Multivariate Location Problem Research Articles

Multivariate tests for the multi‐sample location problem based on depth function

In this paper, a class of affine‐invariant tests is presented for the multi‐sample multivariate location problem. In the procedure to derive the asymptotic distribution of the tests under the null hypothesis, we do not require any symmetric assumption of the distribution functions. The asymptotic relative efficiency of the tests is discussed under the class of elliptically symmetric distributions. Further comparisons are made among several statistics using Monte Carlo results. Asymptotic relative efficiencies along with Monte Carlo results indicate that selected members of the proposed class perform very well for a broad class of distributions. Finally, we apply our proposed tests to Egyptian skulls data for multivariate five different periods comparisons.

Non‐parametric depth‐based tests for the multivariate location problem

SummaryIn this paper, using the notion of data depth, we describe two classes of affine invariant test statistics for the one‐sample location problem. The tests are implemented through the idea of permutation tests. The performance of the test against some competitors is investigated with an extensive simulation study. It is observed that the tests perform well when compared to their competitors for a wide spectrum of alternatives. If the proposed test is defined based on a moment‐free depth function, then it is not inherently required to have finite moments of any order and the tests have broader applicability than some of the existing tests. The robustness property of the proposed tests is considered with a simulation study. Finally, we apply the tests to a real data example.

A New Nonparametric Test for Two Sample Multivariate Location Problem with Application to Astronomy

This paper provides a nonparametric test for the identity of two multivariate continuous distribution functions when they differ in locations. The test uses Wilcoxon rank-sum statistics on distances between observations for each of the components and is unaffected by outliers. It is numerically compared with two existing procedures in terms of power. The simulation study shows that its power is strictly increasing in the sample sizes and/or in the number of components. The applicability of this test is demonstrated by use of two astronomical data sets on early-type galaxies.

Multivariate tests based on interpoint distances with application to magnetic resonance imaging.

The multivariate location problem is addressed. The most familiar method to address the problem is the Hotelling test. When the hypothesis of normal distributions holds, the Hotelling test is optimal. Unfortunately, in practice the distributions underlying the samples are generally unknown and without assuming normality the finite sample unbiasedness of the Hotelling test is not guaranteed. Moreover, high-dimensional data are increasingly encountered when analyzing medical and biological problems, and in these situations the Hotelling test performs poorly or cannot be computed. A test that is unbiased for non-normal data, for small sample sizes as well as for two-sided alternatives and that can be computed for high-dimensional data has been recently proposed and is based on the ranks of the interpoint Euclidean distances between observations. Five modifications of this test are proposed and compared to the original test and the Hotelling test. Unbiasedness and consistency of the tests are proven and the problem of power computation is addressed. It is shown that two of the modified interpoint distance-based tests are always more powerful than the original test. Particularly, the modified test based on the Tippett criterium is suggested when the assumption of normality is not tenable and/or in case of high-dimensional data with complex dependence structure which are typical in molecular biology and medical imaging. A practical application to a case-control study where functional magnetic resonance imaging is used is discussed.

A weighted multivariate sign test for cluster-correlated data

We consider the multivariate location problem with cluster-correlated data. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Under weak assumptions, the test statistic is asymptotically distributed as a chi-squared random variable as the number of clusters goes to infinity. The asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality. Optimal weights maximizing Pitman asymptotic efficiency are provided. These weights depend on the cluster sizes and on the intracluster correlation. Several approaches for estimating these weights are presented. Using Pitman asymptotic efficiency, we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster. A multivariate weighted t-test is also introduced. The finite-sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive. A real data example illustrates the practical application of the methodology.

Combining Univariate Tests for Multivariate Location Problem

In this article, three methods of combining dependent univariate tests are studied. The Bahadur approximate efficiencies are derived under the asymptotic normal assumption. These procedures are applied to the multivariate location problem and compared with two Hotelling-type tests. A Monte Carlo study indicates that in certain cases the powers of the combination methods are much better than Hotelling's T 2 and other multivariate nonparametric tests.

SAS/IMLMacros for a Multivariate Analysis of Variance Based on Spatial Signs

Recently, new nonparametric multivariate extensions of the univariate sign methods have been proposed. Randles (2000) introduced an affine invariant multivariate sign test for the multivariate location problem. Later on, Hettmansperger and Randles (2002) considered an affine equivariant multivariate median corresponding to this test. The new methods have promising efficiency and robustness properties. In this paper, we review these developments and compare them with the classical multivariate analysis of variance model. A new SAS/IML tool for performing a spatial sign based multivariate analysis of variance is introduced.

A multivariate signed sum test for theone-sample location problem

An affine-invariant signed sum test based on interdirections is proposed for the one-sample multivariate location problem. The test proposed, including the two-sided univariate Wilcoxon signed-rank test as a special case, is somewhat like applying the interdirection sign test to the sums of pairs of observed vectors. The proposed statistic is shown to have a limiting x 2 p null distribution when the underlying distribution is elliptically symmetric. In addition, the asymptotic distribution of the statistic under certain contiguous alternatives is obtained for elliptically symmetric distributions with a particular density function form. Comparisons made between the proposed test and Hotelling's T 2 via Pitman asymptotic relative efficiencies show the signed sum test performs better than Hotelling's T 2 when the underlying distribution is heavy-tailed.

A Multivariate Signed-Rank Test for the One-Sample Location Problem

Abstract An affine-invariant signed-rank test is proposed for the one-sample multivariate location problem. The test suggested is a modification of Randles's multivariate sign test based on interdirections, which extends Blumen's bivariate procedure to the multidimensional setting. Comparisons are made between the proposed statistic and several competitors via Pitman asymptotic relative efficiencies and Monte Carlo results. The signed-rank statistic appears to be robust. It performs better than its competitors when the distribution is light-tailed, and virtually as well as Hotelling's T 2 under multivariate normality. For heavy-tailed distributions the signed-rank statistic performs better than Hotelling's T 2 but not as well as Randles's statistic.

A Multivariate Signed-Rank Test for the One-Sample Location Problem

Abstract An affine-invariant signed-rank test is proposed for the one-sample multivariate location problem. The test suggested is a modification of Randles's multivariate sign test based on interdirections, which extends Blumen's bivariate procedure to the multidimensional setting. Comparisons are made between the proposed statistic and several competitors via Pitman asymptotic relative efficiencies and Monte Carlo results. The signed-rank statistic appears to be robust. It performs better than its competitors when the distribution is light-tailed, and virtually as well as Hotelling's T 2 under multivariate normality. For heavy-tailed distributions the signed-rank statistic performs better than Hotelling's T 2 but not as well as Randles's statistic.

A Distribution-Free Multivariate Sign Test Based on Interdirections

Abstract Distribution-free tests are investigated for the one-sample multivariate location problem. Counts, called interdirections, which measure the angular distance between two observation vectors relative to the positions of the other observations, are introduced. These counts are invariant under nonsingular linear transformations and have a small-sample distribution-free property over a broad class of population models, called distributions with elliptical directions, which includes all elliptically symmetric populations and many skewed populations. A sign test based on interdirections is described, including, as special cases, the two-sided univariate sign test and Blumen's bivariate sign test. The statistic is shown to have a limiting χ2 p null distribution and, because it is based on interdirections, it is also seen to be invariant and to have a small-sample distribution-free property. Pitman asymptotic relative efficiencies and a Monte Carlo study show the test to perform well compared with Hotell...