Nonparametric Statistical Inference

arXiv: Statistics Theory, 2020

Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks, hence carry all the “distribution-free information" available in the sample. We derive here the Hajek representation and asymptotic normality results required in the construction of center-outward rank tests for multiple-output regression and MANOVA. When based on appropriate spherical scores, thes...

Review of Regional …, 2007

1 Department of Spatial Economics, Free University of Amsterdam, The Netherlands; email: rpatuelli@feweb.vu.nl; pnijkamp@feweb.vu.nl 2 ISER, University of Essex, UK; email: slonghi@essex.ac.uk 3 Department of Economics, Faculty of Statistics, University of ...

1997

: In Balanced Incomplete Block Designs the presence of outliers in the data not only indicate non-normality, so that classical methods of analysis on the data are inappropriate, but also diminish the power of classical methods of analysis that assume normality of the data. One alternative is to use Durbin's nonparametric test. This research examined a rank transformation approach where all the data are ranked from smallest to largest, over all treatments and blocks. Then the usual F-test is computed on the ranks. This rank transformation approach is asymptotically distribution-free, and is very powerful in cases where outliers are present as compared with both the parametric F-test and Durbin's nonparametric test. In the case of normality of the data it has only a slight loss of power. Thus it is a good alternative procedure to consider when analyzing data from a Balanced Incomplete Block Design.

PLoS ONE, 2014

We propose a new nonparametric test for ordered alternative problem based on the rank difference between two observations from different groups. These groups are assumed to be independent from each other. The exact mean and variance of the test statistic under the null distribution are derived, and its asymptotic distribution is proven to be normal. Furthermore, an extensive power comparison between the new test and other commonly used tests shows that the new test is generally more powerful than others under various conditions, including the same type of distribution, and mixed distributions. A real example from an anti-hypertensive drug trial is provided to illustrate the application of the tests. The new test is therefore recommended for use in practice due to easy calculation and substantial power gain.

Journal of Modern Applied Statistical Methods, 2004

A rank method is presented for estimating regression parameters in the linear model when observations are correlated. This correlation is accounted for by including a random effect term in the linear model. A method is proposed that makes few assumptions about the random effect and error distribution. The main goal of this article is to determine the distributions for which this method performs well relative to existing methods.

Journal of the American Statistical Association, 2000

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Bernoulli, 2012

Although unbiasedness is a basic property of a good test, many tests on vector parameters or scalar parameters against two-sided alternatives are not finite-sample unbiased. This was already noticed by Sugiura [Ann. Inst. Statist. Math. 17 (1965) 261-263]; he found an alternative against which the Wilcoxon test is not unbiased. The problem is even more serious in multivariate models. When testing the hypothesis against an alternative which fits well with the experiment, it should be verified whether the power of the test under this alternative cannot be smaller than the significance level. Surprisingly, this serious problem is not frequently considered in the literature. The present paper considers the two-sample multivariate testing problem. We construct several rank tests which are finite-sample unbiased against a broad class of location/scale alternatives and are finite-sample distribution-free under the hypothesis and alternatives. Each of them is locally most powerful against a specific alternative of the Lehmann type. Their powers against some alternatives are numerically compared with each other and with other rank and classical tests. The question of affine invariance of two-sample multivariate tests is also discussed.

2010

We propose a winsorized adaptative signed rank test for the location alternative for samples coming from asymmetric distributions. We give conditions under which the proposed test can have either greater or smaller acceptance breakdown, than the acceptance breakdown of a recently appeared adaptative rank test for location under asymmetry. Moreover, when symmetry of the sampled distribution can be justified the proposed test has greater acceptance breakdown than the winsorized signed rank test.

Journal of Statistical Planning and Inference, 2005

In this paper aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity in a partially linear model. It turns out that the centered design matrix is not of full rank, and therefore doesn't quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-squared distribution of the test statistic under the null hypothesis, howerer, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff-Savage approach. In passing it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.

Journal of Modern Applied Statistical Methods

The Type I Error Rate of the Robust Rank Order test under various population symmetry conditions is explored through Monte Carlo simulation. Findings indicate the test has difficulty controlling Type I error under generalized Behrens-Fisher conditions for moderately sized samples.