Slightly shrinking an exponential mean estimate

Question

Waiting times X1, \dots, Xn are i.i.d. Exponential with unknown mean 	heta. The sample mean \bar X is unbiased for 	heta, but you consider a scaled version \hat	hetac = c\,\bar X. Which constant c minimizes the mean squared error of \hat	hetac?

Accepted Answer

Gather the pieces. For an exponential, \operatorname{Var}(X) = 	heta^2, so the sample mean has \mathbb{E}[\bar X] = 	heta, \qquad \operatorname{Var}(\bar X) = \frac{	heta^2}{n}. Then \hat	hetac = c\bar X has bias (c-1)	heta and variance c^2	heta^2/n, giving 	ext{MSE}(c) = (c-1)^2	heta^2 + c^2\frac{	heta^2}{n}. Minimize. Factor out 	heta^2 and differentiate: \frac{d}{dc}\Big[(c-1)^2 + \frac{c^2}{n}\Big] = 2(c-1) + \frac{2c}{n} = 0 \implies c\Big(1 + \frac{1}{n}\Big) = 1. Hence c^\star = \frac{n}{n+1}. Interpret. The best estimator is \hat	heta = \frac{n}{n+1}\bar X, a slight shrink …

Slightly shrinking an exponential mean estimate

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