# Basu's theorem

﻿
Basu's theorem

In statistics, Basu's theorem states that any complete sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu.

It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem.

Example

Independence of sample mean and sample variance

Let "X"1, "X"2, ..., "X""n" be independent, identically distributed normal random variables with mean "&mu;" and variance "σ"2.

Then with respect to the parameter "&mu;", one can show that

:$widehat\left\{mu\right\}=frac\left\{sum X_i\right\}\left\{n\right\},,$

the sample mean, is a complete sufficient statistic, and

:

the sample variance, is an ancillary statistic.

Therefore, from Basu's theorem it follows that these statistics are independent.

References

*Basu, D., "On Statistics Independent of a Complete Sufficient Statistic," "Sankhya", Ser. A, 15 (1955), 377-380

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Debabrata Basu — (Bengali: দেবব্রত বসু) (5 July 1924 – 24 March 2001) was a mathematical statistician who made fundamental contributions to the foundations of statistics. Basu invented simple examples that displayed some difficulties of likelihood based… …   Wikipedia

• Rao–Blackwell theorem — In statistics, the Rao–Blackwell theorem is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean squared error criterion or any of a variety of similar criteria. The Rao… …   Wikipedia

• Cochran's theorem — In statistics, Cochran s theorem, devised by William G. Cochran, is a theorem used in to justify results relating to the probability distributions of statistics that are used in the analysis of variance. Contents 1 Statement 2 …   Wikipedia

• Completeness (statistics) — In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be… …   Wikipedia

• List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

• Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• Sufficient statistic — In statistics, a sufficient statistic is a statistic which has the property of sufficiency with respect to a statistical model and its associated unknown parameter, meaning that no other statistic which can be calculated from the same sample… …   Wikipedia

• Sufficiency (statistics) — In statistics, sufficiency is the property possessed by a statistic, with respect to a parameter, when no other statistic which can be calculated from the same sample provides any additional information as to the value of the parameter cite… …   Wikipedia

• Ancillary statistic — In statistics, an ancillary statistic is a statistic whose probability distribution does not depend on which of the probability distributions among those being considered is the distribution of the statistical population from which the data were… …   Wikipedia