Which estimator wins on MSE, the unbiased or the biased one?

Question

Estimator A is unbiased with variance 10. Estimator B has a bias of 1 and a variance of 4. Which has the lower mean squared error? And how large would B's bias have to be for A to win instead?

Accepted Answer

Use \operatorname{MSE} = \operatorname{Var} + \operatorname{Bias}^2 for each. Compare. - \operatorname{MSE}(A) = 10 + 0^2 = 10. - \operatorname{MSE}(B) = 4 + 1^2 = 5. Estimator B wins, with half the MSE, despite being biased. Its 6-unit variance saving (10 	o 4) dwarfs the 1-unit squared-bias cost. Break-even bias. A beats B when \operatorname{MSE}(A) < \operatorname{MSE}(B) \;\Longleftrightarrow\; 10 < 4 + 	ext{Bias}^2 \;\Longleftrightarrow\; 	ext{Bias}^2 6 \;\Longleftrightarrow\; 	ext{Bias} \sqrt{6} \approx 2.45. So B stays the better estimator until its bias exceeds about 2.45. Any bia…

Which estimator wins on MSE, the unbiased or the biased one?

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