[A characterization of simple quadratic natural exponential families with a reverse martingale property]
López-Blázquez and Salamanca-Miño [7] obtained a characterization of natural exponential families with quadratic variance functions via a reverse martingale property of certain orthogonal polynomials. We extend this result to the multidimensional case yielding a new characterization of natural exponential families with simple quadratic variance functions.
Dans cette Note, nous étendons un résultat de López-Blázquez et Salamanca-Miño [7] en obtenant une caractérisation des familles exponentielles naturelles quadratiques simples sur . Cette caractérisation repose sur une propriété de martingale inverse vérifiée par des familles de polynômes multivariés.
Revised:
Published online:
David Gutierrez-Rubio 1; Fernando López-Blázquez 2; Denys Pommeret 2
@article{CRMATH_2002__334_5_405_0, author = {David Gutierrez-Rubio and Fernando L\'opez-Bl\'azquez and Denys Pommeret}, title = {Une caract\'erisation des familles exponentielles naturelles quadratiques simples par une propri\'et\'e de martingale inverse}, journal = {Comptes Rendus. Math\'ematique}, pages = {405--409}, publisher = {Elsevier}, volume = {334}, number = {5}, year = {2002}, doi = {10.1016/S1631-073X(02)02244-6}, language = {fr}, }
TY - JOUR AU - David Gutierrez-Rubio AU - Fernando López-Blázquez AU - Denys Pommeret TI - Une caractérisation des familles exponentielles naturelles quadratiques simples par une propriété de martingale inverse JO - Comptes Rendus. Mathématique PY - 2002 SP - 405 EP - 409 VL - 334 IS - 5 PB - Elsevier DO - 10.1016/S1631-073X(02)02244-6 LA - fr ID - CRMATH_2002__334_5_405_0 ER -
%0 Journal Article %A David Gutierrez-Rubio %A Fernando López-Blázquez %A Denys Pommeret %T Une caractérisation des familles exponentielles naturelles quadratiques simples par une propriété de martingale inverse %J Comptes Rendus. Mathématique %D 2002 %P 405-409 %V 334 %N 5 %I Elsevier %R 10.1016/S1631-073X(02)02244-6 %G fr %F CRMATH_2002__334_5_405_0
David Gutierrez-Rubio; Fernando López-Blázquez; Denys Pommeret. Une caractérisation des familles exponentielles naturelles quadratiques simples par une propriété de martingale inverse. Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 405-409. doi : 10.1016/S1631-073X(02)02244-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02244-6/
[1] Information and Exponential Families, Wiley, New York, 1978
[2] Fundamental of Statistical Exponential Families, Lecture Notes – Monograph Series, 1986
[3] The 2d+4 simple quadratic natural exponential families on , Ann. Statist., Volume 24 (1996), pp. 1828-1854
[4] Some classes of orthogonal polynomials associated with martingales, Proc. Amer. Math. Soc., Volume 98 (1986), pp. 298-302
[5] Exponential Families of Stochastic Processes, Springer Series Statist., 1997
[6] Lectures on natural Exponential Families and their Variance Functions, Monografias de Matemática, 50, Instituto de Matemática Pura e Aplicada, Río de Janeiro, Brésil, 1992
[7] A characterization of natural exponential families with quadratic variance functions by a reverse martingale property, Statist. Probab. Lett., Volume 49 (2000), pp. 63-68
[8] Natural exponential families with quadratic variance functions, Ann. Statist., Volume 10 (1982), pp. 65-82
[9] Natural exponential families with quadratic variance functions: statistical theory, Ann. Statist., Volume 11 (1983), pp. 512-529
[10] Orthogonal polynomials and natural exponential families, Test, Volume 5 (1996), pp. 77-111
[11] Orthogonality of the Sheffer system associated to a Lévy process, J. Statist. Plann. Inference, Volume 86 (2000), pp. 1-10
[12] Diagonality of the Bhattacharyya matrix as a characterization, Theory Probab. Appl., Volume 24 (1979), pp. 430-433
[13] Lévy processes, polynomials and martingales, Stochastic Models, Volume 14 (1998), pp. 335-349
Cited by Sources:
Comments - Policy