Refer to Problem 15.14. For N = 2 groups with n₁ and n₂ independent observations, find the minimum modified chi-squared estimator of π. Compare it to the ML estimator.

Data from Problem 15.14:

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 = (Σ_i y_i)/(Σ_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).

[(:/m,) 1] N [(y:/n;) ] + TT i=1 i=1