# Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

• Volume: 139, Issue: 1, page 129-144
• ISSN: 0037-9484

top

## Abstract

top
If the space $𝒬$ of quadratic forms in ${ℝ}^{n}$ is splitted in a direct sum ${𝒬}_{1}\oplus ...\oplus {𝒬}_{k}$ and if $X$ and $Y$ are independent random variables of ${ℝ}^{n}$, assume that there exist a real number $a$ such that $E\left(X|X+Y\right)=a\left(X+Y\right)$ and real distinct numbers ${b}_{1},...,{b}_{k}$ such that $E\left(q\left(X\right)|X+Y\right)={b}_{i}q\left(X+Y\right)$ for any $q$ in ${𝒬}_{i}.$ We prove that this happens only when $k=2$, when ${ℝ}^{n}$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

## How to cite

top

Letac, G., and Wesołowski, J.. "Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions." Bulletin de la Société Mathématique de France 139.1 (2011): 129-144. <http://eudml.org/doc/272502>.

@article{Letac2011,
abstract = {If the space $\mathcal \{Q\}$ of quadratic forms in $\mathbb \{R\}^n$ is splitted in a direct sum $\mathcal \{Q\}_1\oplus \ldots \oplus \mathcal \{Q\}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb \{R\}^n$, assume that there exist a real number $a$ such that $E(X|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal \{Q\}_i.$ We prove that this happens only when $k=2$, when $\mathbb \{R\}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.},
author = {Letac, G., Wesołowski, J.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {symmetric cones; random matrices; characterization of Wishart laws},
language = {eng},
number = {1},
pages = {129-144},
publisher = {Société mathématique de France},
title = {Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions},
url = {http://eudml.org/doc/272502},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Letac, G.
AU - Wesołowski, J.
TI - Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 1
SP - 129
EP - 144
AB - If the space $\mathcal {Q}$ of quadratic forms in $\mathbb {R}^n$ is splitted in a direct sum $\mathcal {Q}_1\oplus \ldots \oplus \mathcal {Q}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb {R}^n$, assume that there exist a real number $a$ such that $E(X|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal {Q}_i.$ We prove that this happens only when $k=2$, when $\mathbb {R}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.
LA - eng
KW - symmetric cones; random matrices; characterization of Wishart laws
UR - http://eudml.org/doc/272502
ER -

## References

top
1. [1] S. Andersson – « Invariant normal models », Ann. Statist.3 (1975), p. 132–154. Zbl0373.62029MR362703
2. [2] D. J. Bartlett – « On the theory of the statistical regression », Proc. Royal Soc. Edinburgh53 (1933), p. 260–283. Zbl0008.02402JFM59.0513.04
3. [3] K. Bobecka & J. Wesołowski – « The Lukacs-Olkin-Rubin theorem without invariance of the “quotient” », Studia Math.152 (2002), p. 147–160. Zbl0993.62043MR1916547
4. [4] E. M. Carter – « Characterization and testing problems in the complex Wishart distribution », Thèse, University of Toronto, 1975. MR2627434
5. [5] M. Casalis – « Les familles exponentielles à variance quadratique homogène sont des lois de Wishart sur un cône symétrique », C. R. Acad. Sci. Paris Sér. I Math.312 (1991), p. 537–540. Zbl0745.62051MR1099688
6. [6] M. Casalis & G. Letac – « Characterization of the Jørgensen set in generalized linear models », Test3 (1994), p. 145–162. Zbl0815.62030MR1293112
7. [7] —, « The Lukacs-Olkin-Rubin characterization of Wishart distributions on symmetric cones », Ann. Statist.24 (1996), p. 763–786. Zbl0906.62053MR1394987
8. [8] M. L. Eaton – Multivariate statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., 1983. Zbl0587.62097MR716321
9. [9] J. Faraut & A. Korányi – Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 1994, Oxford Science Publications. Zbl0841.43002MR1446489
10. [10] N. R. Goodman – « Statistical analysis based on a certain multivariate complex Gaussian distribution. (An introduction) », Ann. Math. Statist.34 (1963), p. 152–177. Zbl0122.36903MR145618
11. [11] S. Gyndikin – « Invariant generalized functions in homogeneous spaces », J. Funct. Anal. Appl.9 (1975), p. 50–52. Zbl0332.32022
12. [12] S. T. Jensen – « Covariance hypotheses which are linear in both the covariance and the inverse covariance », Ann. Statist.16 (1988), p. 302–322. Zbl0653.62042MR924873
13. [13] R. G. Laha & E. Lukacs – « On a problem connected with quadratic regression », Biometrika47 (1960), p. 335–343. Zbl0093.16002MR121922
14. [14] M. Lassalle – « Algèbre de Jordan et ensemble de Wallach », Invent. Math.89 (1987), p. 375–393. Zbl0622.22008MR894386
15. [15] G. Letac – « Le problème de la classification des familles exponentielles naturelles de ${𝐑}^{d}$ ayant une fonction variance quadratique », in Probability measures on groups, IX (Oberwolfach, 1988), Lecture Notes in Math., vol. 1379, Springer, 1989, p. 192–216. Zbl0679.62010MR1020532
16. [16] G. Letac & H. Massam – « Quadratic and inverse regressions for Wishart distributions », Ann. Statist.26 (1998), p. 573–595. Zbl1073.62536MR1626071
17. [17] G. Letac & J. Wesołowski – « Laplace transforms which are negative powers of quadratic polynomials », Trans. Amer. Math. Soc.360 (2008), p. 6475–6496. Zbl1152.60019MR2434295
18. [18] E. Lukacs – « A characterization of the gamma distribution », Ann. Math. Statist.26 (1955), p. 319–324. Zbl0065.11103MR69408
19. [19] M. L. Mehta – Random matrices, third éd., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. Zbl1107.15019MR2129906
20. [20] R. J. Muirhead – Aspects of multivariate statistical theory, John Wiley & Sons Inc., 1982. Zbl0556.62028MR652932
21. [21] I. Olkin & H. Rubin – « A characterization of the Wishart distribution », Ann. Math. Statist.33 (1962), p. 1272–1280. Zbl0111.34202MR141186
22. [22] S. D. Peddada & D. S. P. Richards – « Proof of a conjecture of M. L. Eaton on the characteristic function of the Wishart distribution », Ann. Probab.19 (1991), p. 868–874. Zbl0728.62053MR1106290
23. [23] D. N. Shanbhag – « The Davidson–Kendall problem and related results on the structure of the Wishart distribution », Austr. J. Statist. 30A (1988), p. 272–280. Zbl0694.62024
24. [24] Y. H. Wang – « Extensions of Lukacs’ characterization of the gamma distribution », in Analytical methods in probability theory (Oberwolfach, 1980), Lecture Notes in Math., vol. 861, Springer, 1981, p. 166–177. Zbl0459.60012MR655271
25. [25] J. Wishart – « The generalised product moment distribution in samples from a normal multivariate population », Biometrika 20A (1928), p. 32–52. JFM54.0565.02

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.