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The polar-generalized normal distribution: properties, Bayesian estimation and applications. (English) Zbl 07504805

Summary: This paper introduces an extension to the normal distribution through the polar method to capture bimodality and asymmetry, which are often observed characteristics of empirical data. The later two features are entirely controlled by a separate scalar parameter. Explicit expressions for the cumulative distribution function, the density function and the moments were derived. The stochastic representation of the distribution facilitates implementing Bayesian estimation via the Markov chain Monte Carlo methods. Some real-life data as well as simulated data are analyzed to illustrate the flexibility of the distribution for modeling asymmetric bimodality.

MSC:

62-XX Statistics

Software:

hypergeo; R; JAGS; Stan
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References:

[1] Alexander, C.; Cordeiro, GM; Ortega, EM; Sarabia, JM, Generalized beta-generated distributions, Comput Stat Data Anal, 56, 6, 1880-97 (2012) · Zbl 1245.60015 · doi:10.1016/j.csda.2011.11.015
[2] Alleva, G.; Giommi, A., Topics in theoretical and applied statistics (2016), Cham: Springer International Publishing, Cham · Zbl 1345.62013 · doi:10.1007/978-3-319-27274-0
[3] Azzalini, A., A class of distributions which includes the normal ones, Scand J Stat, 1, 171-8 (1985) · Zbl 0581.62014
[4] Contreras-Reyes, JE, An asymptotic test for bimodality using the Kullback-Leibler divergence, Symmetry, 12, 6, 1013 (2020) · doi:10.3390/sym12061013
[5] Cover, TM; Thomas, JA, Elements of information theory (2006), New York: Wiley, New York · Zbl 1140.94001
[6] Dahdouh, O.; Khaledi, MJ, Generalized spatial stick-breaking processes, Commun Stat-Simul Comput, 31, 1-20 (2020) · Zbl 07584570 · doi:10.1080/03610918.2020.1746805
[7] Dierickx, D.; Basu, B.; Vleugels, J.; Van der Biest, O., Statistical extreme value modeling of particle size distributions: experimental grain size distribution type estimation and parameterization of sintered zirconia, Mater Charact, 45, 1, 61-70 (2000) · doi:10.1016/S1044-5803(00)00049-8
[8] Ertel, A., Bimodal gene expression and biomarker discovery, Cancer Inf, 9, 11-14 (2010)
[9] Eugene, N.; Lee, C.; Famoye, F., Beta-normal distribution and its application, Commun Stat Theory Methods, 31, 4, 497-512 (2002) · Zbl 1009.62516 · doi:10.1081/STA-120003130
[10] Famoye, F.; Lee, C.; Eugene, N., Beta-normal distribution: bimodality properties and applications, J Modern Appl Stat Methods, 3, 1, 85-103 (2004) · doi:10.22237/jmasm/1083370200
[11] Genç, İ., A skew extension of the slash distribution via beta-normal distribution, Stat Pap, 54, 2, 427-442 (2013) · Zbl 1364.62032 · doi:10.1007/s00362-012-0439-0
[12] Gómez, HW; Elal-Olivero, D.; Salinas, HS; Bolfarine, H., Bimodal extension based on the skew-normal distribution with application to pollen data, Environmetrics, 22, 1, 50-62 (2011) · doi:10.1002/env.1026
[13] Hankin, RK, Numerical evaluation of the gauss hypergeometric function with the hypergeo package, R J., 7, 2, 81 (2015) · doi:10.32614/RJ-2015-022
[14] Jafari Khaledi, M.; Rivaz, F., Empirical Bayes spatial prediction using a Monte Carlo EM algorithm, Stat Methods Appl, 18, 35-47 (2009) · doi:10.1007/s10260-007-0081-5
[15] Jamalizadeh, A.; Arabpour, AR; Balakrishnan, N., A generalized skew two-piece skew-normal distribution, Stat Pap, 52, 2, 431-446 (2011) · Zbl 1247.60017 · doi:10.1007/s00362-009-0240-x
[16] Kristensen, PL; Pedersen-Bjergaard, U.; Schalkwijk, C.; Olsen, NV; Thorsteinsson, B., Erythropoietin and vascular endothelial growth factor as risk markers for severe hypoglycaemia in type 1 diabetes, Eur J Endocrinol, 163, 3, 391-398 (2010) · doi:10.1530/EJE-10-0464
[17] Kruschke, J., Doing Bayesian data analysis: a tutorial with R, JAGS, and Stan (2014), London: Academic Press, London · Zbl 1300.62001
[18] Mameli, V.; Musio, M., A generalization of the skew-normal distribution: the beta skew-normal, Commun Stat Theory Methods, 42, 12, 2229-2244 (2013) · Zbl 1287.60022 · doi:10.1080/03610926.2011.607530
[19] Mameli, V.; Musio, M.; Alleva, G.; Giommi, A., Some new results on the beta skew-normal distribution, Topics in theoretical and applied statistics, 25-35 (2016), Cham: Springer International Publishing, Cham · Zbl 1364.62036 · doi:10.1007/978-3-319-27274-0_3
[20] Marin, JM; Mengersen, K.; Robert, C., Bayesian modeling and inference on mixtures of distributions, Handbook Stat, 25, 459-503 (2005) · doi:10.1016/S0169-7161(05)25016-2
[21] McLachlan, GJ; Peel, D., Finite mixture models (2004), London: Wiley, London · Zbl 0963.62061
[22] North, GR; Wang, J.; Genton, MG, Correlation models for temperature fields, J Clim, 24, 5850-5862 (2011) · doi:10.1175/2011JCLI4199.1
[23] R Core Team (2019) A language and environment for statistical computing. R foundation for statistical computing, Vienna, Austria. http://www.R-project.org
[24] Shannon CE (1961) Two-way communication channels. In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics, The Regents of the University of California · Zbl 0106.11509
[25] Wang, J.; Wen, S.; Symmans, WF; Pusztai, L.; Coombes, KR, The bimodality index: a criterion for discovering and ranking bimodal signatures from cancer gene expression profiling data, Cancer Inf, 7, 199-216 (2009)
[26] Wolfram Research (2020) Wolfram alpha. Wolfram Research, Inc., Champaign, IL
[27] Xu, G.; Genton, MG, Tukey g-and-h random fields, J Am Stat Assoc, 112, 519, 1236-1249 (2017) · doi:10.1080/01621459.2016.1205501
[28] Zareifard, H.; Jafari Khaledi, M., Non-Gaussian modelling of spatial data using scale mixing of a unified skew Gaussian process, J Multivar Anal, 114, 16-28 (2013) · Zbl 1255.62303 · doi:10.1016/j.jmva.2012.07.003
[29] Zareifard, H.; Jafari Khaledi, M.; Dahdouh, O., Multivariate spatial modelling through a convolution-based skewed process, Stoch Environ ResRisk Assess, 33, 657-671 (2019) · doi:10.1007/s00477-019-01657-3
[30] Zhang C, Mapes BE, Soden BJ (2004) Bimodality in tropical water vapor, AGUSM, A21C-03
[31] Zhu, X.; Genton, MG, Short-term wind speed forecasting for power system operation, Int Stat Rev, 38, 2-23 (2012) · Zbl 1422.62339 · doi:10.1111/j.1751-5823.2011.00168.x
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