Consistent estimator: Difference between revisions

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Revision as of 06:51, 15 June 2019

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Alternatively, an estimator can be biased but consistent. For example, if the mean is estimated by <math>{1 \over n} \sum x_i + {1 \over n}</math> it is biased, but as <math>n \rightarrow \infty</math>, it approaches the correct value, and so it is consistent.

Important examples include the [[sample variance]] and [[sample standard deviation]]. Without [[Bessel's correction]] (that is, when using the sample size ''<math>n''</math> instead of the [[Degrees of freedom (statistics)|degrees of freedom]] ''<math>n''&nbsp;−&nbsp;-1</math>), these are both negatively biased but consistent estimators. With the correction, the corrected sample variance is unbiased, while the corrected sample standard deviation is still biased, but less so, and both are still consistent: the correction factor converges to 1 as sample size grows.

Here is another example. Let <math>T_n</math> be a sequence of estimators for <math>\theta</math>.

:<math>\Pr(T_n) = \begin{cases}

1 - 1/n, & \mbox{if }\, T_n = \theta \\

1/n, & \mbox{if }\, T_n = n\delta + \theta

\end{cases}</math>

We can see that <math>T_n \xrightarrow{p} \theta</math>, <math>\operatorname{E}[T_n] = \theta + \delta </math>, and the bias doesn'tdoes not converge to zero.

== See also ==