Some computational aspects of the generalized von Mises distribution

This article deals with some important computational aspects of the generalized von Mises distribution in relation with parameter estimation, model selection and simulation. The generalized von Mises distribution provides a flexible model for circular data allowing for symmetry, asymmetry, unimodality and bimodality. For this model, we show the equivalence between the trigonometric method of moments and the maximum likelihood estimators, we give their asymptotic distribution, we provide bias-corrected estimators of the entropy, the Akaike information criterion and the measured entropy for model selection, and we implement the ratio-of-uniforms method of simulation.

[1]  R. Gatto.  Information theoretic results for circular distributions , 2009 .

[2]  B. D. Spurr,et al.  A comparison of various methods for estimating the parameters in mixtures of von Mises distributions , 1991 .

[3]  Zhi Zong.  Information-theoretic methods for estimating complicated probability distributions , 2006 .

[4]  G. Kitagawa,et al.  Akaike Information Criterion Statistics , 1988 .

[5]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[6]  M. Kendall.  Theoretical Statistics , 1956, Nature.

[7]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[8]  N. Fisher,et al.  The BIAS of the maximum likelihood estimators of the von mises-fisher concentration parameters , 1981 .

[9]  Russell C. H. Cheng,et al.  Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin , 1983 .

[10]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[11]  V. M. Maksimov,et al.  Necessary and Sufficient Statistics for the Family of Shifts of Probability Distributions on Continuous BiCompact Groups , 1967 .

[12]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[13]  J. Hartigan.  A failure of likelihood asymptotics for normal mixtures , 1985 .

[14]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[15]  S. Rao Jammalamadaka,et al.  Inference for wrapped symmetric α-stable circular models , 2003 .

[16]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[17]  Russell C. H. Cheng,et al.  Non-regular maximum likelihood problems , 1995 .

[18]  Allan Pinkus,et al.  Fourier Series and Integral Transforms: Fourier Series , 1997 .

[19]  G. Kitagawa,et al.  Akaike Information Criterion Statistics , 1988 .

[20]  S. R. Jammalamadaka,et al.  The generalized von Mises distribution , 2007 .

[21]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[22]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[23]  Russell C. H. Cheng,et al.  Corrected Maximum Likelihood in Non‐Regular Problems , 1987 .

[24]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[25]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[26]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .