Abstract
A polynomial Q = Q(X 1, …, X n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α 1(F), …, α m (F)), where q(u 1, …, u m ) is a polynomial in u 1, …, u m and α j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X 1 + … + X n) = 0 holds if and only if q(α 1(θ), …, α m (θ)) ≡ 0, where α j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X 1 + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F θ(x) = ∫ x−∞ e θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.
Similar content being viewed by others
References
S. K. Bar-Lev, “Discussion on Paper by B. Jorgensen, ‘Exponential Dispersion Models (with Discussion)’”, J. Roy. Statist. Soc. Ser. B 49, 153–154 (1987).
S. K. Bar-Lev, “Methods of Constructing Characterizations by Constancy of Regression on the Sample Mean and Related Problems for NEF’s”, Math. Methods Statist. 16, 96–109 (2007).
S. K. Bar-Lev and D. Bshouty, “A Class of Infinitely Divisible Variance Functions with an Application to the Polynomial Case”, Statist. Probab. Lett. 10, 377–379 (1990).
S. K. Bar-Lev and P. Enis, “Reproducibility and Natural Exponential Families with Power Variance Functions”, Ann. Statist. 14, 1507–1522 (1986).
S. K. Bar-Lev and O. Stramer, “Characterizations of Natural Exponential Families with Power Variance Functions by Zero Regression Properties”, Probab. Theory Rel. Fields 76, 509–522 (1987).
E. B. Fosam and D. N. Shanbhag, “An Extended Laha-Lukacs Characterization Result Based on a Regression Property”, J. Statist. Planning Inference 63, 173–186 (1997).
A. Hassairi and M. Zarai, “Bhattacharyya Matrices and Cubic Exponential Family”, Statist. Methodology 2, 226–232 (2005).
B. Heller, “Special Functions and Characterizations of Probability Distributions by Zero Regression Properties”, J.Multivariate Anal. 13, 473–487 (1983).
J. Hinde and C. G. B. Demétrio, Overdispersion: Models and Estimation. (São Paulo, ABE, 1998).
A.M. Kagan, “Linearity of Regression of the Third Sample Moment on the Sample Average”, Metron XLVIII, 33–37 (1990).
A.M. Kagan, Yu. V. Linnik, and C. R. Rao, Characterization Problems in Mathematical Statistics (Wiley, New York, 1973).
C. C. Kokonendji, C. G. B. Demétrio, and S. S. Zocchi, “On Hinde-Demétrio Regression Models for Overdispersed Count Data”, Statist. Methodology 4, 277–291 (2007).
C. C. Kokonendji, S. Dossou-Gbété, and C. G. B. Demétrio, “Some Discrete Exponential Discrete Models: Poisson-Tweedie and Hinde-Demétrio Classes”, Statist. Oper. Research Trans. 28, 201–214 (2004).
R. G. Laha and E. Lukacs, “On a Problem Connected with Quadratic Regression”, Biometrika 47, 335–345 (1960).
E. Lukacs and R. G. Laha, Applications of Characteristic Functions (Griffin, London, 1964).
G. Letac and M. Mora, “Natural Real Exponential Families with Cubic Variance Functions”, Ann. Statist. 18, 1–37 (1990).
E. Lukacs, “A Characterization of the Normal Distribution”, Ann.Math. Statist. 13, 91–93 (1942).
C. N. Morris, “Natural Exponential Families with Quadratic Variance Functions”, Ann. Statist. 10, 65–82 (1982).
C. R. Rao and D. N. Shanbhag, “Characterizations Based on Regression Properties: Improved Versions of Recent Results”, Sankhya, Ser.A 57, 167–178 (1995).
D. Shanbhag, “Some Characterizations Based on the Bhattacharyya Matrix”, J. Appl. Probab. 9, 580–587 (1972).
D. Shanbhag, “Diagonality of Bhattacharyya Matrix as a Characterization”, Theory Probab. Appl. 24, 430–433 (1979).
M. C. K. Tweedie, “The Regression of the Sample Variance on the Sample Mean”, J. LondonMath. Soc. 21, 22–28 (1946).
M. C. K. Tweedie, “An Index which Distinguishes between Some Important Exponential Families”, in: Statistics: Applications and New Directions, Proc. Indian Statistical Institute, Golden Jubilee International Conference, Ed. by J. K. Ghosh and J. Roy (Indian Statist. Inst., Calcutta, 1984), pp. 579–604.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Bar-Lev, S.K., Kagan, A.M. Regression of polynomial statistics on the sample mean and natural exponential families. Math. Meth. Stat. 18, 201–206 (2009). https://doi.org/10.3103/S1066530709030016
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530709030016