Characterizations via Linear Regression of Ordered Random Variables: A Unifying Approach

Abstract

Characterizations of distributions based on linear regressions

have been investigated quite extensively in the literature of order statistics and record values (r ∈ ℕ, l ≥ 1). Since generalized order statistics provide a unifying approach to these and other models of ordered random variables, this setup is considered in order to present a comprehensive solution of related characterization problems. In the present article, the above characterization problem is solved in the family of continuous distribution functions (h and g are given functions), where denote generalized order statistics. It is shown that the results hold for arbitrary generalized order statistics subject to the existence of the conditional expectations. The investigation starts with the case of adjacent generalized order statistics, i.e., l = 1. In a first step, a simple inversion formula is given that connects an appropriate conditional expectation to a distribution function F. Afterwards, some analytical conditions are presented that lead to characterizations of the distribution function by the conditional expectations of adjacent generalized order statistics. If the gap l is of higher order the calculations become more difficult. In order to get an explicit result, we restrict ourselves to a linear function g. Utilizing a result on integrated Cauchy functional equations, it is shown that the linearity of the conditional expectation yields a characterization of generalized Pareto distributions.

Acknowledgments