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Mean mixtures of normal distributions: properties, inference and application

Author

Listed:
  • Hossein Negarestani

    (Shahid Bahonar University of Kerman)

  • Ahad Jamalizadeh

    (Shahid Bahonar University of Kerman)

  • Sobhan Shafiei

    (Shahid Bahonar University of Kerman)

  • Narayanaswamy Balakrishnan

    (McMaster University)

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.

Suggested Citation

  • Hossein Negarestani & Ahad Jamalizadeh & Sobhan Shafiei & Narayanaswamy Balakrishnan, 2019. "Mean mixtures of normal distributions: properties, inference and application," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(4), pages 501-528, May.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:4:d:10.1007_s00184-018-0692-x
    DOI: 10.1007/s00184-018-0692-x
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    2. Hashemi, Farzane & Naderi, Mehrdad & Jamalizadeh, Ahad & Bekker, Andriette, 2021. "A flexible factor analysis based on the class of mean-mixture of normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    3. Arellano-Valle, Reinaldo B. & Azzalini, Adelchi, 2021. "A formulation for continuous mixtures of multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
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    5. Nicolae Tarbă & Mihai-Lucian Voncilă & Costin-Anton Boiangiu, 2022. "On Generalizing Sarle’s Bimodality Coefficient as a Path towards a Newly Composite Bimodality Coefficient," Mathematics, MDPI, vol. 10(7), pages 1-17, March.

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