Reducing multidimensional two-sample data to one-dimensional interpoint comparisons. (English) Zbl 0862.62047

Maa, Jen-Fue; Pearl, Dennis K.; Bartoszyński, Robert

Ann. Stat. 24, No. 3, 1069-1074 (1996).

Summary: The most popular technique for reducing the dimensionality in comparing two multidimensional samples of \({\mathbf X} \sim F\) and \({\mathbf Y} \sim G\) is to analyze distributions of interpoint comparisons based on a univariate function \(h\) (e.g. the interpoint distances). We provide a theoretical foundation for this technique, by showing that having both i) the equality of the distributions of within sample comparisons \((h({\mathbf X}_1, {\mathbf X}_2) =_{\mathcal L} h({\mathbf Y}_1, {\mathbf Y}_2))\) and ii) the equality of these with the distribution of between sample comparisons \(((h({\mathbf X}_1, {\mathbf X}_2) =_{\mathcal L} h({\mathbf X}_3, {\mathbf Y}_3))\) is equivalent to the equality of the multivariate distributions \((F=G)\).

MSC:

62H05

Characterization and structure theory for multivariate probability distributions; copulas

Keywords:

characterization of distributional equality; multivariate distances; distributions of interpoint comparisons; within sample comparisons; between sample comparisons

Software:

pyuvdata

References:

[1]	ATKINSON, E. N., BROWN, B. W. and THOMPSON, J. R. 1989. Parallel algorithms for fixed seed simulation based parameter estimation. In Computer Science and Statistics: Proceedings of the 21st Sy mposium on the Interface 259 261. Z.
[2]	DAVID, H. T. 1958. A three-sample Kolmogorov Smirnov test. Ann. Math. Statist. 29 842 851. Z. · Zbl 0089.14901 · doi:10.1214/aoms/1177706540
[3]	FRIEDMAN, J. H. and RAFSKY, L. C. 1979. Multivariate generalizations of the Wald Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7 697 717. Z. · Zbl 0423.62034 · doi:10.1214/aos/1176344722
[4]	HENZE, N. 1988. A multivariate two-sample test based on the number of nearest neighbor ty pe coincidences. Ann. Statist. 16 772 783. · Zbl 0645.62062 · doi:10.1214/aos/1176350835
[5]	MAA, J. 1993. Simulation-based parameter estimation for multivariate distributions. Ph.D. dissertation, Dept. Statistics, Ohio State Univ. Z.
[6]	PEARL, D. K., BARTOSZy NSKI, R. and HORN, D. J. 1989. A stochastic model for simulation of ínteractions between phy tophagous spider mites and their phy toseiid predators. Exp. Appl. Acarol. 7 143 151. Z.
[7]	SCHILLING, M. F. 1986. Multivariate two-sample tests based on nearest neighbors. J. Amer. Statist. Assoc. 81 779 806.Z. JSTOR: · Zbl 0612.62081 · doi:10.2307/2289012
[8]	WHEEDEN, R. L. and Zy GMUND, A. 1977. Measure and Integral: An Introduction to Real Analy sis. Dekker, New York. · Zbl 0362.26004
[9]	CORNING HAZELTON, INC. R. BARTOSZy NSKI ṔO BOX 7545 DEPARTMENT OF STATISTICS MADISON, WISCONSIN 53707 OHIO STATE UNIVERSITY COLUMBUS, OHIO 43210 E-MAIL: [e-mail] bartoszy [e-mail]