THE ZERO, ONE AND ZERO-AND-ONE-INFLATED NEW UNIT-LINDLEY DISTRIBUTIONS
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Abstract
In this paper, we propose the zero, one and zero-and-one-inflated New unit-Lindley distributions as natural extensions of the New unit-Lindley distribution to model continuous responses measured at the following intervals [0, 1), (0, 1] and [0, 1]. They were constructed based on convex combinations between the New unit-Lindley distribution and the distributions degenerate at zero, one, and Bernoulli distribution. They also have a number of interesting properties, such as being members of the exponential family. Besides, they have closed forms for the cumulative distribution functions, quantiles, and moments. Inferential aspects and regression structures are
discussed in this work as well as a Monte Carlo simulation study to evaluate the
performance of the regressors. Finally, we bring an application to real data on the suicide rate in the year 2016.
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References
ABRAMOWITZ, M.; STEGUN, I. A. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: United States Department of Commerce, National Bureau of Standards, 1974
BAPAT, S. R.; BHARDWAJ, R. On an Inflated Unit-Lindley Distribution. arXivpreprint arXiv:2102.04687, 2021.
CASELLA, G.; BERGER, R. L. Statistical inference. Califórnia: Duxbury Pacific Grove, 2002. v. 2.
CHAI, H. et al. A marginalized two-part beta regression model for microbiome compositional data. PLoS computational biology, v. 14, n. 7, p. e1006329, 2018.
CORLESS, R. M. et al. On the Lambert W function. Advances in Computational Mathematics, v. 5, n. 1, p. 329–359, 1996.
CRIBARI-NETO, F.; SANTOS, J. Inflated Kumaraswamy distributions. Anais da Academia Brasileira de Ciências, v. 91, n. 2, 2019.
ENEA, M. et al.gamlss.inf: Fitting Mixed (Inflated and Adjusted) Distributions. [S.l.], 2019. R package version 1.0-1. Disponível em: <https://CRAN.R-project.org/package=gamlss.inf>.
GHITANY, M. E. et al. The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval. Communications in Statistics-Theory and Methods, v. 48, n. 14, p. 3423–3438, 2019.
JODRÁ, P. Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function. Mathematics and Computers inSimulation, v. 81, n. 4, p. 851–859, 2010.
LINDLEY, D. V. Fiducial distributions and bayes’ theorem. Journal of the RoyalStatistical Society: Series B (Methodological), v. 20, n. 1, p. 102–107, 1958.
LIU, P. et al. Zero-one-inflated simplex regression models for the analysis of continuous proportion data. Statistics and Its Interface, v. 13, n. 2, p. 193–208,2020.
MAZUCHELI, J.; BAPAT, S. R.; MENEZES, A. F. B. A new one-parameter unit-lindley distribution.Chilean Journal of Statistics (ChJS), v. 11, n. 1, 2020.
MAZUCHELI, J.; MENEZES, A. F. B.; CHAKRABORTY, S. On the one parameter unit-Lindley distribution and its associated regression model for proportion data. Journal of Applied Statistics, v. 46, n. 4, p. 700–714, 2019.
MAZUCHELI, J.; MENEZES, A. F. B.; DEY, S. Unit-Gompertz distribution with applications. Statistica, v. 79, n. 1, p. 25–43, 2019.
MAZUCHELI, J.; MENEZES, A. F. B.; GHITANY, M. E. The unit-Weibulldistribution and associated inference. Journal of Applied Probability and Statistics, v. 13, p. 1–22, 2018
MENEZES, A. F.; MAZUCHELI, J.; BOURGUIGNON, M. A parametric quantile regression approach for modelling zero-or-one inflated double bounded data. Biometrical Journal, v. 63, n. 4, p. 841–858, 2021.
NELDER, J. A.; WEDDERBURN, R. W. Generalized linear models. Journal of the Royal Statistical Society: Series A (General), Wiley Online Library, v. 135, n. 3, p.370–384, 1972.
OSPINA, R.; FERRARI, S. L. A general class of zero-or-one inflated beta regression models. Computational Statistics & Data Analysis, v. 56, n. 6, p. 1609–1623, 2012.
OSPINA, R.; FERRARI, S. L. P. Inflated beta distributions. Statistical Papers, v. 51, n. 1, p. 111, 2010.
QUEIROZ, F. F.; LEMONTE, A. J. A broad class of zero-or-one inflated regression models for rates and proportions. Canadian Journal of Statistics, v. 49, n. 2, p.566–590, 2021.R Core Team.R: A Language and Environment for Statistical Computing. Vienna, Austria, 2019. Disponível em: <https://www.R-project.org/>.
RIGBY, R. A.; STASINOPOULOS, D. M. Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (AppliedStatistics), v. 54, n. 3, p. 507–554, 2005.
RIVAS, L.; CAMPOS, F. Zero inflated waring distribution. Communications inStatistics-Simulation and Computation, p. 1–16, 2021.
SANTOS, B.; BOLFARINE, H. Bayesian analysis for zero-or-one inflated proportion data using quantile regression. Journal of Statistical Computation and Simulation, v. 85, n. 17, p. 3579–3593, 2015.
SANTOS, B.; BOLFARINE, H. Bayesian quantile regression analysis for continuous data with a discrete component at zero. Statistical Modelling, v. 18, n. 1, p. 73–93,2018.
SILVA, A. R. et al. Augmented-limited regression models with an application to the study of the risk perceived using continuous scales. Journal of Applied Statistics, v. 48, n. 11, p. 1998–2021, 2021.
TOMARCHIO, S. D.; PUNZO, A. Modelling the loss given default distribution via a family of zero-and-one inflated mixture models. Journal of the Royal Statistical Society: Series A (Statistics in Society), v. 182, n. 4, p. 1247–1266, 2019.
XIE, M.; HE, B.; GOH, T. Zero-inflated poisson model in statistical process control. Computational Statistics & data Analysis, v. 38, n. 2, p. 191–201, 2001.