Comparison of Three Methods

To compare the three methods (estimate Binomial n, MLE, and Method of Moments) I generated a set of random variables $(X_1, \dots, X_{10})$ independent identically distributed Bin(100,0.7). From these data, I calculated $\hat p_1, \dots, \hat p_{10}$ in addition to the sample mean and variances $\bar p = 0.721$ and s_p² = 0.0013211. The following table gives the estimates for a and b, including the mean and variance of a Beta random variable with those parameters.

		Maximum Likelihood	Method of
	Scaled Binomial	Estimates	Moments Estimates
a	109.316	110.653	109.062
b	42.9142	42.8638	42.2029
mean	0.7181	0.7208	0.7210
variance	0.0013211	0.0013025	0.0013211