Sufficient statistics and Sufficient Estimators

An estimator $\hat{\theta}$ is sufficient if it makes so much use of the information in the sample that no other estimator could extract from the sample, additional information about the 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 estimator that has minimum variance among all unbiased estimators (MVUE).

If sufficient estimator exists, no other estimator from the sample can provide additional information about the population being estimated.

If there is a sufficient estimator, then there is no need to consider any of the non-sufficient estimators. A good estimator is a function of 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
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:

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

x Logo: Shield Security
This Site Is Protected By
Shield Security