## The Annals of Statistics

### Quadratic and inverse regressions for Wishart distributions

#### Abstract

If $U$ and $V$ are independent random variables which are gamma distributed with the same scale parameter, then there exist $a$ and $b$ in $\mathbb{R}$ such that $$\mathbb{E}(U|U + V) = a(U + V)$$ and $$\mathbb{E}(U^2|U + V) = b(U + V)^2$$. This, in fact, is characteristic of gamma distributions. Our paper extends this property to the Wishart distributions in a suitable way, by replacing the real number $U^2$ by a pair of quadratic functions of the symmetric matrix $U$. This leads to a new characterization of the Wishart distributions, and to a shorter proof of the 1962 characterization given by Olkin and Rubin. Similarly, if $\mathbb{E}(U^{-1})$ exists, there exists $c$ in $\mathbb{R}$ such that $$\mathbb{E}(U^{-1}|U + V) = c(U + V)^{-1}$$. Wesołowski has proved that this also is characteristic of gamma distributions. We extend it to the Wishart distributions. Finally, things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebras.

#### Article information

Source
Ann. Statist., Volume 26, Number 2 (1998), 573-595.

Dates
First available in Project Euclid: 31 July 2002

https://projecteuclid.org/euclid.aos/1028144849

Digital Object Identifier
doi:10.1214/aos/1028144849

Mathematical Reviews number (MathSciNet)
MR1626071

Zentralblatt MATH identifier
1073.62536

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 60E10: Characteristic functions; other transforms

#### Citation

Letac, Gérard; Massam, Hélène. Quadratic and inverse regressions for Wishart distributions. Ann. Statist. 26 (1998), no. 2, 573--595. doi:10.1214/aos/1028144849. https://projecteuclid.org/euclid.aos/1028144849

#### References

• CASALIS, M. 1990. Familles exponentielles naturelles invariantes par un groupe. These, Univ. Paul Sabatier. Z.
• CASALIS, M. 1992. Un estimateur de la variance pour une famille exponentielle a fonction variance quadratique. C.R. Acad. Sci. Paris Ser. I, Math. 314 143 146. ´ Z.
• CASALIS, M. and LETAC, G. 1996. The Lukacs Olkin Rubin characterization of the Wishart distributions on sy mmetric cones. Ann. Statist. 24 763 786. Z.
• DAS GUPTA, S. 1968. Some aspects of discrimination function coefficients. Sankhy a Ser. A 30 387 400. Z.
• FARAUT, J. and KORANy I, A. 1994. Analy sis on Sy mmetric Cones. Oxford Univ. Press.
• HUANG, W.-J., LI, S.-H. and HUANG, M.-N. L. 1994. Characterizations of the Poisson process as a renewal process via two conditional moments. Ann. Inst. Statist. Math. 46 351 360. Z.
• KAWATA, Y. 1972. Fourier Analy sis and Probability. Wiley, New York. Z. Z
• LETAC, G. and MASSAM, H. 1997. Representations of an exponential dispersion model with an. appendix by E. Neher. Technical report, Univ. Paul Sabatier. Z.
• LUKACS, E. 1955. A characterization of the gamma distribution. Ann. Math. Statist. 26 319 324. Z.
• MASSAM, H. 1994. An exact decomposition theorem and a unified view of some related distributions for a class of exponential transformation models on sy mmetric cones. Ann. Statist. 22 369 394. Z.
• MASSAM, H. and NEHER, E. 1997. On transformations and determinants of Wishart variables on sy mmetric cones. J. Theoret. Probab. 10 867 902. Z.
• MUIRHEAD, R. 1982. Aspects of Multivariate Analy sis. Wiley, New York. Z.
• OLKIN, I. and RUBIN, H. 1962. A characterization of the Wishart distribution. Ann. Math. Statist. 33 1272 1280. Z.
• RAO, C. R. 1948. On a problem of Ragnar Frish. Econometrica 15 245 249. Z.
• SRIVASTAVA, M. S. 1965. On complex Wishart distribution. Ann. Math. Statist. 36 313 317. Z.
• VON ROSEN, D. 1988. Moments for the inverted Wishart distribution. Scand. J. Statist. 15 97 109. Z.
• WANG, F. 1981. Extensions of Lukacs' characterization of the gamma distribution. Analy tic Methods in Probability Theory. Lecture Notes in Math. 861 166 177. Springer, New York. Z.
• WESOLOWSKI, J. 1990. A constant regression characterization of the gamma law. Adv. in Appl. Probab. 22 488 489.
• CHARLOTTESVILLE, VIRGINIA 22903