Refer to Problem 15.14. For N = 2 groups with n 1 and n 2 independent observations,
Question:
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 yi be a bin(ni, πi) variate for group i, i = 1,..., N, with {yi} independent. Consider the model that π1 = ... = πN. Denote that common value by π.
a. Show that the ML estimator of π is p = (Σi yi)/(Σi ni).
b. The minimum chi-squared estimator π̃ is the value of π minimizing
The second term results from comparing (1 − yi/ni) to (1 − π), the proportions in the second category. If n1 = ... = nN = 1, show that π̃ minimizes Np(1 − π)/π + N(1 − p)π/(1 − π). Hence show
π̃ = p1/2 / [p1/2 + (1 − p)1/2].
Note the bias toward 1/2 in this estimator.
c. Argue that as N ⇾ ∞ with all ni = 1, the ML estimator is consistent but the minimum chi-squared estimator is not (Mantel 1985).
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