Characterizations based on moments of the number of observations near-order statistics

Abstract

Let X1,..., Xn be independent and identically distributed (iid) real-valued random<br><br>variables with continuous distribution function F, and denote the corresponding<br><br>ith order statistic by Xi:n. Two random variables K+(n;k;a) = #fi : Xi 2 [Xk:n;Xk:n +<br><br>a)g and K(n;k;a) = #fi : Xi 2 (Xk:n a;Xk:n]g have been defined in the literature,<br><br>where k = 1; :::;n and a &gt; 0 is a constant. Several articles have been published to<br><br>study asymptotic behavior of the aforementioned random variables, see for example<br><br>Balakrishnan and Stepanov (2005) and Dembinska et al. (2007). We provide some<br><br>characterization results of F based on moments of K+(n;k;a) and K(n;k;a) using the<br><br>concept of completeness (Higgins, 2004) and the method of general solution of the<br><br>functional equation (Aczél, 1966).