Finding the optimized lower bound for the variance of unbiased estimators in some well-known families of distributions

Finding the optimized lower bound for the variance of unbiased estimators in some well-known families of distributions

One of the most fundamental things in estimation theory about accuracy of an unbiased estimator is computing or approximating its variance. Most of the time, the variance has complicated form or cannot be computed. In this paper, we consider two well-known lower bounds for the variance of unbiased estimators, which are Bhattacharyya (1946, 1947) and Kshirsagar (2000) bounds for some versatile families of distributions in statistics and especially in reliability such as, generalized gamma (GG), inverse Gaussian, Burr type XII and Burr type III distributions. In these distributions, the general forms of Bhattacharyya and Kshirsagar matrices are obtained. In addition, we evaluate different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of some parameter functions and conclude that in each case, which bound has higher convergence and is better to use.