Limit theorems for induced extreme values

Abstract:

For a bivariate sample (Xi, Yi) of size n, let (U(x), V(x)) denote the following pair of induced extreme values: U(x) is the maximum of those Yi-values with corresponding Xi-value less than x and V(x) is the maximum of the remaining Yi-values. In the paper, we study the asymptotic behavior of the (suitably normalized) random vector (U(x), V(x)), and we consider several cases. First, we consider nonrandom x and let x=xn so that as n→∞, xn tends to the endpoint of FX(x), or so that xn tends to x0, a point in the support of FX(x). The second important situation appears when x=Xk:n, i.e., we select Y-values on the basis of the random variable Xk:n, the k-th order-statistic of the X-sample. Here we also consider two cases: (i) k=n-j with fixed j, and (ii) k=[np], where 0<p<1. The paper generalizes the earlier results of David, Joshi, and Nagaraja, where it is assumed that (X, Y) is in the bivariate (max-) domain of attraction of a bivariate stable law with independent marginals. © 1995 Plenum Publishing Corporation.

Año de publicación:

1995

Keywords:

Fuente:

scopus