Lower bounds for the variance of unbiased estimators in generalized beta distribution of the second kind (GB2)

Abstract

In order to give an excellent description of income distributions, although a large number of functional forms have been proposed, but the four-parameter generalized beta model of the second kind (GB2), introduced by J. B. McDonald [18], is now widely acknowledged which is including many other models as special or limiting cases.One of the fundamentals of statistical inference is the estimation problem of a function of unknown parameter in a probability distribution and computing the variance of the estimator or approximating it by lower bounds.In this paper, we consider two famous lower bounds for the variance of any unbiased estimator, which are Bhattacharyya and Kshirsagar bounds. We obtain the general forms of the Bhattacharyya and Kshirsagar matrices in the GB2 distribution. In addition, we compare different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of some parametric functions such as mode, mean, skewness and kurtosis in GB2 distribution and conclude that in each case, which bound is better to use. The results of this paper can be useful for researchers trying to find the accuracy of the estimators.