Biased forever, yet still consistent

Question

A candidate claims that an unbiased estimator is always better than a biased one, and that a consistent estimator must be unbiased.

Give a concrete estimator that is biased at every sample size yet consistent, and one that is unbiased yet inconsistent, to show these are separate properties.

Accepted Answer

They measure different things. Bias is a fixed-sample-size property: does \mathbb{E}[\hat	hetan] equal 	heta? Consistency is a large-sample property: does \hat	hetan 	o 	heta (in probability) as n 	o \infty? Neither implies the other. Biased but consistent. The maximum likelihood variance \hat\sigma^2 = \frac{1}{n}\sumi (xi - \bar x)^2 has expectation \frac{n-1}{n}\sigma^2, so it is biased low by \sigma^2/n at every finite n. But the bias shrinks to zero and its variance also vanishes, so \hat\sigma^2 	o \sigma^2: consistent. Another example is \maxi xi for Uniform(0,	heta): biased lo…

Biased forever, yet still consistent

Your answer

More Statistics questions