Numbers of near-maxima for the bivariate case

Abstract

Let (Z) over bar (1) = (X-1, Y-1)....(Z) over bar (n) = (X-n, Y-n) be independent and identically distributed random vectors with continuous distribution. Let K-n(a, b(1), b(2)) be the number of sample elements that belong to the open rectangle (X-max((n)) - a, X-max((n))) x (Y-max((n)) - b(1), Y-max((n)) + b(2)) - numbers of near-maxima in the bivariate case. in the present paper, we discuss asymptotic properties of K-n (a, b(1), b(2)) and K-n(infinity, 0, infinity). (C) 2009 Elsevier B.V. All rights reserved.