Basu's theorem


In statistics, Basu's theorem states that any boundedly complete minimal 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. An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the [|Example] section below. This property characterizes normal distributions.

Statement

Let be a family of distributions on a measurable space and measurable maps from to some measurable space. If is a boundedly complete sufficient statistic for, and is ancillary to, then is independent of.

Proof

Let and be the marginal distributions of and respectively.
Denote by the preimage of a set under the map. For any measurable set we have
The distribution does not depend on because is ancillary. Likewise, does not depend on because is sufficient. Therefore
Note the integrand is a function of and not. Therefore, since is boundedly complete the function
is zero for almost all values of and thus
for almost all. Therefore, is independent of.

Example

Independence of sample mean and sample variance of a normal distribution (known variance)

Let X1, X2,..., Xn be independent, identically distributed normal random variables with mean μ and variance σ2.
Then with respect to the parameter μ, one can show that
the sample mean, is a complete sufficient statistic – it is all the information one can derive to estimate μ, and no more – and
the sample variance, is an ancillary statistic – its distribution does not depend on μ.
Therefore, from Basu's theorem it follows that these statistics are independent.
This independence result can also be proven by Cochran's theorem.
Further, this property characterizes the normal distribution – no other distribution has this property.