© Piotr Sulewski, Damian Stoltmann. Article available under the CC BY-SA 4.0 licence
This article has two goals. The first (main) goal is to introduce a new flexible distribution defined on an infinite domain (-∞, ∞). This distribution has been named the skew plasticising component normal distribution. The second (additional) goal is to present a chronological overview of distributions belonging to the large family of normal plasticising distributions. Some properties of the proposed distribution such as the PDF, CDF, quantiles, generator, moments, skewness, kurtosis and moments of order statistics are presented. The unknown parameters of the new distribution are estimated by means of the maximum likelihood method. The Shannon entropy, the Hessian Matrix and the Fisher Information Matrix are also presented. The study provides illustrative examples of the applicability and flexibility of the introduced distribution. The most important R codes are provided in Appendix 2.
plasticising component, bimodal model, departure from normality, Azzalini’s transformation
C02, C16, C46
Alavi, M. R. (2012). On a New Bimodal Normal Family. Journal of Statistical Research of Iran, 8(2), 163–176. https://doi.org/10.18869/acadpub.jsri.8.2.163.
Alavi, S. M. R., & Tarhani, M. (2017). On a Skew Bimodal Normal-Normal distribution fitted to the Old-Faithful geyser data. Communications in Statistics – Theory and Methods, 46(15), 7301– 7312. https://doi.org/10.1080/03610926.2016.1148731.
Arellano-Valle, R. B., & Azzalini, A. (2008). The centred parametrization for the multivariate skew-normal distribution. Journal of Multivariate Analysis, 99(7), 1362–1382. https://doi.org/10.1016/j.jmva.2008.01.020.
Arellano-Valle, R. B., Cortés, M. A., & Gómez, H. W. (2010). An extension of the epsilon-skewnormal distribution. Communications in Statistics – Theory and Methods, 39(5), 912–922. https://doi.org/10.1080/03610920902807903.
Arellano-Valle, R. B., Gómez, H. W., & Quintana, F. A. (2004). A new class of skew-normal distributions. Communications in Statistics – Theory and Methods, 33(7), 1465–1480. https://doi.org/10.1081/STA-120037254.
Arnold, B., Beaver, R. J., Azzalini, A., Balakrishnan, N., Bhaumik, A., Dey, D. K., Cuadras, C. M., Sarabia, J. M., Arnold, B. C., & Beaver, R. J. (2002). Skewed multivariate models related to hidden truncation and/or selective reporting. Test, 11, 7–54. https://doi.org/10.1007/BF02595728.
Athayde, E., Azevedo, C., Leiva, V., & Sanhueza, A. (2012). About Birnbaum-Saunders distributions based on the Johnson system. Communications in Statistics – Theory and Methods, 41(11), 2061–2079. http://dx.doi.org/10.1080/03610926.2010.551454.
Azzalini, A. (1985). A Class of Distributions which Includes the Normal Ones. Scandinavian Journal of Statistics, 12(2), 171–178. Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica, 46(2), 199–208. https://doi.org/10.6092/issn.1973-2201/711. Azzalini, A., & Chiogna, M. (2004). Some results on the stress–strength model for skew-normal variates. METRON, 62(3), 315–326.
Azzalini, A., & Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83(4), 715–726. https://doi.org/10.1093/biomet/83.4.715.
Bahrami, W., Agahi, H., & Rangin, H. (2009). A Two-parameter Balakrishnan Skew-normal Distribution. Journal of Statistical Research of Iran, 6(2), 231–242. http://dx.doi.org/10.18869/acadpub.jsri.6.2.231.
Bahrami, W., & Qasemi, E. (2015). A Flexible Skew-Generalized Normal Distribution. Journal of Statistical Research of Iran JSRI, 11(2), 131–145. https://doi.org/10.18869/acadpub.jsri.11.2.131.
Behboodian, J. (1970). On the modes of a mixture of two normal distributions. Technometrics, 12(1), 131–139. https://doi.org/10.2307/1267357.
Birnbaum, Z. W., & Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 319–327. https://doi.org/10.2307/3212003.
Branco, M. D., & Dey, D. K. (2001). A General Class of Multivariate Skew – Elliptical Distributions. Journal of Multivariate Analysis, 79(1), 99–113. https://doi.org/10.1006/jmva.2000.1960.
Chhikara, R. S., & Folks, J. L. (1977). The inverse Gaussian distribution as a lifetime model. Technometrics, 19(4), 461–468. https://doi.org/10.2307/1267886.
Choudhury, K., & Matin, M. A. (2011). Extended skew generalized normal distribution. METRON, 69(3), 265–278. https://doi.org/10.1007/BF03263561.
Das, J., Pathak, D., Hazarika, P. J., Chakraborty, S., & Hamedani, G. G. (2023). A New Flexible Alpha Skew-normal Distribution. Journal of the Indian Society for Probability and Statistics, 24(2), 485–507. https://doi.org/10.1007/s41096-023-00163-8.
Durrans, S. R. (1992). Distributions of fractional order statistics in hydrology. Water Resources Research, 28(6), 1649–1655. https://doi.org/10.1029/92WR00554.
Elal-Olivero, D. (2010). Alpha-skew-normal distribution. Proyecciones. Journal of Mathematics, 29(3), 224–240. http://dx.doi.org/10.4067/S0716-09172010000300006.
Elal-Olivero, D., Gómez, H. W., & Quintana, F. A. (2009). Bayesian modeling using a class of bimodal skew-elliptical distributions. Journal of Statistical Planning and Inference, 139(4), 1484– 1492. https://doi.org/10.1016/j.jspi.2008.07.016.
Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer. https://doi.org/10.1007/978-0-387-35768-3.
Gaddum, J. H. (1945). Lognormal distributions. Nature, 156, 463–466. http://dx.doi.org/10.1038/156463a0.
Gómez, H. W., Elal-Olivero, D., Salinas, H. S., & Bolfarine, H. (2011). Bimodal extension based on the skew-normal distribution with application to pollen data. Environmetrics, 22(1), 50–62. https://doi.org/10.1002/env.1026.
Gómez, H. W., Salinas, H. S., & Bolfarine, H. (2006). Generalized skew-normal models: properties and inference. Statistics. A Journal of Theoretical and Applied Statistics, 40(6), 495–505. https://doi.org/10.1080/02331880600723168.
Gómez, H. W., Varela, H., & Vidal, I. (2013). A new class of skew-symmetric distributions and related families. Statistics. A Journal of Theoretical and Applied Statistics, 47(2), 411–421. https://doi.org/10.1080/02331888.2011.589904.
Gupta, R. C., & Gupta, R. D. (2004). Generalized skew-normal model. Test, 13(2), 501–524. https://doi.org/10.1007/BF02595784.
Gupta, R. D., & Gupta, R. C. (2008). Analyzing skewed data by power normal model. Test, 17(1), 197–210. https://doi.org/10.1007/s11749-006-0030-x.
Handam, A. H. (2012). A note on generalized alpha-skew-normal distribution. International Journal of Pure and Applied Mathematics, 74(4), 491–496. https://www.ijpam.eu/contents/201274-4/6/6.pdf.
Hazarika, P. J., Shah, S., & Chakraborty, S. (2020). Balakrishnan Alpha Skew Normal Distribution: Properties and Applications. Malaysian Journal of Science, 39(2), 71–91. https://doi.org/10.48550/arXiv.1906.07424.
Henze, N. (1986). A Probabilistic Representation of the ‘Skew-normal’ Distribution. Scandinavian Journal of Statistics, 13(4), 271–275.
Jamalizadeh, A., & Arabpour, A. R. N. (2011). A generalized skew two-piece skew-normal distribution. Statistical Papers, 52(2), 431–446. https://doi.org/10.1007/s00362-009-0240-x.
Jamalizadeh, A., & Balakrishnan, N. (2008). On order statistics from bivariate skew-normal and skew-t? distributions. Journal of Statistical Planning and Inference, 138(12), 4187–4197. https://doi.org/10.1016/j.jspi.2008.03.035.
Johnson, N. L. (1949). System of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149–176. https://doi.org/10.2307/2332539.
Kapteyn, J. C. (1916). Skew frequency curves in Biology and Statistics. Recueil des travaux botaniques néerlandais, 13(2), 105–157. https://natuurtijdschriften.nl/pub/552499/RTBN1916013002002.pdf.
Kim, H. J. (2005). On a class of two-piece skew-normal distributions. Statistics, 39(6), 537–553. https://doi.org/10.1080/02331880500366027.
Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. https://doi.org/10.1214/aoms/1177729694.
Kumar, C. S., & Anusree, M. R. (2011). On a generalized mixture of standard normal and skew normal distributions. Statistics & Probability Letters, 81(12), 1813–1821. https://doi.org/10.1016/j.spl.2011.07.009.
Kumar, C. S., & Anusree, M. R. (2013). A generalized two-piece skew normal distribution and some of its properties. Statistics, 47(6), 1370–1380. https://doi.org/10.1080/02331888.2012.697269.
Kumar, C. S., & Anusree, M. R. (2015). On modified generalized skew normal distribution and some of its properties. Journal of Statistical Theory and Practice, 9(3), 489–505. https://doi.org/10.1080/15598608.2014.935617.
Lin, T. I., Lee, J. C., & Yen., S. Y. (2007). Finite mixture modelling using the skew normal distribution. Statistica Sinica, 17(3), 909–927.
Loperfido, N. (2001). Quadratic forms of skew-normal random vectors. Statistics & Probability Letters, 54(4), 381–387. https://doi.org/10.1016/S0167-7152(01)00103-1.
Ma, Y., & Genton, M. G. (2004). Flexible Class of Skew-Symmetric Distributions. Scandinavian Journal of Statistics, 31(3), 459–468. https://doi.org/10.1111/j.1467-9469.2004.03_007.x.
Magnus, G., & Magnus, J. R. (2019). The estimation of normal mixtures with latent variables. Communications in Statistics – Theory and Methods, 48(5), 1255–1269. https://doi.org/10.1080/03610926.2018.1429625.
Mahmoudi, E., Jafari, H., & Meshkat, R. S. (2019). Alpha-Skew Generalized Normal Distribution and its Applications. Applications and Applied Mathematics: An International Journal (AAM), 14(2), 784–804. https://digitalcommons.pvamu.edu/aam/vol14/iss2/10.
Malakhov, A. N. (1978). A cumulant analysis of random non-Gaussian processes and their transformations. Soviet Radio.
Martínez-Flórez, G., Bolfarine, H., & Gómez, H. W. (2014). Skew-normal alpha-power model. Statistics, 48(6), 1414–1428. https://doi.org/10.1080/02331888.2013.826659.
Martínez-Flórez, G., Tovar-Falón, R., & Elal-Olivero, D. (2022). Some new flexible classes of normal distribution for fitting multimodal data. Statistics, 56(1), 182–205. http://dx.doi.org/10.1080/02331888.2022.2041642.
Mudholkar, G. S., & Hutson, A. D. (2000). The epsilon skew-normal distribution for analyzing near-normal data. Journal of Statistical Planning and Inference, 83(1), 291–309. https://doi.org/10.1016/S0378-3758(99)00096-8.
Popović, B. V., Cordeiro, G., Ortega, E. M., & Pascoa, M. A. R. (2017). A new extended mixture normal distribution. Mathematical Communications, 22(1), 53–73. https://www.mathos.unios.hr/mc/index.php/mc/article/view/1409.
Rasekhi, M., Hamedani, G. G., & Chinipardaz, R. (2017). A flexible extension of skew generalized normal distribution. Metron, 75(1), 87–107. https://doi.org/10.1007/s40300-017-0106-2.
Rieck, J. R., & Nedelman, J. R. (1991). A log-linear model for the Birnbaum-Saunders distribution. Technometrics, 33(1), 51–60. https://doi.org/10.1080/00401706.1991.10484769.
Salinas, H. S., Arellano-Valle, R. B., & Gómez, H. W. (2007). The extended skew-exponential power distribution and its derivation. Communications in Statistics – Theory and Methods, 36(9), 1673–1689. https://doi.org/10.1080/03610920601126118.
Salinas, H., Bakouch, H., Qarmalah, N., & Martínez-Flórez, G. (2023). A Flexible Class of Two-Piece Normal Distribution with a Regression Illustration to Biaxial Fatigue Data. Mathematics, 11(5), 1–14. https://doi.org/10.3390/math11051271.
Seijas-Macias, A., Oliveira, A., & Oliveira, T. (2017). The Presence of Distortions in the Extended Skew-normal Distribution. In Proceedings of The International Statistical Institute Regional Statistics Conference 2017 “Enhancing Statistics, Prospering Human Life”. Bali, 20–24 March 2017 (pp. 840–846). Bank Indonesia, International Statistical Institute Regional Statistics Conference. https://isi-web.org/sites/default/files/2025-04/Proceeding-International-StatisticInstitute.pdf.
Shafiei, S., Doostparast, M., & Jamalizadeh, A. (2016). The alpha–beta skew-normal distribution: Properties and applications. Statistics, 50(2), 338–349. http://dx.doi.org/10.1080/02331888.2015.1096938.
Shah, S., Hazarika, P. J., Chakraborty, S., & Ali, M. M. (2021). The Balakrishnan-Alpha-Beta-Skew-Normal Distribution: Properties and Applications. Pakistan Journal of Statistics and Operation Research, 17(2), 367–380. http://dx.doi.org/10.18187/pjsor.v17i2.3731.
Shah, S., Hazarika, P. J., Chakraborty, S., & Ali, M. M. (2023). A generalized-alpha–beta-skew-normal distribution with applications. Annals of Data Science, 10(4), 1127–1155. https://doi.org/10.1007/s40745-021-00325-0.
Shannon, C. E. (1948). A Mathematical Theory of Communication. The Bell System Technical Journal, 27(3), 379–423. https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf.
Sharafi, M., & Behboodian, J. (2008). The Balakrishnan skew–normal density. Statistical Papers, 49(4), 769–778. https://doi.org/10.1007/s00362-006-0038-z.
Stephens, M. (2000). Dealing with label switching in mixture models. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 62(4), 795–809.
Sulewski, P. (2021). DS normal distribution: properties and applications. Lobachevskii Journal of Mathematics, 42(12), 2980–2999. http://dx.doi.org/10.1134/S1995080221120337.
Sulewski, P. (2022a). Modified Lilliefors goodness-of-fit test for normality. Communications in Statistics – Simulation and Computation, 51(3), 1199–1219. https://doi.org/10.1080/03610918.2019.1664580.
Sulewski, P. (2022b). Normal distribution with plasticizing component. Communications in Statistics – Theory and Methods, 51(11), 3806–3835. https://doi.org/10.1080/03610926.2020.1837881.
Sulewski, P. (2023). New Members of The Johnson Family of Probability Distributions: Properties and Application. REVSTAT-Statistical Journal, 21(4), 535–556. https://doi.org/10.57805/revstat.v21i4.429 .
Sulewski, P., & Stoltmann, D. (2023). Modified Cramer-von Mises goodness-of-fit test for normality. Przegląd Statystyczny. Statistical Review, 70(4), 1–36. https://doi.org/10.59139/ps.2023.04.1.
Sulewski, P., & Volodin, A (2022). Sulewski Plasticizing Component Distribution: Properties and Applications. Lobachevskii Journal of Mathematics, 43(8), 2286–2300. http://dx.doi.org/10.1134 /S1995080222110270 .
Wang, Z., & Song, J. (2017). Equivalent linearization method using Gaussian mixture (GM-ELM) for nonlinear random vibration analysis. Structural Safety, 64, 9–19. https://doi.org/10.1016/j.strusafe.2016.08.005.
