Nonparametric tests of independence using copula-based Renyi and Tsallis divergence measures

Keywords: ‎Independence test, Renyi divergence, Tsallis divergence, Copula density, Probit-transformation


‎We introduce new nonparametric independence tests based on R\'enyi and Tsallis divergence measures and copula density function‎. ‎These tests reduce the complexity of calculations because they only depend on the copula density‎. ‎The copula density estimated using the local likelihood probit-transformation method is appropriate for the identification of independence‎. ‎Also‎, ‎we present the consistency of the copula-based R\'enyi and Tsallis divergence measures estimators that are considered as test statistics‎. ‎A simulation study is provided to compare the empirical power of these new tests with the independence test based on the empirical copula‎. ‎The simulation results show that the suggested tests outperform in weak dependency‎. ‎Finally‎, ‎an application in hydrology is presented‎.

Author Biographies

Morteza Mohammadi, Department of Statistics, University of Zabol, Zabol, IRAN
Assistant Professor, Department of Statistics, Faculty of Sciences, University of Zabol, Zabol, Iran
Mahdi Emadi, Department of Statistics‎, ‎Ferdowsi University of Mashhad‎, ‎Mashhad‎, ‎Iran
Associate Professor of Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran


Ahmad, I., and Lin, P. E. (1976). A nonparametric estimation of the entropy for absolutely continuous distributions (corresp.). IEEE Transactions on Information Theory, 22(3), 372-375.

Alajaji, F., Chen, P. N., and Rached, Z. (2004). Csisza/spl acute/r's cutoff rates for the general hypothesis testing problem. IEEE Transactions on Information Theory, 50(4), 663-678.

Arndt, C. (2012). Information measures: information and its description in science and engineering. Springer Science and Business Media.

Belalia, M., Bouezmarni, T., Lemyre, F. C., and Taamouti, A. (2017). Testing independence based on Bernstein empirical copula and copula density. Journal of Nonparametric Statistics, 29(2), 346-380.

Blum, J. R., Kiefer, J., and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function. The Annals of mathematical statistics, 485-498.

Blumentritt, T. and Schmid, F. (2012). Mutual information as a measure of multivariate association: analytical properties and statistical estimation. Journal of Statistical Computation and Simulation, 82(9), 1257-1274.

Charpentier, A., Fermanian, J., and Scaillet, O. (2006). Copulas: From theory to application in finance, chapter The Estimation of Copulas: Theory and Practice. Risk Books, Torquay, UK, 1, 35-62.

Chen, L. and Guo, S. (2019). Copulas and Its Application in Hydrology and Water Resources. Springer.

Csiszar, I. (1995). Generalized cutoff rates and Rnyi's information measures. IEEE Transactions on information theory, 41(1), 26-34.

Deheuvels, P. (1979). La fonction de dependence empirique et ses proprietes, Un test non parametrique d'independance. Bulletin de la classe des sciences, Academie Royale de Belgique, 5e serie, 65, 274-292.

Demarta, S. and McNeil, A. J. (2005). The t copula and related copulas. International statistical review, 73(1), 111-129.

Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873.

Genest, C., Quessy, J. F., and Remillard, B. (2006). Goodness-of-fit procedures for copula models based on the probability integral transformation. Scandinavian Journal of Statistics, 33(2), 337-366.

Genest, C. and Remillard, B. (2004). Test of independence and randomness based on the empirical copula process. Test, 13(2), 335-369.

Gil, M., Alajaji, F., and Linder, T. (2013). Rnyi divergence measures for commonly used univariate continuous distributions. Information Sciences, 249, 124-131.

Guttman, N. B. (1998). Comparing the palmer drought index and the standardized precipitation index1. JAWRA Journal of the American Water Resources Association, 34(1), 113-121.

Havrda, J. and Charvat, F. (1967). Quantification method of classification processes. Concept of structural a-entropy. Kybernetika, 3(1), 30-35.

Hofert, M., Kojadinovic, I., Maechler, M., and Yan, J. (2018). copula: Multivariate dependence with copulas. R package version 0.999-19, URL

Joe, H. (1989). Estimation of entropy and other functionals of a multivariate density. Annals of the Institute of Statistical Mathematics, 41(4), 683-697.

Kullback, S. and Leibler, R. A. (1951). On information and sufficiency. The Annals of mathematical statistics, 22(1), 79-86.

Ma, M., Song, S., Ren, L., Jiang, S., and Song, J. (2013). Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrological Processes, 27(8), 1175-1190.

Ma, J. and Sun, Z. (2011). Mutual information is copula entropy. Tsinghua Science and Technology, 16(1), 51-54.

McKee, T. B., Doesken, N. J., and Kleist, J. (1993). The relationship of drought frequency and duration to time scales. In Proceedings of the 8th Conference on Applied Climatology(Vol. 17, No. 22, pp. 179-183). Boston, MA: American Meteorological Society.

Micheas, A. C. and Zografos, K. (2006). Measuring stochastic dependence using phi-divergence. Journal of Multivariate Analysis, 97(3), 765-784.

Mishra, A. K. and Singh, V. P. (2010). A review of drought concepts. Journal of hydrology, 391(1-2), 202-216.

Mohammadi, M., Emadi, M., and Amini, M. (2022). Testing bivariate independence based on alpha-divergence by improved probit transformation method for copula density estimation. Communications in Statistics-Simulation and Computation, 1-19.

Nagler, T. (2018). kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22.

Nelsen, R. B. (2007). An introduction to copulas. Springer Science and Business Media.

Renyi, A. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics. The Regents of the University of California.

Shiau, J. T. (2006). Fitting drought duration and severity with two-dimensional copulas. Water resources management, 20(5), 795-815.

Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ. inst. statist. univ. Paris, 8, 229-231.

Song, S. and Singh, V. P. (2010). Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm. Stochastic Environmental Research and Risk Assessment, 24(5), 783-805.

Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of statistical physics, 52(1-2), 479-487.

Wong, G., Lambert, M. F., and Metcalfe, A. V. (2008). Trivariate copulas for characterisation of droughts. Anziam Journal, 49, 306-323.

How to Cite
Mohammadi, M., & Emadi, M. (2023). Nonparametric tests of independence using copula-based Renyi and Tsallis divergence measures. Statistics, Optimization & Information Computing, 11(4), 949-962.
Research Articles