In this paper, we propose robust numerical methods for finding the maximum likelihood estimation of the generalized inverse Gaussian distribution. A comparative analysis of the existing algorithms and the results of numerical experiments are presented. Special attention is paid to reproducibility of the tests.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York (1965).
G. E. Alefeld, F.A. Potra, and Yixun Shi, “Algorithm 748: enclosing zeros of continuous functions,” ACM Trans. Math. Softw., 21, 327–344 (1995).
H. Alzer, “On some inequalities for the gamma and psi functions,” Math. Comput., 66, 373–389 (1997).
G.D. Anderson, R.W. Barnard, K.C. Richards, M. K. Vamanamurthy, and M. Vuorinen, “Inequalities for zero-balanced hypergeometric functions,” Trans. Am. Math. Soc., 347, 1713–1723 (1995).
O. Barndorff-Nielsen, “Exponentially decreasing distributions for the logarithm of particle size,” Proc. R. Soc. London Math. Phys. Sci., 353, No. 1674, 401–419 (1977).
O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, University of Aarhus, Aarhus (1977).
R.P. Brent, Algorithms for Minimization without Derivatives, Dover Publications, New York (1973).
J.B. Campbell, “On Temme’s algorithm for the modified Bessel function of the third kind,” ACM Trans. Math. Softw., 6, No. 4, 581–586 (1980).
I. J. Good, “The population frequencies of species and the estimation of population parameters,” Biometrika, 40, No. 3–4, 237–264 (1953).
B. Jorgensen, Statistical Properties of the Generalized Inverse Gaussian Distribution, University of Aarhus, Aarhus (1980).
V.Yu. Korolev, Probabilistic and Statistical Methods for the Decomposition of Volatility of Chaotic Processes, Moscow State University Publishing House, Msocow (2011).
J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J., 7, 308–313 (1965).
M. Paolella, Intermediate Probability: A Computational Approach, Wiley, New York (2007).
W. H. Press, S. A. Teukolsky, W.T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York (2007).
C. J.F. Ridders, “Accurate computation of F′(x) and F′(x)F″(x),” Adv. Eng. Softw., 4, No. 2, 75–76 (1978).
N. M. Temme, “On the numerical evaluation of the modified Bessel function of the third kind,” J. Comput. Phys., 19, No. 3, 324–337 (1975).
I. J. Thompson and A.R. Barnett, “Modified Bessel functions I ν (z) and K ν (z) of real order and complex argument, to selected accuracy,” Comput. Phys. Commun., 47, No. 2–3, 245–257 (1987).
I. Yaroshenko, Atmosphere: statistical package, http://9il.github.io/atmosphere/(2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
* Research supported by the Russian Scientific Foundation, project 14–11–00364.
Rights and permissions
About this article
Cite this article
Yaroshenko, I. On Robust Algorithm for Finding Maximum Likelihood Estimation of the Generalized Inverse Gaussian Distribution*. J Math Sci 218, 354–362 (2016). https://doi.org/10.1007/s10958-016-3035-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-016-3035-3