A set of families of distributions which might be useful for fitting data was described by Burr (1942). Among them, the families type XII (Burr XII) and type III (Burr III), have gathered special attention in physics, actuarial studies, reliability and applied statistics. Estimating a wide range of functions of their parameters such as reliability, hazard rate and mode, under various conditions, have been done. But, the variances of the estimators are not considered precisely yet. In this paper, we consider two well-known lower bounds for the variance of any unbiased estimator, which are Bhattacharyya and Kshirsagar bounds for the Burr XII and Burr III distributions. In these distributions, the general forms of the Bhattacharyya and Kshirsagar matrices are obtained. In addition, we evaluate different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of the reliability, hazard rate, mode and median due to Burr XII and Burr III distributions and conclude that in each case, which bound has higher convergence and is better to use. Also via some figures, we compare the two bounds with bootstrap method in approximating the variance of the unbiased estimator of the reliability, median and mean of the Burr XII distributions.

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