An unbiased estimator that never gets any better

Question

A survey team estimates a population mean $\mu$ but, out of habit, always averages only the first ten responses,

$\hat\mu = \frac{1}{10}\sum_{i=1}^{10} X_i,$

even when the full sample size $n$ grows into the thousands.

Is $\hat\mu$ a consistent estimator of $\mu$ ? Is it unbiased?

Accepted Answer

Unbiased, yes. Each response has mean \mu, so \mathbb{E}[\hat\mu] = \frac{1}{10}\sum{i=1}^{10}\mathbb{E}[Xi] = \mu at every sample size. On average it is spot on. Consistent, no. Consistency requires the estimator to concentrate on \mu as n 	o \infty. But \hat\mu ignores everything past the tenth response, so its variance is stuck: \operatorname{Var}(\hat\mu) = \frac{\sigma^2}{10} \quad	ext{for all } n. It never shrinks. The estimator keeps rattling around \mu with the same spread forever, so it does not converge in probability to \mu. Collecting a million responses buys you nothing because…

An unbiased estimator that never gets any better

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