Probability and Mathematical Statistics: Vol. 9, Fasc. 1, Manfred Denker and Madan L. Puri, An invariance principle for processes indexed by two parameters and some statistical applications

AN INVARIANCE PRINCIPLE FOR PROCESSES INDEXED BY TWO PARAMETERS AND SOME STATISTICAL APPLICATIONS

Manfred Denker
Madan L. Puri

Abstract: Let $D((0,1]2)$ denote the space of all functions on $(0,1]2$ with no discontinuities of the second kind. We prove weak invariance principles in the space $D((0,1]2)$ for processes of the form $integral h(H (t))dF (t), n+m n$ $m, n > 1,$ where $F n$ and $G m$ are two independent empirical distribution functions of independent, identically distributed sequences of random variables,

H = (n +m +1)-1(nF + mG ), n+m n m

and where $h$ belongs to a certain class of functions on the open unit interval. The appropriate topology in $D((0,1]2)$ is given by uniform convergence on compact sets. This type of processes is central in nonparametric statistics having applications to two-sample linear rank statistics and signed rank statistics.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

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