Abstract
Type-I censoring mechanism arises when the number of units experiencing the event is random but the total duration of the study is fixed. There are a number of mathematical approaches developed to handle this type of data. The purpose of the research was to estimate the three parameters of the Frechet distribution via the frequentist Maximum Likelihood and the Bayesian Estimators. In this paper, the maximum likelihood method (MLE) is not available of the three parameters in the closed forms; therefore, it was solved by the numerical methods. Similarly, the Bayesian estimators are implemented using Jeffreys and gamma priors with two loss functions, which are: squared error loss function and Linear Exponential Loss Function (LINEX). The parameters of the Frechet distribution via Bayesian cannot be obtained analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the three parameters is obtained via Metropolis-Hastings algorithm. Comparisons of the estimators are obtained using Mean Square Errors (MSE) to determine the best estimator of the three parameters of the Frechet distribution. The results show that the Bayesian estimation under Linear Exponential Loss Function based on Type-I censored data is a better estimator for all the parameter estimates when the value of the loss parameter is positive.
Highlights
The results show that the Bayesian estimation under Linear Exponential Loss Function based on Type-I censored data is a better estimator for all the parameter estimates when the value of the loss parameter is positive
As shown in the results when the shape paramemter λ was 1 and location pramster α was 0.4 with size 25, the Maximum Likelihood Estimation (MLE) was 1.1042, the Bayesian under squared error loss function (BS) estimation was 1.1486, BL (r = −0.7) estimation was 1.1580 and BL (r = 0.7) estimation was 1.0716, we observed that the BL (r = 0.7) was closer to true value than others estimation
As shown in the results when the shape paramemter λ was 1 and location pramster α was 0.4 with size 25, the Maximum Likelihood Estimation (MLE) was 0 5432, the Bayesian under squared error loss function (BS) estimation was 0.5876, BL (r = −0.7) estimation was 0.5971 and BL (r = 0.7) estimation was 0.5106, we observed that the BL (r = 0.7) was closer to true value than others estimation
Summary
Many studies have been carried out on the Frechet distribution by a lot of researchers with the aim of estimating its parameters using different statistical approaches, some of which include [2] [3] and [4]. [6] derived the reference and matching priors for the Frechet stress-strength model and developed Bayesian approach for Frechet distribution under reference prior, respectively. [7] attained Bayesian estimators of Frechet distribution and their risks by using loss functions under Gumbel Type-II prior and Levy prior. Likewise, [8] estimated the Frechet distribution parameters with an application to the medical field. No previous study dealt with Bayesian estimations of the three-parameter Frechet distribution under Type-I censored data in survival/reliability analysis
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