Comparison of Lower Bounds for the Variance of Unbiased Estimators for some Well-known Families of Distributions

One of the most fundamental issues in estimation theory about accuracy of an unbiased estimator is computing or approximating its variance. Very often, the variance has a complicated form or cannot be computed explicitly. In this paper, we consider two well-known lower bounds for the variance of unbiased estimator, namely the Bhattacharyya (1946, 1947) and the Kshirsagar (2000) bounds for some versatile families of distributions in statistics and especially in reliability analysis. We consider the generalized gamma (GG), inverse Gaussian, Burr type XII and Burr type III distributions, and derive for these distributions, general forms of Bhattacharyya and Kshirsagar matrices. Additionally, we evaluate different Bhattacharyya and Kshirsagar bounds for the variance of estimators of some functions of relevant parameters and arrive at proposal which of the bounds should be used in given situations.