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Geometric Tweedie regression models for continuous and semicontinuous data with variation phenomenon. (English) Zbl 1437.62289
Summary: 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.

MSC:
62J12 Generalized linear models (logistic models)
62F10 Point estimation
62E10 Characterization and structure theory of statistical distributions
62E15 Exact distribution theory in statistics
Software:
R; survival; Tweedie; VGAMdata
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[1] Abid, R.; Kokonendji, Cc; Masmoudi, A., Geometric dispersion models with real quadratic v-functions, Stat. Probab. Lett., 145, 197-204 (2019) · Zbl 1406.60021
[2] Andersen, Da; Bonat, Wh, Double generalized linear compound Poisson models to insurance claims data, Electr. J. Appl. Stat. Anal., 10, 384-407 (2017)
[3] Aryuyuen, S.; Bodhisuwan, W., The negative binomial-generalized exponential (NB-GE) distribution, Appl. Math. Sci., 7, 1093-1105 (2013)
[4] Barlow, R.A., Proschan, F.: Statistical Theory of Reliability and Life Testing: Probability Models. To begin with, Silver Springs, Maryland (1981)
[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)
[6] Bonat, Wh; Kokonendji, Cc, Flexible Tweedie regression models for continuous data, J. Stat. Comput. Simul., 87, 2138-2152 (2017)
[7] Bonat, Wh; Jørgensen, B.; Kokonendji, Cc; Hinde, J.; Demétrio, Cgb, Extended Poisson-Tweedie: properties and regression models for count data, Stat. Model., 18, 24-49 (2018)
[8] Bonat, Wagner H.; Petterle, Ricardo R.; Hinde, John; Demétrio, Clarice Gb, Flexible quasi-beta regression models for continuous bounded data, Statistical Modelling, 19, 6, 617-633 (2018)
[9] Cahoy, Do, Estimation of Mittag-Leffler parameters, Commun. Stat. Simul. Comput., 42, 303-315 (2013) · Zbl 1327.62102
[10] Cahoy, Do; Uhaikin, Vv; Woyczynski, Wa, Parameter estimation for fractional Poisson processes, J. Stat. Plann. Inference, 140, 3106-3120 (2010) · Zbl 1205.62118
[11] Dudenhoeffer, Dd; Gaver, Dp; Jacobs, Pa, Failure, repair and replacement analyses of a navy subsystem: case study of a pump, J. Appl. Stoch. Mod. Data Anal., 13, 369-376 (1998) · Zbl 0913.60078
[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) · Zbl 0773.62069
[14] IwiǹSka, M.; Popowska, B., Characterizations of the exponential distribution by geometric compound, Fasc. Math., 47, 5-10 (2011) · Zbl 1323.62018
[15] IwiǹSka, M.; Szymkowiak, M., Characterizations of the exponential distribution by Pascal compound, Commun. Stat. Theory Methods, 45, 63-70 (2016) · Zbl 1338.60052
[16] IwiǹSka, M.; Szymkowiak, M., Characterizations of distributions through selected functions of reliability theory, Commun. Stat. Theory Methods, 46, 69-74 (2017) · Zbl 1381.62035
[17] Jørgensen, B., The Theory of Dispersion Models (1997), London: Chapman & Hall, London · Zbl 0928.62052
[18] Jørgensen, B.; Knudsen, Sj, Parameter orthogonality and bias adjustment for estimating functions, Scand. J. Stat., 31, 93-114 (2004) · Zbl 1051.62022
[19] Jørgensen, B.; Kokonendji, Cc, Dispersion models for geometric sums, Braz. J. Probab. Stat., 25, 263-293 (2011) · Zbl 1271.62025
[20] Jørgensen, B.; Kokonendji, Cc, Discrete dispersion models and their Tweedie asymptotics, AStA Adv. Stat. Anal., 100, 43-78 (2016) · Zbl 1443.62046
[21] Kalashnikov, V., Geometric Sums: Bounds for Rare Events with Applications (1997), Dordrecht: Kluwer Academic, Dordrecht · Zbl 0881.60043
[22] Kalbfleisch, John D.; Prentice, Ross L., The Statistical Analysis of Failure Time Data (2002), Hoboken, NJ, USA: John Wiley & Sons, Inc., Hoboken, NJ, USA · Zbl 1012.62104
[23] Kemp, Aw, Classes of discrete lifetime distributions, Commun. Stat. Theory Methods, 33, 3069-3093 (2004) · Zbl 1087.62016
[24] Klein, Jp; Moeschberger, Ml, Survival Analysis: Techniques for Censored and Truncated Data (2003), New York: Springer, New York
[25] Kline, Mb, Suitability of the lognormal distribution for corrective maintenance repair times, Reliab. Engine, 9, 65-80 (1984)
[26] Kokonendji, Cc; Balakrishnan, N., Over- and underdispersion models, The Wiley Encyclopedia of Clinical Trials: Methods and Applications of Statistics in Clinical Trials (Chap. 30), 506-526 (2014), New York: Wiley, New York
[27] Kokonendji, Cc; Puig, P., Fisher dispersion index for multivariate count distributions: a review and a new proposal, J. Multivar. Anal., 165, 180-193 (2018) · Zbl 1397.62182
[28] Kotz, S.; Kozubowski, T.; Podgórski, K., The Laplace Distributions and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance (2001), Boston: Birkhuser, Boston · Zbl 0977.62003
[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) · Zbl 1334.90044
[30] Kundu, Debasis, Geometric Skew Normal Distribution, Sankhya B, 76, 2, 167-189 (2014) · Zbl 1329.62073
[31] Liu, X., Survival Analysis: Models and Applications (2012), Chichester: Wiley, Chichester
[32] Mccullagh, P.; Nelder, J., Generalized Linear Models (1989), London: Chapman & Hall, London · Zbl 0744.62098
[33] Nolan, Jp, Stable Distributions: Models for Heavy Tailed Data (2006), Washington, DC: American University, Washington, DC
[34] Pillai, Rn, On Mittag-Leffler functions and related distributions, Ann. Stat., 42, 157-161 (1990) · Zbl 0714.60009
[35] A Language and Environment for Statistical Computing (2018), Vienna: R Foundation for Statistical Computing, Vienna
[36] Smyth, Gk, Generalized linear models with varying dispersion, J. R. Stat. Soc. B, 51, 47-60 (1989)
[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)
[39] Ver Hoef, Jm, Who invented the delta method?, Am. Stat., 66, 124-127 (2012)
[40] Weiss, Ch, An Introduction to Discrete-Valued Time Series (2018), Hoboken: Wiley, Hoboken
[41] Yee, Tw, Vector Generalized Linear and Additive Models: With an Implementation in R (2015), New York: Springer, New York
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