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Mean mixtures of normal distributions: properties, inference and application

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Abstract

The skew normal (SN) distribution of Azzalini (Scand J Stat 12:171–178, 1985) is one of the widely used probability distributions for modelling skewed data. In this article, we introduce a general class of skewed distributions based on mean mixtures of normal distributions, which includes the SN distribution as a special case. Some properties of this new class, such as expressions for mean, variance, skewness and kurtosis coefficients and characteristic function, are derived. Also, estimates of the model parameters are first obtained by the method of moments. Two special cases of this new class are studied in detail. It is shown that the range of skewness and kurtosis coefficients for the special cases is wider than that of the SN distribution, and in addition, unlike the SN distribution, one of these models is an infinitely divisible distribution. For carrying out the maximum likelihood (ML) estimation, an ECM algorithm is developed. This algorithm is analytically simple because closed-form expressions of conditional expectations in the E-step as well as the updating estimators in the CM-step are in explicit form. The observed information matrix is provided for approximating the asymptotic covariance matrix of the ML estimators of the parameters. The usefulness of the proposed distribution is illustrated through simulated as well as two real data sets. Next, a new extension of regression models is constructed by assuming the proposed distributions for the error term. Finally, a multivariate version of the proposed model is discussed.

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References

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold BC, Beaver RJ, Groeneveld RA, Meeker WQ (1993) The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58:471–488

    Article  MathSciNet  MATH  Google Scholar 

  • Arellano-Valle RB, Azzalini A (2006) On the unification of families of skew-normal distributions. Scand J Stat 33:561–574

    Article  MathSciNet  MATH  Google Scholar 

  • Arslan O (2015) Variance-mean mixture of the multivariate skew normal distribution. Stat Pap 56:353–378

    Article  MathSciNet  MATH  Google Scholar 

  • Athayde E, Azevedo A, Barros M, Leiva V (2018) Failure rate of Birnbaum-Saunders distributions: shape, change-point, estimation and robustness. Braz J Probab Statist (to appear)

  • Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Azzalini A (2005) The skew-normal distribution and related multivariate families. Scand J Stat 32:159–188

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat Soc Ser B 61:579–602

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J R Stat Soc Ser B 65:367–389

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (2014) The skew-normal and related families. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen O, Kent J, Sørensen M (1982) Normal variance-mean mixtures and z distributions. Int Stat Rev 50:145–159

    Article  MathSciNet  MATH  Google Scholar 

  • Behboodian J, Jamalizadeh A, Balakrishnan N (2006) A new class of skew-Cauchy distributions. Stat Probab Lett 76:1488–1493

    Article  MathSciNet  MATH  Google Scholar 

  • Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113

    Article  MathSciNet  MATH  Google Scholar 

  • Chandra NK, Roy D (2001) Some results on reverse hazard rate. Probab Eng Inf Sci 15:95–102

    Article  MATH  Google Scholar 

  • Dey S, Alzaatreh A, Zhang C, Kumar D (2017) A new extension of generalized exponential distribution with application to ozone data. Ozone Sci Eng 39:273–285

    Article  Google Scholar 

  • Dominguez-Molina JA, Rocha-Artega A (2007) On the infinite divisibility of some skewed symmetric distributions. Stat Probab Lett 77:644–648

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta PL, Brown N (2001) Reliability studies of skew normal distribution and its application to a strength-stress model. Commun Stat-Theory Methods 30:2427–2445

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta RC, Balakrishnan N (2012) Log-concavity and monotonicity of hazard and reversed hazard functions of univariate and multivariate skew-normal distributions. Metrika 75:181–191

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N (1986) A probabilistic representation of the skew-normal distribution. Scand J Stat 13:271–275

    MathSciNet  MATH  Google Scholar 

  • Jamalizadeh A, Lin TI (2017) A general class of scale-shape mixtures of skew-normal distributions: properties and estimation. Comput Stat 32:451–474

    Article  MathSciNet  MATH  Google Scholar 

  • Kalambet Y, Kozmin Y, Mikhailova K, Nagaev I, Tikhonov P (2011) Reconstruction of chromatographic peaks using the exponentially modified Gaussian function. J Chemom 25:352–356

    Article  Google Scholar 

  • Krupskii P, Huser R, Genton MG (2018) Factor copula models for replicated spatial data. J Am Stat Assoc 113:467–479

    Article  MathSciNet  MATH  Google Scholar 

  • Kozubowski TJ, Nolan JP (2008) Infinite divisibility of skew Gaussian and Laplace laws. Stat Probab Lett 78:654–660

    Article  MathSciNet  MATH  Google Scholar 

  • Lee SX, McLachlan GJ (2013) On mixtures of skew-normal and skew \(t\)-distributions. Adv Data Anal Classif 7:241–266

    Article  MathSciNet  MATH  Google Scholar 

  • Lin TI, Lee JC, Yen SY (2007) Finite mixture modelling using the skew normal distribution. Statistica Sinica 17:909–927

    MathSciNet  MATH  Google Scholar 

  • Louis TA (1982) Finding the observed information when using the EM algorithm. J R Stat Soc Ser B 44:226–232

    MathSciNet  MATH  Google Scholar 

  • Meng XL, Rubin DB (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80:267–278

    Article  MathSciNet  MATH  Google Scholar 

  • Meng XL, Rubin DB (1991) Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm. J Am Stat Assoc 86:899–909

    Article  Google Scholar 

  • Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb–Douglas production functions with composed error. Int Econ Rev 18:435–444

    Article  MATH  Google Scholar 

  • Nadarajah S, Kotz S (2003) Skewed distributions generated by the normal kernel. Stat Probab Lett 65:269–277

    Article  MathSciNet  MATH  Google Scholar 

  • Sadreazami H, Omair Ahmad M, Swamy MNS (2016) A study on image denoising in contourlet domain using the alpha-stable family of distributions. Signal Process 128:459–473

    Article  Google Scholar 

  • Silver JD, Ritchie ME, Smyth GK (2009) Microarray background correction: maximum likelihood estimation for the normal-exponential convolution. Biostatistics 10:352–363

    Article  Google Scholar 

  • Steutel FW, Van Harn K (2003) Infinite divisibility of probability distributions on the real line. Marcel Dekker, New York

    Book  MATH  Google Scholar 

  • Vilca F, Santana L, Leiva V, Balakrishnan N (2011) Estimation of extreme percentiles in Birnbaum–Saunders distributions. Comput Stat Data Anal 55:1665–1678

    Article  MathSciNet  MATH  Google Scholar 

  • Weisberg S (2014) Computing primer for applied linear regression, 4th edn, Using R. http://z.umn.edu/alrprimer

Download references

Acknowledgements

We gratefully acknowledge the incisive comments and suggestions of the Chief Editor, the Associate Editor and two anonymous referees, which led to this greatly improved version of the article.

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Authors and Affiliations

  1. Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Hossein Negarestani, Ahad Jamalizadeh & Sobhan Shafiei

  2. Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

    Narayanaswamy Balakrishnan

Authors
  1. Hossein Negarestani
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  2. Ahad Jamalizadeh
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  3. Sobhan Shafiei
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  4. Narayanaswamy Balakrishnan
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Corresponding author

Correspondence to Ahad Jamalizadeh.

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Negarestani, H., Jamalizadeh, A., Shafiei, S. et al. Mean mixtures of normal distributions: properties, inference and application. Metrika 82, 501–528 (2019). https://doi.org/10.1007/s00184-018-0692-x

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  • DOI: https://doi.org/10.1007/s00184-018-0692-x

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