Cramer-Von Mises Statistics for Testing the Equality of Two Distributions
In the study, two projected integrated empirical processes for testing the equality of two multivariate distributions are introduced. The bootstrap is used for determining the approximate critical values. The result shows that the test statistics and thei
- PDF / 394,354 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 5 Downloads / 204 Views
Cramer-Von Mises Statistics for Testing the Equality of Two Distributions Qun Huang and Ping Jing
Abstract In the study, two projected integrated empirical processes for testing the equality of two multivariate distributions are introduced. The bootstrap is used for determining the approximate critical values. The result shows that the test statistics and their bootstrap version have the same limit if the null hypothesis is true. A number-theoretic method is applied to the simulation of efficient computation of the bootstrap critical values.
Keywords Bootstrap Integrated empirical distribution function empirical process Number-theoretic methods Projection pursuit
Integrated
10.1 Introduction In the field of statistical inference, equality of two distributions testing has been drawing great attention. Præstgaard [10] carried out a study about bootstrap Kolmogorov-Smirnov tests for the equality of two distributions. Jing and Dai [8] discussed PP bootstrap Kolmogorov-Smirnov tests for the equality of two multivariate distributions with projection pursuit technique employed. In this paper the bootstrap to another type of statistics was applied to test the equality of two multivariate distributions. Let X1; . . .; Xm and Y1; . . .; Yn be independent scalar random variables, where X1; . . .; Xm are i.i.d. with continuous distribution function F, and Y1; . . .; Yn are i.i.d. with continuous distribution function G. In [5], the
Q. Huang (&) Beijing City University, Beijing 100083, China e-mail: [email protected] P. Jing China University of Mining and Technology, Beijing 100083, China e-mail: [email protected]
S. Li et al. (eds.), Frontier and Future Development of Information Technology in Medicine and Education, Lecture Notes in Electrical Engineering 269, DOI: 10.1007/978-94-007-7618-0_10, Springer Science+Business Media Dordrecht 2014
93
94
Q. Huang and P. Jing
following integrated empirical process was given consideration to test the hypothesis H0:F = G, recently. rffiffiffiffiffiffiffiffiffiffiffiffi mn bmn ðtÞ ¼ F m ðtÞ Gn ðtÞ ; 1\t\ þ 1 ð10:1Þ mþn where F m ðt Þ ¼
Z
t
Fm ð xÞdFm ð xÞ; 1
Gn ðtÞ ¼
Z
t
Gn ð yÞdGn ð yÞ 1
are the so-called integrated empirical distribution functions. Here Fm ð xÞ ¼ m1
m n X X I Xj x and Gn ð yÞ ¼ n1 I Yj y j¼1
j¼1
are the empirical distribution functions of the X-sample and Y-sample, respectively. Based on (10.1), the same authors defined the integrated KolmogorovSmirnov (K-S) statistic ð10:2Þ Dmn ¼ supbmn ðtÞ t2R
as their test statistic for testing the equality of two univariate distributions. The researcher of the study holds that testing the equality of two multivariate distributions using a bootstrap statistic based on (10.3) should be considered. To describe the test statistics more precisely, X1; . . .; Xm should be pdimensional i.i.d. observations having underlying distribution P on a probability space (S, /) with Y1; . . .; Yn being independent samples from a distribution Q on the same probability space. It is aimed to test the hypothe
Data Loading...
