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. 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. Let ( P θ ; θ ∈ Θ ) {displaystyle (P_{ heta }; heta in Theta )} be a family of distributions on a measurable space ( X , A ) {displaystyle (X,{mathcal {A}})} and T , A {displaystyle T,A} measurable maps from ( X , A ) {displaystyle (X,{mathcal {A}})} to some measurable space ( Y , B ) {displaystyle (Y,{mathcal {B}})} . (Such maps are called a statistic.) If T {displaystyle T} is a boundedly complete sufficient statistic for θ {displaystyle heta } , and A {displaystyle A} is ancillary to θ {displaystyle heta } , then T {displaystyle T} is independent of A {displaystyle A} . Let P θ T {displaystyle P_{ heta }^{T}} and P θ A {displaystyle P_{ heta }^{A}} be the marginal distributions of T {displaystyle T} and A {displaystyle A} respectively. Denote by A − 1 ( B ) {displaystyle A^{-1}(B)} the preimage of a set B {displaystyle B} under the map A {displaystyle A} . For any measurable set B ∈ B {displaystyle Bin {mathcal {B}}} we have The distribution P θ A {displaystyle P_{ heta }^{A}} does not depend on θ {displaystyle heta } because A {displaystyle A} is ancillary. Likewise, P θ ( ⋅ ∣ T = t ) {displaystyle P_{ heta }(cdot mid T=t)} does not depend on θ {displaystyle heta } because T {displaystyle T} is sufficient. Therefore Note the integrand (the function inside the integral) is a function of t {displaystyle t} and not θ {displaystyle heta } . Therefore, since T {displaystyle T} is boundedly complete the function is zero for P θ T {displaystyle P_{ heta }^{T}} almost all values of t {displaystyle t} and thus for almost all t {displaystyle t} . Therefore, A {displaystyle A} is independent of T {displaystyle T} .