One of the main branches of statistical inference is estimation the- ory which introduce different estimators of unknown parameters and verify various properties of estimators such as unbiasedness, consistency, efficiency and minimum error or variance. In the class of unbiased estimators for the parameter functions, the variance of estimator is one of the basic criteria to compare and evaluate the accuracy of the estimators. In addition, the variance of unbiased estimators is one of the main components of statisti- cal inference such as hypothesis testing and confidence intervals. In many cases the variance has complicated form and we can not compute it, so, by lower bounds, we can approximate it. Many studies have been done on the lower bounds for the variance of an unbiased estimator of the parameter. The well-known and applicable lower bounds are Cramer-Rao (by Rao, 1945 and Cramer, 1946), Bhattacharyya (by Bhattacharyya, 1946, 1947), Hammersley- Chapman-Robbins (by Hammersley, 1950 and Chapman and Robbins, 1951) and Kshirsagar (by Kshirsagar, 2000). In this paper, by comparing with boot- strap method, we show that Bhattacharyya and Kshirsagar lower bounds are better that bootsterap method for approximating the variance of the un- biased estimators. Also, we obtain the general form of Bhattacharyya and Kshirsagar matrices in natural exponential family (NEF), whose role and importance in all fields of statistics is clear to everyone. Furthermore by giving two examples of negative binomial and exponential distributions, we compare the Bhattacharyya and Kshirsagar bounds for variances of unbiased estimators of the parameter functions with the bootstrap method.

Keywords