The Annals of Applied Probability

Nonlinear Regression of Stable Random Variables

Clyde D. Hardin Jr, Gennady Samorodnitsky, and Murad S. Taqqu

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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.

Article information

Ann. Appl. Probab., Volume 1, Number 4 (1991), 582-612.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)


Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 62J02: General nonlinear regression 60E10: Characteristic functions; other transforms

Stable random vectors linear regression symmetric $\alpha$-stable


Jr, Clyde D. Hardin; Samorodnitsky, Gennady; Taqqu, Murad S. Nonlinear Regression of Stable Random Variables. Ann. Appl. Probab. 1 (1991), no. 4, 582--612. doi:10.1214/aoap/1177005840.

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