Skip to main content

Univariate Mixture Modeling Using SMSN Distributions

  • Chapter
  • First Online:
Finite Mixture of Skewed Distributions

Abstract

In this chapter we consider a flexible class of probability distributions, convenient for modeling data with skewness behavior, discrepant observations, and population heterogeneity. The elements of this family are convex linear combinations of densities that are scale mixtures of skew-normal distributions. An EM-type algorithm for maximum likelihood estimation is developed and the observed information matrix is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.

    Article  MathSciNet  Google Scholar 

  • Arnold, B. C., Beaver, R. J., Groeneveld, R. A., & Meeker, W. Q. (1993). The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika, 58, 471–488.

    Article  MathSciNet  Google Scholar 

  • Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 32, 159–188.

    Article  MathSciNet  Google Scholar 

  • Bai, Z. D., Krishnaiah, P. R., & Zhao, L. C. (1989). On rates of convergence of efficient detection criteria in signal processing with white noise. IEEE Transactions on Automatic Control, 35, 380–388.

    MathSciNet  MATH  Google Scholar 

  • Basso, R. M., Lachos, V. H., Cabral, C. R. B., & Ghosh, P. (2010). Robust mixture modeling based on scale mixtures of skew-normal distributions. Computational Statistics and Data Analysis, 54, 2926–2941.

    Article  MathSciNet  Google Scholar 

  • Biernacki, C., Celeux, G., & Govaert, G. (2000). Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 719–725.

    Article  Google Scholar 

  • Biernacki, C., & Govaert, G. C. G. (2003). Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Computational Statistics & Data Analysis, 41, 561–575.

    Article  MathSciNet  Google Scholar 

  • Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.

    MathSciNet  MATH  Google Scholar 

  • Dias, J. G., & Wedel, M. (2004). An empirical comparison of EM, SEM and MCMC performance for problematic Gaussian mixture likelihoods. Statistics and Computing, 14, 323–332.

    Article  MathSciNet  Google Scholar 

  • DiCiccio, T. J., & Monti, A. C. (2004). Inferential aspects of the skew exponential power distribution. Journal of the American Statistical Association, 99, 439–450.

    Article  MathSciNet  Google Scholar 

  • Efron, B., & Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 1, 54–75.

    Article  MathSciNet  Google Scholar 

  • Hartigan, J. A., & Wong, M. A. (1979). A k-means clustering algorithm. Applied Statistics, 28, 100–108.

    Article  Google Scholar 

  • Lachos, V. H., Ghosh, P., & Arellano-Valle, R. B. (2010). Likelihood based inference for skew-normal independent linear mixed models. Statistica Sinica, 20, 303–322.

    MathSciNet  MATH  Google Scholar 

  • Lin, T. I., Lee, J. C., & Hsieh, W. J. (2007a). Robust mixture modelling using the skew t distribution. Statistics and Computing, 17, 81–92.

    Article  MathSciNet  Google Scholar 

  • Lin, T. I., Lee, J. C., & Yen, S. Y. (2007b). Finite mixture modelling using the skew normal distribution. Statistica Sinica, 17, 909–927.

    MathSciNet  MATH  Google Scholar 

  • Liu, C., & Rubin, D. B. (1994). The ECME algorithm: A simple extension of EM and ECM with faster monotone convergence. Biometrika, 80, 267–278.

    MathSciNet  MATH  Google Scholar 

  • McLachlan, G. J., & Krishnan, T. (2008). The EM algorithm and extensions, 2nd ed. New York: Wiley.

    Book  Google Scholar 

  • McLachlan, G. J., & Peel, D. (1998). Robust cluster analysis via mixtures of multivariate t-distributions. Lecture Notes in Computer Science, 1451, 658–666.

    Article  MathSciNet  Google Scholar 

  • McLachlan, G. J., & Peel, D. (2000). Finite mixture models. New York: Wiley.

    Book  Google Scholar 

  • Meng, X., & Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika, 81, 633–648.

    MathSciNet  MATH  Google Scholar 

  • Nityasuddhi, D., & Böhning, D. (2003). Asymptotic properties of the EM algorithm estimate for normal mixture models with component specific variances. Computational Statistics & Data Analysis, 41, 591–601.

    Article  MathSciNet  Google Scholar 

  • Peel, D., & McLachlan, G. J. (2000). Robust mixture modelling using the t distribution. Statistics and Computing, 10, 339–348.

    Article  Google Scholar 

  • Prates, M. O., Cabral, C. R. B., & Lachos, V. H. (2013). mixsmsn: Fitting finite mixture of scale mixture of skew-normal distributions. Journal of Statistical Software, 54, 1–20.

    Article  Google Scholar 

  • Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s), under exclusive licence to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Lachos Dávila, V.H., Cabral, C.R.B., Zeller, C.B. (2018). Univariate Mixture Modeling Using SMSN Distributions. In: Finite Mixture of Skewed Distributions. SpringerBriefs in Statistics(). Springer, Cham. https://doi.org/10.1007/978-3-319-98029-4_4

Download citation

Publish with us

Policies and ethics