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On the Applications of Divergence Type Measures in Testing Statistical Hypotheses

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  • Salicru, M.
  • Morales, D.
  • Menendez, M. L.
  • Pardo, L.

Abstract

The fundamentals of information theory and also their applications to testing statistical hypotheses have been known and available for some time. There is currently a new and heterogeneous development of statistical procedures, based on information measures, scattered through the literature. In this paper a unification is attained by consistent application of the concepts and properties of information theory. Our aim is to examine a wide range of divergence type measures and their applications to statistical inferences, with special emphasis on multinomial and multivariate normal distributions, The "maximum likelihood" and the "minimum discrepancy" principles are combined here in order to derive new approaches to the discrimination between two groups or populations. To study the asymptotic properties of divergence statistics, we propose a unified expression, called (, )-divergence, which includes as particular cases most divergences. Under different assumptions it is shown that the asymptotic distributions of the (, )-divergences are either normal or chi square. From the previous results a wide range of statistical hypotheses about the parameters of one or two populations can be tested To help clarify the discussion and provide a simple illustration examples are given.

Suggested Citation

  • Salicru, M. & Morales, D. & Menendez, M. L. & Pardo, L., 1994. "On the Applications of Divergence Type Measures in Testing Statistical Hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 372-391, November.
    • Handle: RePEc:eee:jmvana:v:51:y:1994:i:2:p:372-391
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    Cited by:

    1. Alejandro C. Frery & Juliana Gambini, 2020. "Comparing samples from the $${\mathcal {G}}^0$$G0 distribution using a geodesic distance," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(2), pages 359-378, June.
    2. Valdevino Félix de Lima, Wenia & David Costa do Nascimento, Abraão & José Amorim do Amaral, Getúlio, 2021. "Distance-based tests for planar shape," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    3. W. V. Félix de Lima & A. D. C. Nascimento & G. J. A. Amaral, 2021. "Entropy-based pivotal statistics for multi-sample problems in planar shape," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 153-178, March.
    4. Javier E. Contreras-Reyes & Mohsen Maleki & Daniel Devia Cortés, 2019. "Skew-Reflected-Gompertz Information Quantifiers with Application to Sea Surface Temperature Records," Mathematics, MDPI, vol. 7(5), pages 1-14, May.
    5. Basu, Ayanendranath & Chakraborty, Soumya & Ghosh, Abhik & Pardo, Leandro, 2022. "Robust density power divergence based tests in multivariate analysis: A comparative overview of different approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    6. Mora, Ricardo & Ruiz-Castillo, Javier, 2010. "A Kullback-Leibler measure of conditional segregation," UC3M Working papers. Economics we1015, Universidad Carlos III de Madrid. Departamento de Economía.
    7. Martín, Nirian & Balakrishnan, Narayanaswami, 2011. "Hypothesis testing in a generic nesting framework with general population distributions," DES - Working Papers. Statistics and Econometrics. WS ws113527, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Kakizawa, Yoshihide, 1997. "Parameter estimation and hypothesis testing in stationary vector time series," Statistics & Probability Letters, Elsevier, vol. 33(3), pages 225-234, May.
    9. Basu, A. & Mandal, A. & Pardo, L., 2010. "Hypothesis testing for two discrete populations based on the Hellinger distance," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 206-214, February.
    10. Martín, N. & Balakrishnan, N., 2013. "Hypothesis testing in a generic nesting framework for general distributions," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 1-23.
    11. Domingo Morales & Leandro Pardo, 2001. "Some approximations to power functions of ϕ-divergence tests in parametric models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 249-269, December.
    12. Menéndez, M. L. & Morales, D. & Pardo, L. & Zografos, K., 1999. "Statistical inference for finite Markov chains based on divergences," Statistics & Probability Letters, Elsevier, vol. 41(1), pages 9-17, January.
    13. Morales, D. & Pardo, L. & Vajda, I., 1997. "Some New Statistics for Testing Hypotheses in Parametric Models, ," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 137-168, July.
    14. Abraão Nascimento & Jodavid Ferreira & Alisson Silva, 2023. "Divergence-based tests for the bivariate gamma distribution applied to polarimetric synthetic aperture radar," Statistical Papers, Springer, vol. 64(5), pages 1439-1463, October.
    15. Tomáš Hobza & Domingo Morales & Leandro Pardo, 2014. "Divergence-based tests of homogeneity for spatial data," Statistical Papers, Springer, vol. 55(4), pages 1059-1077, November.
    16. Alexander Katzur & Udo Kamps, 2020. "Classification using sequential order statistics," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(1), pages 201-230, March.
    17. M. Menéndez & J. Pardo & L. Pardo, 2001. "Csiszar’s ϕ-divergences for testing the order in a Markov chain," Statistical Papers, Springer, vol. 42(3), pages 313-328, July.
    18. Alba-Fernández, V. & Jiménez-Gamero, M.D., 2009. "Bootstrapping divergence statistics for testing homogeneity in multinomial populations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(12), pages 3375-3384.
    19. Diédhiou, Alassane & Ngom, Papa, 2009. "Cutoff time based on generalized divergence measure," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1343-1350, May.
    20. Wegenkittl, Stefan, 2002. "A Generalized [phi]-Divergence for Asymptotically Multivariate Normal Models," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 288-302, November.
    21. Conde, J. & Salicrú, M., 1998. "Uniform association in contingency tables associated to Csiszar divergence," Statistics & Probability Letters, Elsevier, vol. 37(2), pages 149-154, February.

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