IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
Let X1,...,Xn be independent with mean μ and variance σ². Consider the estimator T = (X1+...+Xn)/(n+1) of μ. Which expression gives its mean squared error?
The variance is nσ²/(n+1)² and the bias is -μ/(n+1), so the MSE is nσ²/(n+1)² + μ²/(n+1)². The bias term must be squared, which includes the denominator (n+1)².
- Aσ²/(n+1)
- Bnσ²/(n+1)² + μ²/(n+1)²Correct
- Cσ²/n
- Dnσ²/(n+1)²
- nσ²/(n+1)² + μ²/(n+1)
Explanation
E[T] = nμ/(n+1), so bias = -μ/(n+1) and bias² = μ²/(n+1)². Var(T) = nσ²/(n+1)². MSE = sum of these. Dropping the bias gives nσ²/(n+1)².
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