Comparison of the Bootstrap and Delta Method Variances of the Variance Estimator of the Bernoulli Distribution

Abstract

It is interesting to calculate the variance of the variance estimator of the Bernoulli distribution. Therefore, we compare the Bootstrap and Delta Method variances of the variance estimator of the Bernoulli distribution in this paper. Firstly, we provide the correct Bootstrap, Delta Method, and true variances of the variance estimator of the Bernoulli distribution for three parameter values in Table 2.1. Secondly, we obtain the estimates of the variance of the variance estimator of the Bernoulli distribution by the Delta Method (analytically), the true method (analytically), and the Bootstrap Method (algorithmically). Thirdly, we compare the Bootstrap and Delta Methodsin terms of the variance estimates, the errors, and the absolute errors in three gures for 101 parameter values in [0, 1], with the purpose to explain the di erences between the Bootstrap and Delta Methods. Finally, we give three examples of the Bernoulli trials to illustrate the three methods.