Skew multimodal normal distribution

Authors

Department of Statistics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Abstract

In this paper the multimodal normal distribution is introduced. This distribution is a generalization of the standard normal distribution and covers the symmetric distributions with up to three modes. Then the skew multimodal normal distribution is introduced as a skewed-symmetric distribution generated by the standard normal distribution. Some properties of this distribution are investigated. Data generation methods from this distribution are proposed and its location and scale distribution is introduced. The maximum likelihood estimates of the parameters are studied and evaluated using simulations. The famous geyser dataset is fitted to this distribution and compared with several competing distributions.

Keywords

Main Subjects


[1] س.م.ر. علوی، ر. چینی پرداز، استنباط در توزیع نرمال بر اساس نمونه گیری وزنی، مجله علوم آماری  ،۱۳۸۴  .۷۳-۸۸ ،(۱)  1 .
 
[2] Alavi, S. M. R., On a new bimodal normal family, Journal of Statistical Research of Iran, (2011), 8 (2), 163–175.
[3] Alavi, S. M. R. and Tarhani, M., On a Skew Bimodal Normal-Normal distribution fitted to the Old- Faithful geyser data, Communications in Statistics - Theory and Methods, (2016), 46(15), 7301-7312.
[4] Azzalini, A., Bowman, A. W., A Look at Some Data on the Old Faithful Geyser, Journal of Applied Statistics , (1990). 39, 357-365.
[5] Azzalini, A., A Class of Distributions which Includes the Normal Ones, Scandinavian Journal of Statistics, (1985). 12, 171-178.
[6] Azzalini, A., The Skew-normal Distribution and Relative Multivariate Families, Scandinavian Journal of Statistics, (2005). 32, 159-188.
[7] Bolfarine H., Gómez, H.W., Rivas, L. The log-bimodal-skew-normal model. A geochemical application, Journal of Chemometrics, (2011). 25(6), 329–332.
[8] Gómez, H. W., Venegas, O. and Bolfarine, H., Skew-Symmetric Distributions Generated by the Distribution Function of the Normal Distribution, Environmetrics, (2007). 18, 395–407
[9] Gradshteyn, I.S. and Ryzhik, I.M., Tables of Integrals, Seires and Products, Academic press, (1965).
[10] Liseo, B., The Skew-Normal Class of Densities: Aspects of Inference From the Bayesian Point of View, Statistica, (1990), 50(1), 71–82.
[11] Martinez, E. H., Varela, H., Gomez, H. W. andBolfarine, H., A Note on the Likelihood and Moments of the Skew-Normal Distribution, Statistical Operation Research Transaction, (2008), 32(1), 57–66.
[12] Maleki, M. and Nematollahi, A. R., Bayesian Approach to Epsilon-Skew-Normal Family, Communications in Statistics-Theory and Methods, (2016), 46 (15), 7546-7561.
[13] Mameli, V. and Musio, M. , A Generalization of the Skew-Normal Distribution: the Beta-Skew-Normal Distribution, Communications in Statistics: Theory and Methods, (2013), 42, 2229-2242.
[14] Nekoukhou, V, and Alamatsaz, M. H., A family of skew-symmetric-Laplace distributions, Ststistical papers, (2012), 53, 685–696.
[15] Pewsey, A., Problems of Inference for Azzalini’s Skew-Normal Distribution, Journal of Applied Statistics, (2000), 27(7), 859–870.
[16] Rasekhi, M., Chinipardaz, R. and Alavi, S. M. R., A Flexible Generalization of the Skew Normal Distribution Based on a Weighted Normal Distribution, Statistical Methods and Application, (2015), 25 (3), 375-394.
[17] Sharafi, M. and Behboodian, J.,The Balakrishnan Skew-Normal Density, Statistical Papers, (2007), 49, 769-778.
[18] Wang , J. , Boyer , J. and Genton , M. G., A skew-symmetric representation of multivariate distributions, Statist. Sinica, ( 2004 ), 14, 1259 – 1270.
[19] Yadegari, I., Gerami, A. and Khaledi, M. J., A Generalized of the Balakrishnan Skew-normal Distribution, Statistics and Probablity Letters, (2008), 78, 1165- 1167.