Let yi be a bin(ni =, πi) variate for group i,

Let y_i be a bin(n_i =, π_i) variate for group i, i = 1,…, N, with {y_i} independent. Consider the model that π₁ = … = π_N. Denote that common value by π.

a. Show that the ML estimator of π is p = (∑_i y_i)/(∑_i n_i).

b. The minimum chi-squared estimator π̃ is the value of π minimizing

The second term results from comparing (1 – y_i/n_i) to (1 – π), the proportions in the second category. If n₁ = … = n_N = 1, show that π̃ minimizes N_p(1 – π)/π + N(1 – p)π/(1 – π). Hence show

π̃ = p^1/2 / [p^1/2 + (1 – p)^1/2].

Note the bias toward 1/2 in this estimator.

c. Argue that as N → ∞ with all n_i = 1, the ML estimator is consistent but the minimum chi-squared estimator is not (Mantel 1985).