Let y i be a bin(n i =, i ) variate for group i, i =
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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 = (∑i yi)/(∑i ni).
b. The minimum chi-squared estimator π̂ 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 π̂ minimizes Np(1 − π)/π + N(1 − p)π/(1 − π). Hence show
π̂ = 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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