Abstract
In this study, we discussed the Bayesian property of unknown parameter and reliability characteristic of the Shanker distribution. The maximum likelihood estimate is calculated. The approximate confidence interval of the unknown parameter is constructed based on the asymptotic normality of maximum likelihood estimator. Two bootstrap confidence intervals for the unknown parameter are also computed. Bayesian estimates of parameter and reliability characteristic against squared error loss function are obtained. Lindley’s approximation and Metropolis-Hastings algorithm are applied to obtain the Bayes estimates. In consequence, we also construct the highest posterior density intervals. A numerical comparison is also made to compare different methods through a Monte Carlo simulation study. Finally, two real data sets are also analyzed using the proposed methods.
Highlights
In the literature, a continuous one-parameter distribution named the “Shanker distribution” has its origin in the papers by Shanker [1]
Numerical comparison In order to evaluate the performance of all the point estimates and different methods of constructing confidence intervals (CIs) and highest posterior density (HPD) interval discussed in the preceding sections, a Monte Carlo
We found that Bayes estimators have smaller Mean squared error (MSE) values than the ML estimate (MLE) of θ
Summary
A continuous one-parameter distribution named the “Shanker distribution” has its origin in the papers by Shanker [1]. The Bayes estimates of unknown parameter are derived under the squared error loss function. In “The Bayesian estimation” section, the Bayes estimates relative to square error loss function and HPD interval are considered. We obtain asymptotic intervals of θ using asymptotic normality property of MLEs. The asymptotic variance of θfor Shanker distribution is given by Var(θ) =[ IX(θ)]−1 where I(θ) is the observed Fisher’s information which is given by I(θ) d2 log dθ 2
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