Brazilian Journal of Probability and Statistics

Multi-sample Rényi test statistics

Tomáš Hobza, Isabel Molina, and Domingo Morales

Full-text: Open access


This paper focuses on testing composite hypotheses about parameters of s independent samples of different sizes. With this purpose, it introduces test statistics based on the family of Rényi divergences between likelihoods. The asymptotic distributions of the proposed test statistics and of the likelihood ratio statistic are derived under standard regularity assumptions. An application to test the homogeneity of variances in data from families belonging to different populations is described and, under this setup, a simulation experiment compares the small sample performance of the likelihood ratio test and some members of the Rényi family of tests. The experiment indicates that some of the Rényi tests perform better under null hypothesis.

Article information

Braz. J. Probab. Stat., Volume 23, Number 2 (2009), 196-215.

First available in Project Euclid: 26 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Rényi divergence divergence statistics testing composite hypotheses homogeneity of variances


Hobza, Tomáš; Molina, Isabel; Morales, Domingo. Multi-sample Rényi test statistics. Braz. J. Probab. Stat. 23 (2009), no. 2, 196--215. doi:10.1214/08-BJPS008.

Export citation


  • Bhandary, M. and Alam, M. K. (2000). Test for the equality of intraclass correlation coefficients under unequal family sizes for several populations. Communications in Statistics. Theory and Methods 29 755–768.
  • Csiszár, I. (1963). Eine Informationstheoretische Ungleidung und ihre Anwendung auf den Bewis del Ergodizität on Markhoffschen Ketten. In Publications of the Mathematical Institute, Hungarian Academy of Sciences 8 85–108. Budapest, Hungary.
  • Ferguson, T. S. (1988). A Course in Large Sample Theory. Chapman & Hall, New York.
  • Hobza, T., Molina, I. and Morales, D. (2003). Likelihood divergence statistics for testing hypotheses in familial data. Communications in Statistics. Theory and Methods 32 415–434.
  • Kullback, S. (1959). Information Theory and Statistics. Wiley, New York.
  • Kupperman, M. (1957). Further applications of information theory to multivariate analysis and statistical inference. Ph.D. dissertation, The George Washington University.
  • Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer, New York.
  • Liese, F. and Vajda, I. (1987). Convex Statistical Distances. Teubner, Leipzig.
  • Morales, D., Pardo, L. and Pardo, M. C. (2001). Likelihood divergence statistics for testing hypotheses about multiple populations. Communications in Statistics. Simulation and Computation 30 867–884.
  • Morales, D., Pardo, L. and Vajda, I. (1997). Some new statistics for testing composite hypotheses in parametric models. Journal of Multivariate Analysis 62 137–168.
  • Morales, D., Pardo, L. and Vajda, I. (2000). Rényi statistics in directed families of exponential experiments. Statistics 34 151–174.
  • Morales, D., Pardo, L., Pardo, M. C. and Vajda, I. (2004). Rényi statistics for testing composite hypotheses in general exponential models. Statistics 38 133–147.
  • Rényi, A. (1961). On measures of entropy and information. In Fourth Berkeley Symposium on Mathematics, Statistics and Probability 1 547–561. Univ. California Press, Berkeley.
  • Salicrú, M., Menéndez, M. L., Morales, D. and Pardo, L. (1994). On the applications of divergence type measures in testing statistical hypotheses. Journal of Multivariate Analysis 51 372–391.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Srivastava, M. S. (1984). Estimation of interclass correlations in familial data. Biometrika 71 177–185.
  • Srivastava, M. S. and Katapa, R. S. (1986). Comparison of estimators of interclass and intraclass correlations from familial data. La Revue Canadienne de Statistique 14 29–42.