Non-Bayes, Bayes and Empirical Bayes Estimators for the Shape Parameter of Lomax Distribution

Abstract

Point estimation is one of the core topics in mathematical statistics. In this paper we consider the most common methods of point estimation: non-Bayes, Bayes and empirical Bayes methods to estimate the shape parameter of Lomax distribution based on complete data. The maximum likelihood, moment and uniformly minimum variance unbiased estimators are obtained as non-Bayes estimators. Bayes and empirical Bayes estimators are obtained corresponding to three informative priors "gamma, chi-square and inverted Levy" based on symmetric "squared error" and asymmetric "LINEX and general entropy" loss functions. The estimates of the shape parameter were compared empirically via Monte Carlo simulation study based upon the mean squared error. Among the set of conclusions that have been reached, it is observed that, for all sample sizes and different cases, the performance of uniformly minimum variance unbiased estimator is better than other non-Bayes estimators. Further that, Monte Carlo simulation results indicate that the performance of Bayes and empirical Bayes estimator in some cases are better than non-Bayes for some appropriate of prior distribution, loss function, values of parameters and sample size.

Keywords: Lomax distribution; maximum likelihood estimator; moment estimator; uniformly minimum variance unbiased estimator; Bayes estimator; empirical Bayes estimator; informative prior; squared error loss function; LINEX  loss  function;  general  entropy  loss  function;  mean  squared  error.