Limit theorems for functions of marginal quantiles

Abstract

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that √ n(1/nσ n i=1φ(X(1) n : i, ⋯ , X(d) n : i) - ȳ)=1/√nσn i=1 Zn,i + oP (1) as n→ ∞, where ȳ is a constant and Zn,i are i.i.d. random variables for each n. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations. © 2011 ISI/BS.