Consistent but biased, unbiased but inconsistent

Question

Unbiasedness and consistency are the two properties everyone recites about estimators.

Define both precisely, give an estimator that is biased but consistent, another that is unbiased but inconsistent, and state a sufficient condition linking the two notions.

Accepted Answer

Definitions. An estimator \hat{	heta}n is unbiased if \mathbb{E}[\hat{	heta}n] = 	heta for every n and every 	heta, a finite-sample, on-average property. It is consistent if \hat{	heta}n 	o 	heta in probability as n 	o \infty, an asymptotic, per-realization property. One is about the center of the sampling distribution at fixed n; the other about the distribution collapsing onto the truth. Biased but consistent. Three standard examples: - \hat{\sigma}^2{	ext{MLE}} = \frac{1}{n}\sum(Xi - \bar{X})^2 has \mathbb{E}[\hat{\sigma}^2] = \frac{n-1}{n}\sigma^2: biased for every n, but the bia…

Consistent but biased, unbiased but inconsistent

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