A New Powerful Nonparametric Rank Test for Ordered Alternative Problem (original) (raw)

Nonparametric multivariate rank tests and their unbiasedness

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.

Testing nonparametric statistical functionals with applications to rank tests

Journal of Statistical Planning and Inference, 1999

The present paper discusses how nonparametric tests can be deduced from statistical functionals. E cient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one -and two -sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach o ers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated. (A. Janssen) 0378-3758/99/$ -see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 -3 7 5 8 ( 9 9 ) 0 0 0 0 9 -9

A Random-Linear-Extension Test Based on Classic Nonparametric Procedures

2020

Most distribution free nonparametric methods depend on the ranks or orderings of the individual observations. This dissertation develops methods for the situation when there is only partial information about the ranks available. A random-linear-extension exact test and an empirical version of the random-linear-extension test are proposed as a new way to compare groups of data with partial orders. The basic computation procedure is to generate all possible permutations constrained by the known partial order using a randomization method similar in nature to multiple imputation. This random-linear-extension test can be simply implemented using a Gibbs Sampler to generate a random sample of complete orderings. Given a complete ordering, standard nonparametric methods, such as the Wilcoxon rank-sum test, can be applied, and the corresponding test statistics and rejection regions can be calculated. As a direct result of our new method, a single p-value is replaced by a distribution of p-v...

A Rank-Based Permutation Test for Equivalence and Non-Inferiority

2015

Testing for the equivalence of two treatments has received attention in recent literature. Solutions typically considered are based on likelihood methods and the intersection-union (IU) principle. The IU principle focuses on the investigation of the equivalence between two treatments; that is a true equivalence has to be identified with a probability converging to one. The goal of this paper is to propose a rank- based permutation test through the Nonparametric Combination (NPC) of dependent tests, as an alternative to likelihood techniques.

Hypotheses tests based on sequential ranks

Sequential ranks of data X 1 , X 2 , . . . observed sequentially in time are defined as ranks computed from the data observed so far, denoting R ii the rank of X i among the values X 1 , X 2 , . . . , X i for any i. This paper studies tests of various hypotheses based on sequential ranks and derives such tests, which are locally most powerful among all tests based on sequential ranks. Such locally most powerful sequential rank test is derived for the hypothesis of random-ness against a general alternative, including the regression in location as a special case for the alternative hypothesis. Further, the locally most powerful sequential rank tests are derived for independence of two samples. The new tests are suitable for the situation when data are observed sequentially in time and the test is carried out each time after obtaining a new observation. While the classical rank tests require to recalculate all values of the ranks each time, the methods based on sequential ranks only re...

Teaching Rank-Based Tests by Emphasizing Structural Similarities to Corresponding Parametric Tests

Journal of Statistics Education

Students learn to examine the distributional assumptions implicit in the usual t-tests and associated confidence intervals, but are rarely shown what to do when those assumptions are grossly violated. Three data sets are presented. Each data set involves a different distributional anomaly and each illustrates the use of a different nonparametric test. The problems illustrated are well-known, but the formulations of the nonparametric tests given here are different from the large sample formulas usually presented. We restructure the common rank-based tests to emphasize structural similarities between large sample rank-based tests and their parametric analogs. By presenting large sample nonparametric tests as slight extensions of their parametric counterparts, it is hoped that nonparametric methods receive a wider audience.