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Geometric Tweedie regression models for continuous and semicontinuous data with variation phenomenon

  • Rahma AbidEmail author
  • Célestin C. Kokonendji
  • Afif Masmoudi
Original Paper
  • 15 Downloads

Abstract

We introduce a new class of regression models based on the geometric Tweedie models (GTMs) for analyzing both continuous and semicontinuous data, similar to the recent and standard Tweedie regression models. We also present a phenomenon of variation with respect to the equi-varied exponential distribution, where variance is equal to the squared mean. The corresponding power v-functions which characterize the GTMs, obtained in turn by exponential-Tweedie mixture, are transformed into variance to use the conventional generalized linear models. The real power parameter of GTMs works as an automatic distribution selection such for asymmetric Laplace, geometric-compound-Poisson-gamma and geometric-Mittag-Leffler. The classification of all power v-functions reveals only two border count distributions, namely geometric and geometric-Poisson. We establish practical properties, into the GTMs, of zero-mass and variation phenomena, also in connection with some reliability measures. Simulation studies show that the proposed model highlights asymptotic unbiased and consistent estimators, despite the general over-variation. We illustrate two applications, under- and over-varied, on real datasets to a time to failure and time to repair in reliability; one of which has positive values with many achievements in zeros. We finally make concluding remarks, including future directions.

Keywords

Coefficient of variation Exponential mixture Generalized linear models Geometric dispersion models Reliability v-function Zero-mass 

Mathematics Subject Classification

62J12 62F10 62E10 62E15 62J99 

Notes

Acknowledgements

We sincerely thank the Associate Editor and two anonymous referees for their valuable comments and constructive suggestions. Part of this work was performed while the first author was at the Laboratoire de Mathématiques de Besançon as a visiting scientist, partly funded by The University of Sfax.

References

  1. Abid, R., Kokonendji, C.C., Masmoudi, A.: Geometric dispersion models with real quadratic v-functions. Stat. Probab. Lett. 145, 197–204 (2019)MathSciNetCrossRefGoogle Scholar
  2. Andersen, D.A., Bonat, W.H.: Double generalized linear compound Poisson models to insurance claims data. Electr. J. Appl. Stat. Anal. 10, 384–407 (2017)MathSciNetGoogle Scholar
  3. Aryuyuen, S., Bodhisuwan, W.: The negative binomial-generalized exponential (NB-GE) distribution. Appl. Math. Sci. 7, 1093–1105 (2013)MathSciNetCrossRefGoogle Scholar
  4. Barlow, R.A., Proschan, F.: Statistical Theory of Reliability and Life Testing: Probability Models. To begin with, Silver Springs, Maryland (1981)Google Scholar
  5. Beasly, K., Ebeling, C.: The determination of operational and support requirements and costs during the conceptual design of space systems final report. National Aeronautics and Space Administration National; Technical Information Service, Distributor, Washington, DC (1992)Google Scholar
  6. Bonat, W.H., Kokonendji, C.C.: Flexible Tweedie regression models for continuous data. J. Stat. Comput. Simul. 87, 2138–2152 (2017)MathSciNetCrossRefGoogle Scholar
  7. Bonat, W.H., Jørgensen, B., Kokonendji, C.C., Hinde, J., Demétrio, C.G.B.: Extended Poisson–Tweedie: properties and regression models for count data. Stat. Model. 18, 24–49 (2018a)MathSciNetCrossRefGoogle Scholar
  8. Bonat, W.H., Petterle, R.R., Hinde, J., Demétrio, C.G.B.: Flexible quasi-beta regression models for continuous bounded data. Stat. Model. (2018b)  https://doi.org/10.1177/1471082X18790847
  9. Cahoy, D.O.: Estimation of Mittag-Leffler parameters. Commun. Stat. Simul. Comput. 42, 303–315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cahoy, D.O., Uhaikin, V.V., Woyczynski, W.A.: Parameter estimation for fractional Poisson processes. J. Stat. Plann. Inference 140, 3106–3120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dudenhoeffer, D.D., Gaver, D.P., Jacobs, P.A.: Failure, repair and replacement analyses of a navy subsystem: case study of a pump. J. Appl. Stoch. Mod. Data Anal. 13, 369–376 (1998)zbMATHGoogle Scholar
  12. Dunn, P.K.: The R package tweedie: Tweedie exponential family models version 2.1.7. (2013). http://cran.r-project.org/web/packages/tweedie/tweedie. Accessed 15 Feb 2013
  13. Engel, B., te Brake, J.: Analysis of embryonic development with a model for under- or overdispersion relative to binomial variation. Biometrics 49, 269–279 (1993)CrossRefzbMATHGoogle Scholar
  14. Iwiǹska, M., Popowska, B.: Characterizations of the exponential distribution by geometric compound. Fasc. Math. 47, 5–10 (2011)MathSciNetzbMATHGoogle Scholar
  15. Iwiǹska, M., Szymkowiak, M.: Characterizations of the exponential distribution by Pascal compound. Commun. Stat. Theory Methods 45, 63–70 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Iwiǹska, M., Szymkowiak, M.: Characterizations of distributions through selected functions of reliability theory. Commun. Stat. Theory Methods 46, 69–74 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jørgensen, B.: The Theory of Dispersion Models. Chapman & Hall, London (1997)zbMATHGoogle Scholar
  18. Jørgensen, B., Knudsen, S.J.: Parameter orthogonality and bias adjustment for estimating functions. Scand. J. Stat. 31, 93–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Jørgensen, B., Kokonendji, C.C.: Dispersion models for geometric sums. Braz. J. Probab. Stat. 25, 263–293 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Jørgensen, B., Kokonendji, C.C.: Discrete dispersion models and their Tweedie asymptotics. AStA Adv. Stat. Anal. 100, 43–78 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kalashnikov, V.: Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic, Dordrecht (1997)CrossRefzbMATHGoogle Scholar
  22. Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data. Wiley, New York (2002)Google Scholar
  23. Kemp, A.W.: Classes of discrete lifetime distributions. Commun. Stat. Theory Methods 33, 3069–3093 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Klein, J.P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data. Springer, New York (2003)zbMATHGoogle Scholar
  25. Kline, M.B.: Suitability of the lognormal distribution for corrective maintenance repair times. Reliab. Engine 9, 65–80 (1984)CrossRefGoogle Scholar
  26. Kokonendji, C.C.: Over- and underdispersion models. In: Balakrishnan, N. (ed.) The Wiley Encyclopedia of Clinical Trials: Methods and Applications of Statistics in Clinical Trials (Chap. 30), vol. 2, pp. 506–526. Wiley, New York (2014)Google Scholar
  27. Kokonendji, C.C., Puig, P.: Fisher dispersion index for multivariate count distributions: a review and a new proposal. J. Multivar. Anal. 165, 180–193 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kotz, S., Kozubowski, T., Podgórski, K.: The Laplace Distributions and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  29. Kumar, A., Saini, M.: Cost-benefit analysis of a single-unit system with preventive maintenance and Weibull distribution for failure and repair activities. J. Appl. Math. Stat. Inform. 10, 5–19 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Kundu, D.: Geometric skew normal distribution. Sankhya B 76, 176–189 (2014)Google Scholar
  31. Liu, X.: Survival Analysis: Models and Applications. Wiley, Chichester (2012)CrossRefzbMATHGoogle Scholar
  32. McCullagh, P., Nelder, J.: Generalized Linear Models, 2nd edn. Chapman & Hall, London (1989)CrossRefzbMATHGoogle Scholar
  33. Nolan, J.P.: Stable Distributions: Models for Heavy Tailed Data. American University, Washington, DC (2006)Google Scholar
  34. Pillai, R.N.: On Mittag-Leffler functions and related distributions. Ann. Stat. 42, 157–161 (1990)MathSciNetzbMATHGoogle Scholar
  35. R Core Team: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2018)Google Scholar
  36. Smyth, G.K.: Generalized linear models with varying dispersion. J. R. Stat. Soc. B 51, 47–60 (1989)MathSciNetGoogle Scholar
  37. Therneau, T.: The R package Survival: A Package for Survival Analysis in S version 2.37.4. (2013). http://CRAN.R-project.org/package=survival. Accessed 27 Mar 2013
  38. Tweedie, M.C.K.: An index which distinguishes between some important exponential families. In: Ghosh, J.K., Roy, J. (eds.) Statistics: Applications and New Directions. Proceedings of the Indian Statistical Golden Jubilee International Conference, Calcutta, pp. 579–604 (1984)Google Scholar
  39. Ver Hoef, J.M.: Who invented the delta method? Am. Stat. 66, 124–127 (2012)MathSciNetCrossRefGoogle Scholar
  40. Weiss, C.H.: An Introduction to Discrete-Valued Time Series. Wiley, Hoboken (2018)CrossRefzbMATHGoogle Scholar
  41. Yee, T.W.: Vector Generalized Linear and Additive Models: With an Implementation in R. Springer, New York (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Probability and StatisticsUniversity of SfaxSfaxTunisia
  2. 2.Laboratoire de Mathématiques de BesançonUniversité Bourgogne Franche-ComtéBesançonFrance

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