Two biased estimators, which loses less?

Question

Two estimators of the same quantity are both a little off. Estimator X is biased by 1 with variance 9. Estimator Y is biased by 4 with variance 2.

Mean squared error splits as $\text{MSE} = \text{Variance} + \text{Bias}^2$ .

Which estimator has the smaller MSE, and what is that value?

Accepted Answer

The idea. Neither is unbiased, so both pieces matter. Square each bias, add its variance, compare. Estimator X. Small bias, large scatter: 	ext{MSE}X = 9 + 1^2 = 9 + 1 = 10. Estimator Y. Large bias, small scatter: 	ext{MSE}Y = 2 + 4^2 = 2 + 16 = 18. The winner. Estimator X wins, 10 versus 18. Estimator Y is very steady, but a bias of 4 becomes 16 once squared, and that dominates its total error. The moral is that bias is dangerous precisely because it is squared: doubling the bias quadruples its contribution. A steady estimator that is badly off-center is worse than a jumpy one that is near…

Two biased estimators, which loses less?

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