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.[1]
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.[2] 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.
The following theorem, due to Basu ... helps us in proving independence between certain types of statistics, without actually deriving the joint and marginal distributions of the statistics involved. This is a very powerful tool and it is often used ...