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.

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, one can show 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.