Open Access
Translator Disclaimer
November, 1991 Nonlinear Regression of Stable Random Variables
Clyde D. Hardin Jr, Gennady Samorodnitsky, Murad S. Taqqu
Ann. Appl. Probab. 1(4): 582-612 (November, 1991). DOI: 10.1214/aoap/1177005840

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.

Citation

Download Citation

Clyde D. Hardin Jr. Gennady Samorodnitsky. Murad S. Taqqu. "Nonlinear Regression of Stable Random Variables." Ann. Appl. Probab. 1 (4) 582 - 612, November, 1991. https://doi.org/10.1214/aoap/1177005840

Information

Published: November, 1991
First available in Project Euclid: 19 April 2007

MathSciNet: MR1129776
Digital Object Identifier: 10.1214/aoap/1177005840

Subjects:
Primary: 60E07
Secondary: 60E10 , 62J02

Keywords: Linear regression , Stable random vectors , symmetric $\alpha$-stable

Rights: Copyright © 1991 Institute of Mathematical Statistics

JOURNAL ARTICLE
31 PAGES


SHARE
Vol.1 • No. 4 • November, 1991
Back to Top