Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

  • Ibrahim A. Ahmad


LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by:\(\lambda = \lambda (F,G) = 2\int {f(x)g(x)dx} /\left[ {\int {f^2 (x)dx + \int {g^2 (x)dx} } } \right]\). Based on two independent samples it is proposed to estimate λ by\(\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]\), whereFn(x) andGn(x) are the empirical distribution functions ofF(x) andG(x) respectively and\(\hat f(x)\) and\(\hat g(x)\) are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of\(\hat \lambda \) is presented and a two sample goodness-of-fit test is presented based on\(\hat \lambda \). Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f0(x), withf0(x) completely known and for testing symmetry, i.e., testingH0:f(x)=f(−x).


Probability Density Function Central Limit Theorem Bounded Variation Kernel Estimate Lebesgue Dominate Convergence Theorem 


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  1. [1]
    Ahmad, I. A. and Lin, P. E. (1977). Non parametric density estimation for dependent variables with application, under revision.Google Scholar
  2. [2]
    Ahmad, I. A. and Van Belle, G. (1974). Measuring affinity of distributions.Reliability and Biometry, Statistical Analysis of Life Testing, (eds., Proschan and R. J. Serfling), SIAM, Philadelphia, 651–668.Google Scholar
  3. [3]
    Bhattacharayya, G. K. and Roussas, G. (1969). Estimation of certain functional of probability density function,Skand. Aktuarietidskr.,52, 203–206.MathSciNetGoogle Scholar
  4. [4]
    Billingsley, P. (1968).Convergence of Probability Measures, John Wiley and Sons, New York.zbMATHGoogle Scholar
  5. [5]
    Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics,Ann. Math. Statist.,29, 972–994.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator.Ann. Math. Statist.,27, 642–669.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Matusita, K. (1955). Decision rules based on the distance for the problems of fit, two samples, and estimation,Ann. Math. Statist.,26, 631–640.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Matusita, K. (1964). Distance and decision rules,Ann. Inst. Statist. Math.,16, 305–315.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Matusita, K. (1966). A distance and related statistics in multivariate analysis,Multivariate Analysis I, (ed. P. R. Krishnaiah), Academic Press, New York, 187–200.Google Scholar
  10. [10]
    Matusita, K. (1967a). Classification Based on Distance in Multivariate Gaussian Cases,Proc. Fifth Berkeley Symp. Math. Statist. Prob., Vol. I, 299–304.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Matusita, K. (1967b). On the notion of affinity of several distributions and some of its applications,Ann. Inst. Statist. Math.,19, 181–192.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Matusita, K. (1971). Some properties of affinity and applications,Ann. Inst. Statist. Math.,23, 137–155.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Matusita, K. (1973). Correlation and affinity in Gaussian cases,Multivariate Analysis III, (ed., P. R. Krishnaiah), Academic Press, New York, 345–349.CrossRefGoogle Scholar
  14. [14]
    Matusita, K. and Akaike, H. (1956). Decision rules based on the distance for the problem of independence, invariance, and two samples,Ann. Inst. Statist. Math.,7, 67–80.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Nadaraya, E. A. (1965). On nonparametric estimation of density function and regression curve,Theory Prob. Appl.,10, 186–190.CrossRefGoogle Scholar
  16. [16]
    Parzen, E. (1962). On the estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Philipp, W. (1969). The central limit theorem for mixing sequences of random variables.Z. Wahrscheinlickeitsth.,12, 155–171.CrossRefGoogle Scholar
  18. [18]
    Resenblatt, M. (1956a). Remarks on some nonparametric estimates of a density function.Ann. Math. Statist.,27, 832–837.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Rosenblatt, M. (1956b). A central limit theorem and a strong mixing condition,Proc. Nat. Acad. Sci. USA,42, 43–47.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Royden, H. L. (1968),Real Analysis (Second Edition), Macmillan, New York.zbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1980

Authors and Affiliations

  • Ibrahim A. Ahmad
    • 1
  1. 1.McMaster UniversityCanada

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