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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.

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 [

A generalization of Type II censoring is progressive Type II censoring. Under this scheme,

In this paper, we will make inference on the parameters of three-parameter Frèchet distribution under progressive type II censoring with binomial removals. The maximum likelihood estimators (MLEs) for the parameters in an explicit and implicit form are obtained in Section

Let random variable

Suppose

In this section, we are going to derive the sampling distributions of the MLE’s and obtain the confidence intervals for the parameters [

Let

Suppose that

Suppose that,

Such that,

Such that,

In reliability analysis [

We develop some results on a three-parameter Frèchet distribution when progressive Type II censoring with binomial removals is performed. We derive the MLEs and confidence for the parameters. The MLE and confidence interval for the percentiles of failure time distribution are obtained. In practice, it is often useful to have an idea of the duration of a life test. Therefore, it is important to compute the expected time required to complete a life test. In the case of progressively type II-censored sampling plan with binomial removals, one can obtain this information by calculating the expectation of the

Probability density function

Cumulative distribution function

Three-parameter Frèchet distribution

Number of removals

Removal probability

Chi-square

Confidence interval

100-th percentile of the failure time distribution.