Solution
Constant-Risk Shrinkage
Show the problem again
Let X ~ Binomial(9, p) with p unknown. Consider the estimator p̂ = (X + 3/2)/12 = (X + √n/2)/(n + √n) with n = 9. Compute its mean squared error E[(p̂ − p)²] when the true value is p = 1/10. Give a reduced fraction.
Worked solution
Write p̂ = (X + a)/(n + b) with a = 3/2 and b = 3, so n + b = 12. Bias = (np + a)/(n + b) − p = (a − bp)/(n + b), and Var = np(1 − p)/(n + b)². The MSE is [np(1 − p) + (a − bp)²]/(n + b)². Expanding the numerator: np − np² + a² − 2abp + b²p² = (n − 2ab)p + (b² − n)p² + a². With a = √n/2 and b = √n, we get n − 2ab = 0 and b² − n = 0, so the numerator is just a² = n/4 for every p. The MSE is therefore the constant (n/4)/(n + √n)² = (9/4)/144 = 1/64, regardless of p (check at p = 1/10: variance 9 · 0.09/144 = 0.005625, bias² = (1.5 − 0.3)²/144 = 0.01, sum 0.015625). This constant-risk property is what makes the estimator minimax under squared error. The answer is 1/64.
Source: The minimax estimator of a binomial proportion is a standard result in decision theory (Hodges and Lehmann, 1950). Statement written for AxiomIQ.