Consistent estimator: Difference between revisions

Revision as of 05:26, 16 March 2022

Revision as of 05:27, 16 March 2022

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=== Unbiased but not consistent ===

An estimator can be [[biased estimator|unbiased]] but not consistent. For example, for an [[iid]] sample {''x''{{su|b=1}},..., ''x{{su|b=n}}''} one can use ''T{{su|b=n}}''(''X'') = ''x''{{su|b=n}} as the estimator of the mean E[''xX'']. Note that here the sampling distribution of ''T{{su|b=n}}'' is the same as the underlying distribution (for any ''n,'' as it ignores all points but the last), so E[''T{{su|b=n}}''(''X'')] = E[''X''] and it is unbiased, but it does not converge to any value.

However, if a sequence of estimators is unbiased ''and'' converges to a value, then it is consistent, as it must converge to the correct value.