Comparing Bhattacharyya and Kshirsagar bounds with bootstrap method

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 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. Another  common and popular method that is used in many statistical problems such as variance estimation, is bootstrap method. This method has some advantages and disadvantages that must be careful when using them. In this paper, first we briefly introduce the two famous lower bounds named `Kshirsagar` insert ignore into journalissuearticles values(one parameter case); and `Bhattacharyya` insert ignore into journalissuearticles values(one and multi parameter case); bounds and then we extend the Kshirsagar bound in multi parameter case.  Also, by giving some examples in different distributions, we  compare one  and multi parameter  Bhattacharyya and Kshirsagar lower bounds with bootstrap method for approximating  the variance of the unbiased estimators  and show  that the mentioned  bounds have a better performance than bootstrap method.