Shrinking the sample mean to cut its error

Question

You estimate a Normal mean \mu from n observations but consider shrinking the sample mean toward zero: \hat\muc = c\,\bar X for some constant 0 \le c \le 1. Which shrinkage factor c minimizes the mean squared error of \hat\muc as an estimator of \mu?

Accepted Answer

Write the error as bias-squared plus variance. Since \mathbb{E}[\bar X] = \mu and \operatorname{Var}(\bar X) = \sigma^2/n, \mathbb{E}[\hat\muc] = c\mu, \qquad 	ext{bias} = c\mu - \mu = (c-1)\mu, \qquad \operatorname{Var}(\hat\muc) = c^2\frac{\sigma^2}{n}. So 	ext{MSE}(c) = (c-1)^2\mu^2 + c^2\frac{\sigma^2}{n}. Minimize. Differentiate and set to zero: \frac{d}{dc}	ext{MSE} = 2(c-1)\mu^2 + 2c\frac{\sigma^2}{n} = 0 \implies c\Big(\mu^2 + \frac{\sigma^2}{n}\Big) = \mu^2. Hence c^\star = \frac{\mu^2}{\mu^2 + \sigma^2/n}. Read it. The optimal factor is strictly between 0 and 1, so some shrinkage…

Shrinking the sample mean to cut its error

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