SOME ASYMPTOTIC RESULTS FOR A BROAD CLASS OF NONPARAMETRIC STATISTICS

This paper gives a brief survey of some of the recent results dealing with the rates of convergence, large deviation probabilities and asymptotic normality for broad classes of nonparametric statistics. The statistics are SIGMA(j=1)(N)c(Nj) a(NRNj)g(Z(Nj)) and SIGMA(j=1)(N)c(Nj) a(NRNj)(g)H(Z(Nj)) where where Z(Nj), 1 less-than-or-equal-to j less-than-or-equal-to N, are independent p-variate random variables, g and H are functions defined on R(p), R(Nj)g is the rank of X(Nj)g (=g(Z(Nj))) among X(Nk)g, 1 less-than-or-equal-to k less-than-or-equal-to N, c(Nj), 1 less-than-or-equal-to j less-than-or-equal-to N, are known constants, and a(Nj), 1 less-than-or-equal-to j less-than-or-equal-to N, are scores; and they include as special cases, the unsigned linear rank statistic, signed rank statistic, linear combinations of functions of order statistics, linear functions of concomitants of order statistics and rank combinatorial statistic, among possibly others.