The Power M-Gaussian Distribution: An R-Symmetric Analog of the Exponential-Power Distribution

  • Saria Salah Awadalla
  • Govind S. Mudholkar
  • Ziji Yu


The mode-centric M-Gaussian distribution, which may be considered a fraternal twin of the Gaussian distribution, is an attractive alternative for modeling non-negative, unimodal data, which are often right-skewed. In this paper, we aim to expand upon the existing theory and utility of R-symmetric distributions by introducing a three-parameter generalization of the M-Gaussian distribution, namely the Power M-Gaussian distribution. The basic distributional character of this R-symmetric analog of the exponential-power distribution will be studied extensively. Estimation of the mode, dispersion, and kurtosis parameters will be developed based on both moments and maximum likelihood methods. Simulation and real data examples will be used to evaluate the model.


  1. 1.
    Awadalla, S.S. 2012. Some contributions to the theory and applications of R-symmetry. Ph.D. thesis, Department of Computational Biology, University of Rochester, Rochester, NY, USA.Google Scholar
  2. 2.
    Baker, R. 2008. Probabilistic applications of the Schlömilch transformation. Communications in Statistics - Theory and Methods. 37: 2162–2176.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Box, G.E.P. 1953. A note on regions for tests of kurtosis. Biometrika 40: 465–468.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Box, G.E.P., G.C. Tiao. 1973. Bayesian inference in statistical analysis. Wiley Online Library.Google Scholar
  5. 5.
    Broyden, C.G. 1970. The convergence of a class of double-rank minimization algorithms 1. general considerations. IMA Journal of Applied Mathematics 6: 76–90.CrossRefMATHGoogle Scholar
  6. 6.
    Chaubey, Y.P., G.S. Mudholkar, and M.C. Jones. 2010. Reciprocal symmetry, unimodality and Khintchine’s theorem. Proceedings of the Royal Society of London A 466: 2079–2096.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Diananda, P.H. 1949. Note on some properties of maximum likelihood estimates. Mathematical Proceedings of the Cambridge Philosophical Society 45: 536–544.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fletcher, R. 1970. A new approach to variable metric algorithms. The Computer Journal 13: 317–322.CrossRefMATHGoogle Scholar
  9. 9.
    Folks, J.L., and R.S. Chhikara. 1978. The inverse Gaussian distribution and its statistical application—a review. Journal of the Royal Statistical Society Series B 40: 263–289.MathSciNetMATHGoogle Scholar
  10. 10.
    Goldfarb, D. 1970. A family of variable metric methods derived by variational means. Mathematics of Computation 24: 23–26.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gradshtein, I.S., I.M. Ryzhik. 2007. Table of integrals, series and products, 7th ed. Academic Press.Google Scholar
  12. 12.
    Jones, M.C. 2010. Distributions generated by transformations of scale using an extended Schlomilch transformation. Sankhya A: The Indian Journal of Statistics 72: 359–375.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jones, M.C. 2012. Relationships between distributions with certain symmetries. Statistical and Probability Letters 82: 1737–1744.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jones, M.C., and B.C. Arnold. 2012. Distributions that are both log-symmetric and R-symmetric. Electronic Journal of Statistics 2: 1300–1308.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Khintchine, A.Y. 1938. On unimodal distributions. Izvestiya Nauchno-Issledovatel’skogo Instituta Matematiki i Mekhaniki 2: 1–7.Google Scholar
  16. 16.
    Lange, T.R., H.E. Royals, and L.L. Connor. 1993. Influence of water chemistry on mercury concentration in largemouth bass from Florida lakes. Transactions of the American Fisheries Society 122: 74–84.CrossRefGoogle Scholar
  17. 17.
    Mineo, A.M., and M. Ruggieri. 2005. A software tool for the exponential power distribution: The normalp package. Journal of Statistical Software 12 (4): 1–24.CrossRefGoogle Scholar
  18. 18.
    Mudholkar, G.S., and H. Wang. 2007. IG-symmetry and R-symmetry: Interrelations and applications to the inverse Gaussian theory. Journal of Statistical Planning and Inference 137: 3655–3671.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mudholkar, G.S., and H. Wang. 2007. Product-Convolution of R-symmetric unimodal distributions: An analogue to Wintner’s Theorem. Journal of Statistical Theory and Practice 4: 803–811.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mudholkar, G.S., Z. Yu, and S.S. Awadalla. 2015. The mode-centric M-Gaussian distribution: A model for right skewed data. Statistics & Probability Letters 107: 1–10.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Seshadri, V. 1994. The Inverse Gaussian distribution: A case study in exponential families. Oxford University Press.Google Scholar
  22. 22.
    Seshadri, V. 1999. The inverse Gaussian distribution: Statistical theory and applications. Springer.Google Scholar
  23. 23.
    Shanno, D.F. 1970. Conditioning of quasi-Newton methods for function minimization. Mathematics of Computation 24: 647–656.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Subbotin, M.T. 1923. On the law of frequency of error. Matematicheskii Sbornik 31 (2): 296–301.MATHGoogle Scholar
  25. 25.
    Turner, M.E. 1960. 150. Note: On heuristic estimation methods. Biometrics. 16: 299–301.Google Scholar
  26. 26.
    Tweedie, M.C.K. 1947. Functions of a statistical variate with given means, with special reference to Laplacian distributions. Mathematical Proceedings of the Cambridge Philosophical Society 43: 41–49.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Watson, G.N. 1958. A treatise on the theory of Bessel functions. Cambridge Press.Google Scholar
  28. 28.
    Wintner, A. 1938. Asymptotic distributions and infinite convolutions. Edwards Brothers.Google Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Saria Salah Awadalla
    • 1
  • Govind S. Mudholkar
    • 2
  • Ziji Yu
    • 3
  1. 1.Division of Epidemiology and BiostatisticsUIC School of Public Health (SPH-PI)ChicagoUSA
  2. 2.Department of Statistics and BiostatisticsUniversity of RochesterRochesterUSA
  3. 3.Biostatistics DepartmentJazz PharmaceuticalsPalo AltoUSA

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