A generalized Φ-divergence for asymptotically multivariate normal models

I. Csiszar's (Magyar. Tud. Akad. Mat. Kutato Int. Kozl8 (1963), 85-108) [phi]-divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131-142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized [phi]-divergence that unifies the [phi]-divergence approach with that of C. R. Rao and S. K. Mitra ("Generalized Inverse of Matrices and Its Applications," Wiley, New York, 1971) and derive weak convergence to a [chi]2 distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.

[1]  J. Neyman,et al.  Contribution to the Theory of the {χ superscript 2} Test , 1949 .

[2]  V. I. Romanovsky.  Discrete Markov Chains , 1970 .

[3]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .

[4]  I. Vajda,et al.  Asymptotic divergence of estimates of discrete distributions , 1995 .

[5]  C. R. Rao,et al.  Generalized Inverse of Matrices and its Applications , 1972 .

[6]  Timothy R. C. Read,et al.  Multinomial goodness-of-fit tests , 1984 .

[7]  S. Kullback,et al.  Information Theory and Statistics , 1959 .

[8]  Pierre L'Ecuyer,et al.  Random Number Generators: Selection Criteria and Testing , 1998 .

[9]  J. Gani,et al.  Contributions to Probability. , 1984 .

[10]  Leandro Pardo,et al.  On the applications of divergence type measures in testing statistical hypotheses , 1994 .

[11]  S. M. Ali,et al.  A General Class of Coefficients of Divergence of One Distribution from Another , 1966 .

[12]  K. Matusita.  Distance and decision rules , 1964 .

[13]  Peter Hellekalek,et al.  On the assessment of random and quasi-random point sets , 1998 .

[14]  On the construction of least favourable distributions , 1978 .

[15]  T. Snijders.  Multivariate Statistics and Matrices in Statistics , 1995 .

[16]  K. Pearson.  On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling , 1900 .

[17]  T. Papaioannou,et al.  Divergence statistics: sampling properties and multinomial goodness of fit and divergence tests , 1990 .

[18]  Timothy R. C. Read,et al.  Goodness-Of-Fit Statistics for Discrete Multivariate Data , 1988 .

[19]  Leandro Pardo,et al.  Asymptotic behaviour and statistical applications of divergence measures in multinomial populations: a unified study , 1995 .

[20]  I. Vincze.  On the Concept and Measure of Information Contained in an Observation , 1981 .

[21]  Leandro Pardo,et al.  Divergence-based estimation and testing of statistical models of classification , 1995 .

[22]  Makoto Matsumoto,et al.  Getting rid of correlations among pseudorandom numbers: discarding versus tempering , 1999, TOMC.

[23]  S. Kullback,et al.  Minimum discrimination information estimation. , 2006, Biometrics.

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

[25]  Stefan Wegenkittl.  Monkeys, gambling, and return times: assessing pseudorandomness , 1999, WSC '99.

[26]  Leandro Pardo,et al.  Asymptotic properties of divergence statistics in a stratified random sampling and its applications to test statistical hypotheses , 1994 .

[27]  D. E. Boekee,et al.  A generalization of the Fisher information measure , 1977 .

[28]  I. Vajda,et al.  Convex Statistical Distances , 2018, Statistical Inference for Engineers and Data Scientists.

[29]  C. R. Rao,et al.  An Alternative to Correspondence Analysis Using Hellinger Distance. , 1997 .

[30]  K. Matusita.  Decision Rules, Based on the Distance, for Problems of Fit, Two Samples, and Estimation , 1955 .

[31]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .