On the discrete quasi xgamma distribution.

*(English)*Zbl 1448.62030Summary: Methods to obtain discrete analogs of continuous distributions have been widely applied in recent years. In general, the discretization process provides probability mass functions that can be competitive with traditional models used in the analysis of count data. The discretization procedure also avoids the use of continuous distribution to model strictly discrete data. In this paper, we propose two discrete analogs for the quasi xgamma distribution as alternatives to model under- and overdispersed datasets. The methods of infinite series and survival function have been considered to derive the models and, despite the difference between the methods, the resulting distributions are interchangeable. Several statistical properties of the proposed models have been derived. The maximum likelihood theory has been considered for estimation and asymptotic inference concerns. An intensive simulation study has been carried out in order to evaluate the main properties of the maximum likelihood estimators. The usefulness of the proposed models has been assessed by using two real datasets provided by literature. A general comparison of the proposed models with some well-known discrete distributions has been provided.

##### MSC:

62E15 | Exact distribution theory in statistics |

62F10 | Point estimation |

62P20 | Applications of statistics to economics |

##### Keywords:

count data; discretization methods; quasi xgamma distribution; data dispersion; maximum likelihood estimation; simulation study
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\textit{J. Mazucheli} et al., Methodol. Comput. Appl. Probab. 22, No. 2, 747--775 (2020; Zbl 1448.62030)

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##### References:

[1] | Bakouch, HS; Jazi, MA; Nadarajah, S., A new discrete distribution, Statistics, 48, 1, 200-240 (2014) · Zbl 1367.60011 |

[2] | Bi Z, Faloutsos C, Korn F (2001) The DGX distribution for mining massive, skewed data. In: Proceedings of the seventh ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 17-26 |

[3] | Bliss, CI; Fisher, RA, Fitting the negative binomial distribution to biological data, Biometrics, 9, 2, 176-200 (1953) |

[4] | Bracquemond, C.; Gaudoin, O., A survey on discrete lifetime distributions, Int J Reliab Qual Saf Eng, 10, 1, 69-98 (2003) |

[5] | Castillo, JD; Pérez-Casany, M., Weighted poisson distributions for overdispersion and underdispersion situations, Ann Inst Stat Math, 50, 3, 567-585 (1998) · Zbl 0912.62019 |

[6] | Chakraborty, Subrata, A New Discrete Distribution Related to Generalized Gamma Distribution and Its Properties, Communications in Statistics - Theory and Methods, 44, 8, 1691-1705 (2013) · Zbl 1319.62030 |

[7] | Chakraborty S (2015b) Generating discrete analogues of continuous probability distributions - a survey of methods and constructions. Journal of Statistical Distributions and Applications 2(1):1-30 · Zbl 1359.62053 |

[8] | Chakraborty, S.; Chakravarty, D., Discrete Gamma distributions: properties and parameter estimation, Communications in Statistics - Theory and Methods, 41, 18, 3301-3324 (2012) · Zbl 1296.62032 |

[9] | Chakraborty, S.; Chakravarty, D., A new discrete probability distribution with integer support on (−∞, +∞), Communications in Statistics - Theory and Methods, 45, 2, 492-505 (2016) · Zbl 1338.62053 |

[10] | Chakraborty, S.; Gupta, RD, Exponentiated Geometric distribution: another generalization of geometric distribution, Communications in Statistics - Theory and Methods, 44, 6, 1143-1157 (2015) · Zbl 1325.62036 |

[11] | Collett, D., Modelling survival data in medical research (2003), New York: Chapaman and Hall, New York |

[12] | Doornik, JA, Object-oriented matrix programming using Ox (2007), London: Timberlake Consultants Press and Oxford, London |

[13] | Doray, LG; Luong, A., Efficient estimators for the Good family, Commun Stat Simul Comput, 26, 3, 1075-1088 (1997) · Zbl 0925.62098 |

[14] | Gómez-Déniz, E.; Calderín-ojeda, E., The discrete Lindley distribution: properties and applications, J Stat Comput Simul, 81, 11, 1405-1416 (2011) · Zbl 1270.60022 |

[15] | Good, IJ, The population frequencies of species and the estimation of population parameters, Biometrika, 40, 3-4, 237-264 (1953) · Zbl 0051.37103 |

[16] | Grandell J (1997) Mixed Poisson processes, vol 77. Chapman and Hall/CRC · Zbl 0922.60005 |

[17] | Haight, FA, Queueing with balking, Biometrika, 44, 3-4, 360-369 (1957) · Zbl 0085.34703 |

[18] | Hamada, MS; Wilson, AG; Reese, CS; Martz, HF, Bayesian reliability. Springer series in statistics (2008), New York: Springer, New York |

[19] | Hussain, T.; Ahmad, M., Discrete inverse Rayleigh distribution, Pakistan Journal of Statistics, 30, 2, 203-222 (2014) |

[20] | Inusah, S.; Kozubowski, TJ, A discrete analogue of the Laplace distribution, Journal of Statistical Planning and Inference, 136, 3, 1090-1102 (2006) · Zbl 1081.60011 |

[21] | Jazi, MA; Lai, CD; Alamatsaz, MH, A discrete inverse Weibull distribution and estimation of its parameters, Stat Methodology, 7, 2, 121-132 (2010) · Zbl 1230.62130 |

[22] | Kalbfleisch, JD; Prentice, RL, The statistical analysis of failure time data (2002), New York: Wiley, New York |

[23] | Keilson, J.; Gerber, H., Some results for discrete unimodality, J Am Stat Assoc, 66, 334, 386-389 (1971) · Zbl 0236.60017 |

[24] | Kemp, AW, Characterizations of a discrete normal distribution, Journal of Statistical Planning and Inference, 63, 2, 223-229 (1997) · Zbl 0902.62020 |

[25] | Kemp, AW, Classes of discrete lifetime distributions, Communications in Statistics - Theory and Methods, 33, 12, 3069-3093 (2004) · Zbl 1087.62016 |

[26] | Kemp, Adrienne W., The Discrete Half-Normal Distribution, Advances in Mathematical and Statistical Modeling, 353-360 (2008), Boston: Birkhäuser Boston, Boston |

[27] | Kendall, MG, Natural law in social sciences, J R Stat Soc Ser A, 124, 1-19 (1961) |

[28] | Klein, JP; Moeschberger, ML, Survival analysis: techniques for censored and truncated data (1997), New York: Springer, New York |

[29] | Kozubowski, TJ; Inusah, S., A skew Laplace distribution on integers, Ann Inst Stat Math, 58, 3, 555-571 (2006) · Zbl 1100.62010 |

[30] | Krishna, H.; Pundir, PS, Discrete Burr and discrete Pareto distributions, Stat Methodology, 6, 2, 177-188 (2009) · Zbl 1220.62013 |

[31] | Kulasekera, KB; Tonkyn, DW, A new discrete distribution, with applications to survival, dispersal and dispersion, Commun Stat Simul Comput, 21, 2, 499-518 (1992) · Zbl 0850.62164 |

[32] | Lawless, JF, Statistical models and methods for lifetime data (2003), Hoboken: Wiley, Hoboken |

[33] | Lee, ET; Wang, JW, Statistical methods for survival data analysis (2003), Hoboken: Wiley, Hoboken |

[34] | Lisman, JHC; Van Zuylen, MCA, Note on the generation of most probable frequency distributions, Statistica Neerlandica, 26, 1, 19-23 (1972) · Zbl 0298.62006 |

[35] | Meeker, WQ; Escobar, LA, Statistical methods for reliability data (1998), New York: Wiley, New York |

[36] | Nakagawa, T.; Osaki, S., The discrete Weibull distribution, IEEE Trans Reliab, R-24, 5, 300-301 (1975) |

[37] | Nekoukhou, V.; Alamatsaz, MH; Bidram, H., A discrete analog of the generalized exponential distribution, Communication in Statistics - Theory and Methods, 41, 11, 2000-2013 (2012) · Zbl 1253.60017 |

[38] | Nekoukhou, V.; Alamatsaz, MH; Bidram, H., Discrete generalized Exponential distribution of a second type, Statistics - A Journal of Theoretical and Applied Statistics, 47, 4, 876-887 (2013) · Zbl 1440.62051 |

[39] | R Development Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria |

[40] | Ridout, MS; Besbeas, P., An empirical model for underdispersed count data, Stat Model, 4, 1, 77-89 (2004) · Zbl 1111.62017 |

[41] | Roy, D., The discrete normal distribution, Communication in Statistics - Theory and Methods, 32, 10, 1871-1883 (2003) · Zbl 1155.60302 |

[42] | Roy, D., Discrete Rayleigh distribution, IEEE Trans Reliab, 53, 2, 255-260 (2004) |

[43] | Rubinstein, RY; Kroese, DP, Simulation and the Monte Carlo method. Wiley series in probability and statistics (2008), Hoboken: Wiley, Hoboken |

[44] | Saha, KK, Analysis of one-way layout of count data in the presence of over or under dispersion, Journal of Statistical Planning and Inference, 138, 7, 2067-2081 (2008) · Zbl 1134.62009 |

[45] | Sato, H.; Ikota, M.; Sugimoto, A.; Masuda, H., A new defect distribution metrology with a consistent discrete exponential formula and its applications, IEEE Trans Semicond Manuf, 12, 4, 409-418 (1999) |

[46] | Sen, S.; Chandra, N., The quasi xgamma distribution with application in bladder cancer data, Journal of Data Science, 15, 61-76 (2017) |

[47] | Siromoney, G., The general Dirichlet’s series distribution, Journal of the Indian Statistical Association, 2-3, 2, 1-7 (1964) |

[48] | Slater, LJ, Generalized hypergeometric functions (1966), Cambridge: Cambridge University Press, Cambridge |

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