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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


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.


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Clyde D. Hardin Jr. Gennady Samorodnitsky. Murad S. Taqqu. "Nonlinear Regression of Stable Random Variables." Ann. Appl. Probab. 1 (4) 582 - 612, November, 1991.


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

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

Primary: 60E07
Secondary: 60E10 , 62J02

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

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.1 • No. 4 • November, 1991
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