Munsup Seoh

Edgeworth expansion for signed linear rank statistics with regression constants

Abstract An Edgeworth expansion with remainder o(N−1) is obtained for signed linear rank statistics under suitable assumptions. The theorem is proved for a wide class of score generating functions including the Chi-quantile function by adapting van Zwets methodand Doess conditioning arguments.

Edgeworth expansions for signed linear rank statistics under near location alternatives

Abstract Edgeworth expansions with the uniform remainder of order o ( N −1 ) are established for signed linear rank statistics with regression constants under near location alternatives. The results are obtained both with exact scores and with approximate scores, normalized with natural parameters as well as with simple constants.

Central limit theorems under alternatives for a broad class of nonparametric statistics

Abstract Asymptotic normality, under the assumption that underlying distributions are only independent, is established for a class of nonparametric statistics which includes as special cases unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear functions of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

Asymptotic Normality of a Class of Nonparametric Statistics

Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

Berry-Esséen rate in asymptotic normality for perturbed sample quantiles

In estimating quantiles with a sample of sizeN obtained from a distributionF, the perturbed sample quantiles based on a kernel functionk have been investigated by many authors. It is well known that their behaviour depends on the choices of “window-width”, saywN. Under suitable and reasonably mild assumptions onF andk, Ralescu and Sun (1993) have recently proven that limN→∞N1/4wN=0 is the necessary and sufficient condition for the asymptotic normality of the perturbed sample quantiles. In this paper, their rate of convergence is investigated. It turns out that the optimal Berry-Esséen rate ofO(N−1/2) can be achieved by choosing the window-width suitably, saywN=O(N−1/2). The obtained results, in addition to being explicit enough to verify the sufficient condition for the asymptotic normality, improve Ralescus (1992) result of which the rate is of order (logN)N−1/2.

On the relative error in normal approximation for signed rank statistics with a broad range of scores

Abstract The Cramer-type large deviation results in the optimal range of 0 1 4 ) are obtained on the relative error in normal approximation for the signed rank statistics under the symmetry hypothesis. The methods used are applications of elementary conditional probability and Fellers Theorem 1 [Trans. Amer. Math. Soc. 54 (1943), 361–372] (as Kallenberg, Z. Wahrsch. Verw. Geb. 60 (1982), 403–409, did to obtain the optimal range of 0 1 6 ) for the simple linear rank statistics). The results obtained are valid with a broad range of regression constants and scores (allowed to be generated by discontinuous functions, but not necessarily) restricted by mild conditions, while most of previous results for the problem dealing with rank statistics are obtained with a narrow range of scores generated by differentiable and bounded functions, and/or with severely restricted regression constants.

An elementary proof of CLT for signed rank statistics with regression constants under hypothesis of symmetry

Abstract This note gives a short proof, based on the conditioning arguments, of the well-known asymptotic normality of the signed rank statistics with regression constants under the hypothesis of symmetry.