## Sufficient statistics and Sufficient Estimators

An * estimator* $\hat{\theta}$ is

*if it makes so much use of the information in the sample that no other*

**sufficient***could extract from the sample, additional information about the*

**estimator***population*parameter being estimated.

The** sample mean **$\overline{X}$ utilizes all the values included in the sample so it is * sufficient estimator *of

*the population mean*$\mu$.

* Sufficient estimators* are often used to develop the

**that has**

*estimator**minimum variance among all unbiased estimators*(

**MVUE**).

If * sufficient estimator* exists, no other

*from the sample can provide additional information about the population being estimated.*

**estimator**If there is a* sufficient estimator*, then there is no need to consider any of the

*non-sufficient estimators*. A good

*is a function of*

**estimator***.*

**sufficient statistics**Let $X_1,X_2,\cdots,X_n$ be a random sample from a *probability distribution* with unknown parameter $\theta$, then this *statistic (estimator)* $U=g(X_1,X_,\cdots,X_n)$ observation gives $U=g(X_1,X_2,\cdots,X_n)$ does not depend upon *population parameter* $\Theta$.

## Sufficient Statistic Example

The * sample mean *$\overline{X}$ is a

*sufficient*for the

*population mean*$\mu$ of a

*normal distribution*with known variance. Once the

*sample mean*is known, no further information about the

*population mean*$\mu$ can be obtained from the sample itself, while the median is not

*sufficient*for the mean; even if the median of the sample is known, knowing the sample itself would provide further information about the population mean $\mu$.

## Mathematical Definition of Sufficiency

Suppose that $X_1,X_2,\cdots,X_n \sim p(x;\theta)$. $T$ is *sufficient* for $\theta$ if the conditional distribution of $X_1,X_2,\cdots, X_n|T$ does not depend upon $\theta$. Thus

\[p(x_1,x_2,\cdots,x_n|t;\theta)=p(x_1,x_2,\cdots,x_n|t)\]

This means that we can replace $X_1,X_2,\cdots,X_n$ with $T(X_1,X_2,\cdots,X_n)$ without losing information.

For further reading visit: https://en.wikipedia.org/wiki/Sufficient_statistic