Asymptotic Normality in Nonparametric Methods

Abstract

Let $U_1, U_2, \cdots, U_N$ be a random sample from a population with a continuous distribution function and $R_i, i = 1, \cdots, N,$ be the rank of $U_i$ among the $N$ observations. Asymptotic normality is studied for the statistics of the type \begin{equation*}\tag{0.1} \sum^N_{i=1} \sum^N_{j=1} c_{ij}a_N(R_i/N, R_j/N),\end{equation*} where constants $c_{ij}$ satisfy certain negligibility conditions and the score function $a_N(\cdot, \cdot)$ is derived from a function $a(\cdot, \cdot)$ satisfying certain monotonicity and integrability conditions. It is shown that the statistic (0.1) is asymptotically equivalent to \begin{equation*}\tag{0.2} \sum^N_{i=1} \sum^N_{j=1} c_{ij}a(U_i, U_j),\end{equation*} so that the problem is reduced to a simpler one, viz. studying the asymptotic distribution of (0.2). Similar results are obtained for the two sample analog of (0.1) viz. \begin{equation*}\tag{0.3} \sum^N_{i=1} \sum^M_{j=1} c_{ij}a_{NM}(R_i/N, S_j/M)\end{equation*} where $S_j, j = 1, \cdots, M$, are the ranks corresponding to another independent random sample of size $M$ from some other population. Few more variants of the above and applications of these statistics are given. The present study is a generalization of a paper by Hajek (1961).