Stefan S. Ralescu

Necessary and sufficient conditions for the asymptotic normality of perturbed sample quantiles

Abstract We deal with perturbed sample quantiles based on a kernel k and a sequence of window-width an > 0. Under minimal assumptions on the underlying cumulative distribution and the kernel k, necessary and sufficient conditions for the central limit theorem to hold for these quantiles are found for the sequence {an}. Our results (i) generalize the central limit theorem of Nadaraya (1964), and (ii) extend results of Chanda (1975) and Falk (1985). Several applications are included.

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.

Asymptotic expansions for sums of nonidentically distributed Bernoulli random variables

This paper concerns an asymptotic expansion for the distribution of the sum of independent zero-one random variables in case where this surn has variance [sigma]n2 --> [infinity]. The expansion presented is given to the order O([sigma]n-2). An application to the study of the exact rate of convergence in the central limit theorem for intermediate order statistics is included.

Stein estimation for non-normal spherically symmetric location families in three dimensions

We consider estimation of a location vector in the presence of known or unknown scale parameter in three dimensions. The technique of proof is Steins integration by parts and it is used to cover several cases (e.g., non-unimodal distributions) for which previous results were known only in the cases of four and higher dimensions. Additionally, we give a necessary and sufficient condition on the shrinkage constant for improvement on the usual estimator for the spherical uniform distribution.