Basu's theorem

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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 showing one is complete sufficient and the other is ancillary.

[edit] Example

[edit] Independence of sample mean and sample variance

Let X1, X2, ..., Xn be independent, identically distributed normal random variables with mean θ and variance σ2.

Then with respect to the parameter θ, we see that

\hat{\mu}=\frac{\sum X_i}{n}\,, the sample mean, is a complete sufficient statistic, and
\hat{\sigma}^2=\frac{\sum \left(X_i-\bar{X}\right)^2}{N-1}\,, the sample variance, is an ancillary statistic.

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

[edit] Reference

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