5 Large Sample Properties of U statistics

Mr Taranga Mukherjee

1 Large Sample Properties

Unbiasedness and possessing minimum variance are the small sample properties of an esti-mator. To judge the performance of an estimator for large sample sizes, one uses Consistency and asymptotic normality. Consistency gives the limiting value whereas asymptotic normal-ity speci es the rate of convergence to the limiting value. A consistent estimator having asymptotic normality is known as Consistent Asymptotic Normal(CAN). U statistic is a CAN estimator.

The rst term in the RHS of (*) converges in distribution to a 2 21 distribution. The second term converges to 2 in probability. Now an application of Slutsky’s Theorem gives that the LHS of (*) converges to a 2( 21 1) distribution. Thus we need n as the normalizing factor to get a non-degenerate limiting distribution. However, the limiting distribution under degeneracy is no longer normal.