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

- 28 Downloads
- 11 Citations

## Abstract

Let*F* and*G* denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functions*f* and*g*, respectively. A measure of the closeness between*F* and*G* 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\(\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]\), where*F*_{n}*(x)* and*G*_{n}*(x)* are the empirical distribution functions of*F(x)* and*G(x)* respectively and\(\hat f(x)\) and\(\hat g(x)\) are taken to be the so-called kernel estimates of*f(x)* and*g(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., when*g(x)=f*_{0}*(x)*, with*f*_{0}*(x)* completely known and for testing symmetry, i.e., testing*H*_{0}:*f(x)=f(−x)*.

## Keywords

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

Unable to display preview. Download preview PDF.

## References

- [1]Ahmad, I. A. and Lin, P. E. (1977). Non parametric density estimation for dependent variables with application, under revision.Google Scholar
- [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]Bhattacharayya, G. K. and Roussas, G. (1969). Estimation of certain functional of probability density function,
*Skand. Aktuarietidskr.*,**52**, 203–206.MathSciNetGoogle Scholar - [4]Billingsley, P. (1968).
*Convergence of Probability Measures*, John Wiley and Sons, New York.zbMATHGoogle Scholar - [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]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]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]Matusita, K. (1964). Distance and decision rules,
*Ann. Inst. Statist. Math.*,**16**, 305–315.MathSciNetCrossRefGoogle Scholar - [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]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]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]Matusita, K. (1971). Some properties of affinity and applications,
*Ann. Inst. Statist. Math.*,**23**, 137–155.MathSciNetCrossRefGoogle Scholar - [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]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]Nadaraya, E. A. (1965). On nonparametric estimation of density function and regression curve,
*Theory Prob. Appl.*,**10**, 186–190.CrossRefGoogle Scholar - [16]Parzen, E. (1962). On the estimation of a probability density function and mode,
*Ann. Math. Statist.*,**33**, 1065–1076.MathSciNetCrossRefGoogle Scholar - [17]Philipp, W. (1969). The central limit theorem for mixing sequences of random variables.
*Z. Wahrscheinlickeitsth.*,**12**, 155–171.CrossRefGoogle Scholar - [18]Resenblatt, M. (1956a). Remarks on some nonparametric estimates of a density function.
*Ann. Math. Statist.*,**27**, 832–837.MathSciNetCrossRefGoogle Scholar - [19]Rosenblatt, M. (1956b). A central limit theorem and a strong mixing condition,
*Proc. Nat. Acad. Sci. USA*,**42**, 43–47.MathSciNetCrossRefGoogle Scholar - [20]Royden, H. L. (1968),
*Real Analysis*(Second Edition), Macmillan, New York.zbMATHGoogle Scholar