Brazilian Journal of Probability and Statistics

Dispersion models for geometric sums

Bent Jørgensen and Célestin C. Kokonendji

Full-text: Open access

Abstract

A new class of geometric dispersion models associated with geometric sums is introduced by combining a geometric tilting operation with geometric compounding, in much the same way that exponential dispersion models combine exponential tilting and convolution. The construction is based on a geometric cumulant function which characterizes the geometric compounding operation additively. The so-called v-function is shown to be a useful characterization and convergence tool for geometric dispersion models, similar to the variance function for natural exponential families. A new proof of Rényi’s theorem on convergence of geometric sums to the exponential distribution is obtained, based on convergence of v-functions. It is shown that power v-functions correspond to a class of geometric Tweedie models that appear as limiting distributions in a convergence theorem for geometric dispersion models with power asymptotic v-functions. Geometric Tweedie models include geometric tiltings of Laplace, Mittag-Leffler and geometric extreme stable distributions, along with geometric versions of the gamma, Poisson and gamma compound Poisson distributions.

Article information

Source
Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 263-293.

Dates
First available in Project Euclid: 22 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1313973395

Digital Object Identifier
doi:10.1214/10-BJPS136

Mathematical Reviews number (MathSciNet)
MR2832887

Zentralblatt MATH identifier
1271.62025

Keywords
Geometric compounding geometric cumulants geometric infinite divisibility geometric tilting geometric Tweedie model v-function

Citation

Jørgensen, Bent; Kokonendji, Célestin C. Dispersion models for geometric sums. Braz. J. Probab. Stat. 25 (2011), no. 3, 263--293. doi:10.1214/10-BJPS136. https://projecteuclid.org/euclid.bjps/1313973395


Export citation

References

  • Barndorff-Nielsen, O. E. (1978). Information and Exponential Families in Statistical Theory. Chichester: Wiley.
  • Blanchet, J. and Glynn, P. (2007). Uniform renewal theory with applications to expansions of random geometric sums. Adv. in Appl. Probab. 39, 1070–1097.
  • Bryc, W. (2009). Free exponential families as kernel families. Demonstratio Math. 42, 657–672.
  • Jensen, S. T. and Nielsen, B. (1997). On convergence of multivariate Laplace transforms. Statist. Probab. Lett. 33, 125–128.
  • Jørgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall.
  • Jørgensen, B., Martínez, J. R. and Tsao, M. (1994). Asymptotic behaviour of the variance function. Scand. J. Stat. 21, 223–243.
  • Jørgensen, B., Martínez, J. R. and Vinogradov, V. (2009). Domains of attraction to Tweedie distributions. Lith. Math. J. 49, 399–425.
  • Jørgensen, B., Goegebeur, Y. and Martínez, J. R. (2010). Dispersion models for extremes. Extremes 13, 399–437.
  • Jose, K. K. and Seetha Lakshmy, V. (1999). On geometric exponential distributions and its applications. J. Indian Statist. Assoc. 37, 51–58.
  • Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Dordrecht: Kluwer Academic.
  • Klebanov, L. B., Maniya, G. M. and Melamed, I. A. (1985). A problem of Zolotarev and analogues of infinitely divisible and stable distributions in the scheme of summing a random number of random variables. Theory Probab. Appl. 29, 791–794.
  • Kotz, S., Kozubowski, T. J. and Podgórski, K. (2001). The Laplace Distributions and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Boston: Birkhäuser.
  • Kozubowski, T. J. (2000). Exponential mixture representation of geometric stable distributions. Ann. Inst. Statist. Math. 52, 231–238.
  • Kozubowski, T. J. and Rachev, S. T. (1999). Univariate geometric stable laws. J. Comput. Anal. Appl. 1, 177–217.
  • Mittnik, S. and Rachev, S. T. (1991). Alternative multivariate stable distributions and their applications to financial modelling. In Stable Processes and Related Topics (S. Cambanis, G. Samorodnitsky and M. S. Taqqu, eds.) 107–119. Boston: Birkhäuser.
  • Mora, M. (1990). La convergence des fonctions variance des familles exponentielles naturelles. Ann. Fac. Sci. Toulouse Math. (5) 11, 105–120.
  • Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65–80.
  • Pillai, R. N. (1990a). Harmonic mixtures and geometric infinite divisibility. J. Indian Statist. Assoc. 28, 87–98.
  • Pillai, R. N. (1990b). On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42, 157–161.
  • Pistone, G. and Wynn, H. P. (1999). Finitely generated cumulants. Statist. Sinica 9, 1029–1052.
  • Rényi, A. (1956). A Poisson-folyamat egy jellemzíse [A characterization of Poisson processes]. Magyar Tud. Akad. Mat. Kutató, Int. Közl. 1, 519–527.
  • Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill.
  • Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference (J. K. Ghosh and J. Roy, eds.) 579–604. Calcutta: Indian Statistical Institute.
  • Vinogradov, V. (2000). On a conjecture of B. Jørgensen and A. D. Wentzell: From extreme stable laws to Tweedie exponential dispersion models. In Proceedings Volume for the International Conference on Stochastic Models in Honour of D. A. Dawson (L. Gorostiza and G. Ivanoff, eds.). Canadian Math. Society Conference Proceedings Series 26, 435–443. Providence, RI: Amer. Math. Soc.
  • Vinogradov, V. (2007). On structural and asymptotic properties of some classes of distributions. Acta Appl. Math. 97, 335–351.