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 (independence of sample mean and sample variance ) characterizes normal distributions. Statement. Let formula_1 be a family of distributions on a measurable space formula_2 and formula_3 measurable maps from formula_2 to some measurable space formula_5. (Such maps are called a statistic.) If formula_6 is a boundedly complete sufficient statistic for formula_7, and formula_8 is ancillary to formula_7, then formula_6 is independent of formula_8. Proof. Let formula_12 and formula_13 be the marginal distributions of formula_6 and formula_8 respectively. Denote by formula_16 the preimage of a set formula_17 under the map formula_8. For any measurable set formula_19 we have The distribution formula_13 does not depend on formula_7 because formula_8 is  ancillary. Likewise, formula_24 does not depend on formula_7 because formula_6 is sufficient. Therefore Note the integrand (the function inside the integral) is a function of formula_28 and not formula_7. Therefore, since formula_6 is boundedly complete the function is zero for formula_12 almost all values of formula_28 and thus for almost all formula_28. Therefore, formula_8 is independent of formula_6. Example. Independence of sample mean and sample variance of a normal distribution (known variance). Let "X", "X", ..., "X" be independent, identically distributed normal random variables with mean "μ" and variance "σ". 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 (that the sample mean and sample variance of the normal distribution are independent) "characterizes" the normal distribution – no other distribution has this property.