Startseite Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling
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Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling

  • Zakaria Alizadeh Ghajari , Karim Zare ORCID logo EMAIL logo und Soheil Shokri ORCID logo
Veröffentlicht/Copyright: 11. April 2024

Abstract

In this paper, we study linear regression models in which the error term has shape mixtures of skew-normal distribution. This type of distribution belongs to the skew-normal (SN) distribution class that can be used for heavy tails and asymmetry data. For the first time, for the classical (non-Bayesian) estimation of the parameters of the SN family, we apply the Markov chains Monte Carlo ECM (MCMC-ECM) algorithm where the samples are generated by Gibbs sampling, denoted by Gibbs-ECM, and also, we extend two other types of the EM algorithm for the above model. Finally, the proposed method is evaluated through a simulation and compared with the Numerical Math-ECM algorithm and Monte Carlo ECM (MC-ECM) using a real data set.

MSC 2020: 65C05; 65C40; 65Z05

References

[1] S. Allassonnière, E. Kuhn and A. Trouvé, Construction of Bayesian deformable models via a stochastic approximation algorithm: A convergence study, Bernoulli 16 (2010), no. 3, 641–678. 10.3150/09-BEJ229Suche in Google Scholar

[2] D. F. Andrews and C. L. Mallows, Scale mixtures of normal distributions, J. Roy. Statist. Soc. Ser. B 36 (1974), 99–102. 10.1111/j.2517-6161.1974.tb00989.xSuche in Google Scholar

[3] R. B. Arellano-Valle, R. B. Bolfarine and H. Lachos, Skew-normal linear mixed models, J. Data Sci. 3 (2005), 415–438. 10.6339/JDS.2005.03(4).238Suche in Google Scholar

[4] R. B. Arellano-Valle, L. M. Castro, M. G. Genton and H. W. Gómez, Bayesian inference for shape mixtures of skewed distributions, with application to regression analysis, Bayesian Anal. 3 (2008), no. 3, 513–539. 10.1214/08-BA320Suche in Google Scholar

[5] R. B. Arellano-Valle, C. S. Ferreira and M. G. Genton, Scale and shape mixtures of multivariate skew-normal distributions, J. Multivariate Anal. 166 (2018), 98–110. 10.1016/j.jmva.2018.02.007Suche in Google Scholar

[6] R. B. Arellano-Valle, M. G. Genton and R. H. Loschi, Shape mixtures of multivariate skew-normal distributions, J. Multivariate Anal. 100 (2009), no. 1, 91–101. 10.1016/j.jmva.2008.03.009Suche in Google Scholar

[7] R. B. Arellano-Valle, H. W. Gómez and F. A. Quintana, A new class of skew-normal distributions, Comm. Statist. Theory Methods 33 (2004), no. 7, 1465–1480. 10.1081/STA-120037254Suche in Google Scholar

[8] B. C. Arnold and R. J. Beaver, The skew-Cauchy distribution, Statist. Probab. Lett. 49 (2000), no. 3, 285–290. 10.1016/S0167-7152(00)00059-6Suche in Google Scholar

[9] A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), no. 2, 171–178. Suche in Google Scholar

[10] A. Azzalini and A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. R. Stat. Soc. Ser. B Stat. Methodol. 61 (1999), no. 3, 579–602. 10.1111/1467-9868.00194Suche in Google Scholar

[11] A. Azzalini, T. Dal Capello and S. Kotz, Log-skew-normal and log-skew-t distributions as models for family income data, JIncomDistrib. 11 (2003), 13–21. 10.25071/1874-6322.1249Suche in Google Scholar

[12] A. Azzalini and A. Dalla Valle, The multivariate skew-normal distribution, Biometrika 83 (1996), no. 4, 715–726. 10.1093/biomet/83.4.715Suche in Google Scholar

[13] R. M. Basso, V. H. Lachos, C. R. Cabral and P. Ghosh, Robust mixture modeling based on scale mixtures of skew-normal distributions, Comput. Statist. Data Anal. 54 (2010), 2926–2941. 10.1016/j.csda.2009.09.031Suche in Google Scholar

[14] M. D. Branco and D. K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivariate Anal. 79 (2001), no. 1, 99–113. 10.1006/jmva.2000.1960Suche in Google Scholar

[15] R. L. Butler, J. B. Mcdonald, R. D. Nelson and S. B. White, Robust and partly adaptive estimation of regression models, Rev. Econ. Statist. 72 (1990), 321–327. 10.2307/2109722Suche in Google Scholar

[16] G. Celeux and J. Diebolt, The SEM algorithm: A probabilistic teacher algorithm derived from the EM algorithm for the mixture problem, Comput. Statist. Quart. 2 (1985), 73–82. Suche in Google Scholar

[17] K. S. Chan and J. Ledolter, Monte Carlo EM estimation for time series models involving counts, J. Amer. Statist. Assoc. 90 (1995), no. 429, 242–252. 10.1080/01621459.1995.10476508Suche in Google Scholar

[18] R. D. Cook and S. Weisberg, An Introduction to Regression Graphics, Wiley Ser. Probab. Stat., John Wiley & Sons, New York, 1994. 10.1002/9780470316863Suche in Google Scholar

[19] C. da Silva Ferreira, H. Bolfarine and V. H. Lachos, Skew scale mixtures of normal distributions: Properties and estimation, Stat. Methodol. 8 (2011), no. 2, 154–171. 10.1016/j.stamet.2010.09.001Suche in Google Scholar

[20] B. Delyon, M. Lavielle and E. Moulines, Convergence of a stochastic approximation version of the EM algorithm, Ann. Statist. 27 (1999), no. 1, 94–128. 10.1214/aos/1018031103Suche in Google Scholar

[21] A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 1, 1–38. 10.1111/j.2517-6161.1977.tb01600.xSuche in Google Scholar

[22] T. do Bem Mattos, A. M. Garay and V. H. Lachos, Likelihood-based inference for censored linear regression models with scale mixtures of skew-normal distributions, J. Appl. Stat. 45 (2018), no. 11, 2039–2066. 10.1080/02664763.2017.1408788Suche in Google Scholar

[23] C. S. Ferreira, V. H. Lachos and H. Bolfarine, Inference and diagnostics in skew scale mixtures of normal regression models, J. Stat. Comput. Simul. 85 (2015), no. 3, 517–537. 10.1080/00949655.2013.828057Suche in Google Scholar

[24] C. S. Ferreira, V. H. Lachos and H. Bolfarine, Likelihood-based inference for multivariate skew scale mixtures of normal distributions, AStA Adv. Stat. Anal. 100 (2016), no. 4, 421–441. 10.1007/s10182-016-0266-zSuche in Google Scholar

[25] C. E. Galarza, V. H. Lachos and D. Bandyopadhyay, Quantile regression in linear mixed models: A stochastic approximation EM approach, Stat. Interface 10 (2017), no. 3, 471–482. 10.4310/SII.2017.v10.n3.a10Suche in Google Scholar PubMed PubMed Central

[26] H. W. Gómez, O. Venegas and H. Bolfarine, Skew-symmetric distributions generated by the distribution function of the normal distribution, Environmetrics 18 (2007), no. 4, 395–407. 10.1002/env.817Suche in Google Scholar

[27] W. Jank, Implementing and diagnosing the stochastic approximation EM algorithm, J. Comput. Graph. Statist. 15 (2006), no. 4, 803–829. 10.1198/106186006X157469Suche in Google Scholar

[28] F. Kahrari, C. S. Ferreira and R. B. Arellano-Valle, Skew-normal-Cauchy linear mixed models, Sankhya B 81 (2019), no. 2, 185–202. 10.1007/s13571-018-0173-2Suche in Google Scholar

[29] F. Kahrari, M. Rezaei, F. Yousefzadeh and R. B. Arellano-Valle, On the multivariate skew-normal Cauchy distribution, Statist. Probab. Lett. 117 (2016), 80–88. 10.1016/j.spl.2016.05.005Suche in Google Scholar

[30] M. Khounsiavash, M. Ganjali and T. Baghfalaki, A stochastic version of the EM algorithm to analyze multivariate skew-normal data with missing responses, Appl. Appl. Math. 6 (2011), no. 12, 412–427. Suche in Google Scholar

[31] E. Kuhn and M. Lavielle, Coupling a stochastic approximation version of EM with an MCMC procedure, ESAIM Probab. Stat. 8 (2004), 115–131. 10.1051/ps:2004007Suche in Google Scholar

[32] E. Kuhn and M. Lavielle, Maximum likelihood estimation in nonlinear mixed effects models, Comput. Statist. Data Anal. 49 (2005), 1020–1038. 10.1016/j.csda.2004.07.002Suche in Google Scholar

[33] F. V. Labra, A. M. Garay, V. H. Lachos and E. M. M. Ortega, Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions, J. Statist. Plann. Inference 142 (2012), no. 7, 2149–2165. 10.1016/j.jspi.2012.02.018Suche in Google Scholar

[34] V. H. Lachos, H. Bolfarine, R. B. Arellano-Valle and L. C. Montenegro, Likelihood-based inference for multivariate skew-normal regression models, Comm. Statist. Theory Methods 36 (2007), no. 9–12, 1769–1786. 10.1080/03610920601126241Suche in Google Scholar

[35] T. I. Lin, J. C. Lee and S. Y. Yen, Finite mixture modelling using the skew normal distribution, Statist. Sinica 17 (2007), no. 3, 909–927. Suche in Google Scholar

[36] C. Liu and D. B. Rubin, The ECME algorithm; A simple extension of EM and ECM with faster monotone convergence, Biometrica 80 (1993), 267–278. 10.1093/biomet/80.3.543Suche in Google Scholar

[37] I. Meilijson, A fast improvement to the EM algorithm on its own terms, J. Roy. Statist. Soc. Ser. B 51 (1989), no. 1, 127–138. 10.1111/j.2517-6161.1989.tb01754.xSuche in Google Scholar

[38] X.-L. Meng and D. B. Rubin, Maximum likelihood estimation via the ECM algorithm: A general framework, Biometrika 80 (1993), no. 2, 267–278. 10.1093/biomet/80.2.267Suche in Google Scholar

[39] C. P. Robert and G. Casella, Introducing Monte Carlo Methods with R, Springer, New York, 2010. 10.1007/978-1-4419-1576-4Suche in Google Scholar

[40] G. C. G. Wei and M. A. Tanner, A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm, J. Amer. Statist. Assoc. 85 (1990), 699–704. 10.1080/01621459.1990.10474930Suche in Google Scholar

[41] R. Zhou, J. Liu, S. Kumar and D. P. Palomar, Student’s 𝑡 VAR modeling with missing data via stochastic EM and Gibbs sampling, IEEE Trans. Signal Process. 68 (2020), 6198–6211. 10.1109/TSP.2020.3033378Suche in Google Scholar

Received: 2023-08-06
Revised: 2024-02-09
Accepted: 2024-02-19
Published Online: 2024-04-11
Published in Print: 2024-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 26.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/mcma-2024-2003/html
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