- Basu's theorem
In

statistics ,**Basu's theorem**states that any complete sufficient statistic is independent of anyancillary statistic . This is a 1955 result ofDebabrata 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 normalrandom variable s withmean "μ" andvariance "σ"^{2}.Then with respect to the parameter "μ", one can show that

:$widehat\{mu\}=frac\{sum\; X\_i\}\{n\},,$

the sample mean, is a complete sufficient statistic, and

:$widehat\{sigma\}^2=frac\{sum\; left(X\_i-ar\{X\}\; ight)^2\}\{n-1\},,$

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,[1] is a theorem used in to justify results relating to the probability distributions of statistics that are used in the analysis of variance.[2] 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