A Characterization of the Exponential and Related Distributions by Linear Regression

Abstract

Let $X_1, \cdots, X_n (n \geqslant 2)$ be a random sample on a rv $X$, and $Y_1 < \cdots < Y_n$ be the corresponding order statistics. Define $Z_k = \frac{1}{n-k}\Sigma^n_{i=k+1}(Y_i - Y_k), 1 \leqslant k \leqslant n - 1, W_k = \frac{1}{k-1}\Sigma^{k-1}_{i=1}(Y_k - Y_i),$ $2 \leqslant k \leqslant n$. Using the properties $E(Z_k\mid Y_k = y) = \alpha y + \beta$ and $E(W_k\mid Y_k = y) = \alpha y + \beta$, a.e. $(dF)$, where $\alpha$ and $\beta$ are constants, we obtain characterizations of several distributions which include the exponential, the Pearson (type I) and the Pareto (of the second kind) distributions.