The Continuous Hidden Threshold Mixed Skew-Symmetric Distribution

Bouaddi, Mohammed and Belhachemi, Rachid and Douch, Mohamed (2013): The Continuous Hidden Threshold Mixed Skew-Symmetric Distribution.

Abstract

This paper explores a way to construct a new family of univariate probability distributions where the parameters of the distribution capture the dependence between the variable of interest and the continuous latent state variable (the regime). The distribution nests two well known families of distributions, namely, the skew normal family of Azzalini (1985) and a mixture of two Arnold et al. (1993) distribution. We provide a stochastic representation of the distribution which enables the user to easily simulate the data from the underlying distribution using generated uniform and normal variates. We also derive the moment generating function and the moments. The distribution comprises eight free parameters that make it very flexible. This flexibility allows the user to capture many stylized facts about the data such as the regime dependence, the asymmetry and fat tails as well as thin tails.

• Item Type: MPRA Paper
• Language: English
• Keywords: Continuous Hidden threshold, Mixture Distribution, Skew-Symmetric distribution, Split Distribution.
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling

References

Arellano-Valle, R. B., H.W. Gomez and F. A. Quintana, (2004). A new classof skew-normal distributions. Communications in Statistics : Theory and Methods, 33(7), 1465-1480.

Arellano-Valle, R. B., H. W. Gomez and F. A. Quintana, (2005). Statistical inference for a general class of asymmetric distributions. Journal of Statistical Planning and Inference, 128, 427-443.

Arnold, B.C., R.J. Beaver, R.A. Groeneveld and W.Q. Meeker, (1993). The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika, 58, 47-478.

Arnold, B. C. and R.J. Beaver, (2000). Hidden truncation models. Sankhya Series A, 62, 23-35.

Azzalini, A., (1985). A class of distribution which includes the normal ones. Scandinavian Journal of Statistics. 12, pp. 171-178.

Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica., 46, 199-208.

Fernandez, C. J. O. and M.F.J. Steel, (1995). Modeling and inference with v-spherical distributions. Journal of the American Statistical Association, 90, 1331-1340.

Fernandez, C. J. O. and M.F.J. Steel, (1998). On Bayesian modelling of fat tails and skewness. Journal of the American Statistical Association, 93, 359-371.

Genton, M. G., (2004). Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC.

Gupta .K., F.C. Chang and W.C. Huang, (2002). Some skew-symmetric models. Random Oper. Stochastic Equations 10, 133-140.

Henze, N., (1986). A probabilistic representation of the skew-normal distribution. Scandinavian Journal of Statistics, 13, 271-275.

Jones, M.C., (2006). A note on rescalings, reparametrizations and classes of distributions. Journal of Statistical Planning and Inference, 136, 3730-3733.

Jones, M.C. and M.J. Faddy, (2003). A skew extension of the t-distribution, with applications. Journal of the Royal Statistical Society, Series B, 65, 159-174. Ma Y. and M.G. Genton, (2004). A flexible class of skew-symmetric distributions. Scandinavian Journal of Statistics, 31, 459-468.

Mudholkar G.S. and A. D. Hutson, (2000). The epsilon-skew-normal distribution for analysing nearnormal data. Journal of Statistical Planning and Inference, 83, 291-309.