the estimators of parameters and derive the sampling distributions of the estimators, and we also construct the confidence intervals for the parameters and percentile of the failure time distribution. 1. Introduction Recently, the extreme value distribution is becoming increasingly important in engineering statistics as a suitable model to represent phenomena with usually large maximum observations. In engineering circles, this distribution is often called the Frèchet model. It is one of the pioneers of extreme value statistics. The Frèchet (extreme value type II) distribution is one of the probability distributions used to model extreme events. The generalization of the standard Frèchet distribution has been introduced by Nadarajah and Kotz [1] and Abd-Elfattah and omima[2]. There are over fifty applications ranging from accelerated life testing through to earthquakes, floods, rain fall, queues in supermarkets, sea currents, wind speeds, and track race records, see Kotz and Nadarajah [3]. Censoring arises in a life test when exact lifetimes are known for only a portion of test units and the remainder of the lifetimes are known only to exceed certain values under an experiment. There are several types of censored test. One of the most common censoring schemes is Type II censoring. In a Type II censoring, a total of units is placed on test, but instead of continuing until all units have failed, the test is terminated at the time of the th unit failure. Type II censoring with different failure time distribution has been studied by many authors including Mann et al. [4], Lawless [5], and Meeker and Escobar [6]. If an experiment desires to remove live units at points other than the final termination point of the life test, the above described scheme will not be of use to experimenter. Type II censoring does not allow for units to be lost or removed from the test at points other than the final termination point see Balakrishnan and Aggarwala [7, Chapter 1]. A generalization of Type II censoring is progressive Type II censoring. Under this scheme, units are placed on test at time zero, and failure are going to be observed. When the first failure is observed, of surviving units are randomly selected, removed, and so on. This experiment terminates at the time when the th failures is observed and