Nonlinear Regression of Stable Random Variables

Abstract

Let $(X_1, X_2)$ be an $\alpha$-stable random vector, not necessarily symmetric, with $0 < \alpha < 2$. This article investigates the regression $E(X_2 \mid X_1 = x)$ for all values of $\alpha$. We give conditions for the existence of the conditional moment $E(|X_2|^p|X_1 = x)$ when $p \geq \alpha$, and we obtain an explicit form of the regression $E(X_2 \mid X_1 = x)$ as a function of $x$. Although this regression is, in general, not linear, it can be linear even when the vector $(X_1, X_2)$ is skewed. We give a necessary and sufficient condition for linearity and characterize the asymptotic behavior of the regression as $x \rightarrow \pm \infty$. The behavior of the regression functions is also illustrated graphically.