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

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