4 Small Sample Properties of U Statistics

Mr Taranga Mukherjee

1 A few concepts

Suppose X = (X1; X2; ::; Xn) are iid observations from an absolutely continuous distribution F . F is the class of all absolutely continuous DFs. Y = (X(1); X(2); ::; X(n)) is the full set of order statistic corresponding to X. Pn is the set of n! permutations of the rst n natural numbers. The joint pdf of (X1; X2; ::; Xn) given the full set of order statistics is

4 U Statistic & MVUE

Suppose F is the class of all absolutely continuous DFs. Then Y = (X(1); X(2); ::; X(n)) is complete su cient for F(see, Fraser,1957). Un is symmetric in the observations and hence is a function of the full set of order statistics. Thus Un is Unbiased and function of complete su cient statistics. Therefore Un is MVUE for F

5 Exact variance of U Statistic

6 Derivation of V ar(Un)

First of all note that Un is a sum of dependent variables. Therefore the covariance terms are not always zero. Thus