Basu's theorem

In statistics, Basu's theorem states that any boundedly 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. 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 (independence of sample mean and sample variance) characterizes normal distributions.

Let P_θ be a family of distributions on a measurable space (X, Σ). Then if T is a boundedly complete sufficient statistic for θ, and A is ancillary to θ, then T is independent of A.

Let P_θ^T and P_θ^A be the marginal distributions of T and A respectively.

Define B here and the meaning of A⁻¹B .....

The P_θ^A does not depend on θ because A is ancillary. Likewise, P_θ(·|T = t) does not depend on θ because T is sufficient. Therefore:

Note the integrand (the function inside the integral) is a function of t and not θ. Therefore, since T is boundedly complete:

Therefore, A is independent of T.

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

...
Wikipedia