Permutation Tests for Multivariate Location Problems

The paper presents some permutation test procedures for multivariate location. The tests are based on projected univariate versions of multivariate data. For one-sample cases, the tests are affine invariant and strictly distribution-free for the symmetric null distribution with elliptical direction and their permutation counterparts are conditionally distribution-free when the underlying null distribution of the sample is angularly symmetric. For multi-sample cases, the tests are also affine invariant and permutation counterparts of the tests are conditionally distribution-free for any null distribution with certain continuity. Hence all of the tests in this paper are exactly valid. Furthermore, the equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and of their permutation counterparts under local alternative are investigated. A simulation study shows the tests to perform well compared with some existing tests in the literature, particularly when the underlying null distribution is symmetric whether light-tailed or heavy-tailed. For revealing the influence of data sparseness on the effect of the test, some simulations with different dimensions are also performed.

[1]  I. Blumen,et al.  A New Bivariate Sign Test , 1958 .

[2]  J. Hartigan,et al.  An omnibus test for departures from constant mean , 1990 .

[3]  Kanwar Sen,et al.  A bivariate signed rank test for two sample location problem , 1997 .

[4]  Ronald H. Randles,et al.  Multivariate rank tests for the two-sample location problem , 1990 .

[5]  Michael Buckley,et al.  Detecting a smooth signal: Optimality of cusum based procedures , 1991 .

[6]  Hannu Oja,et al.  On Certain Bivariate Sign Tests and Medians , 1992 .

[7]  P. Munson,et al.  A cubic spline extension of the Durbin-Watson test , 1989 .

[8]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[9]  B. M. Brown,et al.  An Affine Invariant Bivariate Version of the Sign Test , 1989 .

[10]  Ronald H. Randles,et al.  A Bivariate Signed Rank Test for the Two-Sample Location Problem , 1991 .

[11]  E. Giné,et al.  Some Limit Theorems for Empirical Processes , 1984 .

[12]  謙太郎 野間口,et al.  仮説に制約条件がある場合の Bivariate Sign Test , 1986 .

[13]  J. Tukey.  Mathematics and the Picturing of Data , 1975 .

[14]  Hannu Oja,et al.  Bivariate Sign Tests , 1989 .

[15]  R. L. Eubank,et al.  Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests , 1993 .

[16]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[17]  J. Praestgaard.  Permutation and bootstrap Kolmogorov-Smirnov tests for the equality of two distributions , 1995 .

[18]  Hannu Oja,et al.  Affine Invariant Multivariate Multisample Sign Tests , 1994 .

[19]  Joseph P. Romano.  Bootstrap and randomization tests of some nonparametric hypotheses , 1989 .

[20]  P. Sen,et al.  Nonparametric methods in multivariate analysis , 1974 .

[21]  T. P. Hettmansperger,et al.  Statistical Inference Based on Ranks. , 1985 .

[22]  Ronald H. Randles,et al.  A Multivariate Signed-Rank Test for the One-Sample Location Problem , 1990 .

[23]  E. Giné,et al.  Lectures on the central limit theorem for empirical processes , 1986 .

[24]  P. Bickel.  A Distribution Free Version of the Smirnov Two Sample Test in the $p$-Variate Case , 1969 .

[25]  R. Randles.  A Distribution-Free Multivariate Sign Test Based on Interdirections , 1989 .

[26]  S. Hora.  Statistical Inference Based on Ranks , 1986 .

[27]  B. M. Brown,et al.  Affine Invariant Rank Methods in the Bivariate Location Model , 1987 .

[28]  Regina Y. Liu,et al.  A Quality Index Based on Data Depth and Multivariate Rank Tests , 1993 .

[29]  K. Alexander,et al.  Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm , 1984 .

[30]  J. L. Hodges.  A Bivariate Sign Test , 1955 .

[31]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[32]  Chris Beaumont,et al.  Distribution-Free Statistical Methods , 1982 .

[33]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

[34]  Hannu Oja,et al.  Affine Invariant Multivariate One‐Sample Sign Tests , 1994 .

[35]  D. Pollard.  Convergence of stochastic processes , 1984 .