Multivariate signed rank statistics for shift alternatives

Let X ₁,X ₂,…,X _N be successive independent random P-vectors drawn from some continuous diagonally symmetric distribution. The problem of detecting a shift in level of the sequence at an unknown time point M, ≦M ≦ N-1, is studied. Test statistics based on multivariate analogues of the rank statistics derived by BHATTACHARYYA and JOHNSON (1888) are proposed, and their asymptotic properties are investigated.     

Edgeworth expansions for signed linear rank statistics under near location alternatives

Edgeworth expansions with the uniform remainder of order o(N⁻¹) are established for signed linear rank statistics with regression constants under near location alternatives. The results are obtained both with exact scores and with approximate scores, normalized with natural parameters as well as with simple constants.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

Uniform asymptotic linearity of a process based on a signed rank statistic

Consider the process with, cf. (1.2) on page 265 in B1, X₁, …, X_N a sample from a distribution F and, for i = 1, …, N, R${}_{\text{|x}{}_{\text{1}}\text{, - q}{}_{\text{1}}\text{ø|}}$ , the rank of |X₁ - q₁ø| among |X₁ - q₁ø|, …, |X_N - q_Nø|. It is shown that, under certain regularity conditions on F and on the constants p_i and q_i, T^ø_N(t) is asymptotically approximately a linear function of ø uniformly in t and in ø for |ø| ≤ C. The special case where the p_i and the q_i, are independent of i is considered.     

A note on the multivariate multisample rank sum and median tests

The muitivariate nonparametric tests analogous to the univar-iate rank sum test and median test are contained in Puri and Sen (1970). These tests provided a practical alternative for the analysis of multivariate data when the assumptions of parametric methods are not satisfied.

In this paper maximum values for L_Nthe asymptotic chi-Square test statistic for both the Multivariate Multisample Rank Sum Test (MMRST) and the Multivariate Multisample Median Test (MMMT) are developed.     

Adjusted Wilcoxon signed rank test tables for ratio of percentiles

The ordinary Wilcoxon signed rank test table provides confidence intervals for the median of one population. Adjusted Wilcoxon signed rank test tables which can provide confidence intervals for the median and the 10th percentile of one population are created in this paper. Base-(n + 1) number system and theorems about property of symmetry of the adjusted Wilcoxon signed rank test statistic are derived for programming. Theorem 1 states that the adjusted Wilcoxon signed rank test statistic are symmetric around n(n + 1)/4. Theorem 2 states that the adjusted Wilcoxon signed rank test statistic with the same number of negative ranks m are symmetric around m(n+1)/2. 87.5% and 85% confidence intervals of the median are given in the table for n = 12, 13,…, 29 to create approximated 95% confidence intervals of the ratio of medians for two independent populations. 95% and 92.5% confidence intervals of the 10th percentile are given in the table for n = 26, 27, 28, 29 to create approximated 95% confidence regions of the ratio of the 10th percentiles for two independent populations. Finally two large datasets from wood industry will be partitioned to verify the correctness of adjusted Wilcoxon signed rank test tables for small samples.     

Nonparametric tests for scale shift at an unknown time point

A class of linear rank tests is suggested for testing a shift in scale at an unknown time point in a sequence of independent observations. The tests,based on inverse normal scores and on ordered exponential scores,are shown to be asymptotically as efficient as their distribution-oriented competitors. Critical values and powers for these two rank tests are also discussed.     

A rank test for stochastically ordered alternatives

A rank test based on the number of ‘near-matches’ among within-block rankings is proposed for stochastically ordered alternatives in a randomized block design with t treatments and b blocks. The asymptotic relative efficiency of this test with respect to the Page test is computed as number of blocks increases to infinity. A sequential analog of the above test procedure is also considered. A repeated significance test procedure is developed and average sample number is computed asymptotically under the null hypothesis as well as under a sequence of contiguous alternatives.     

Multivariate rank tests for the two-sample location problem

A multivariate affinc-invariant family of rank tests is proposed for the two sample location problem. The class of statistics introduced is built upon Randles' multivariate one-sample sign statistic based on interdirections and the multivariate one-sample signed-rank statistic of Peters and Randles. Asymptotic relative efficiencies are obtained which indicate that selected members of the class perform very well for a broad class of distributions. Further comparisons are made among several statistics using Monte Carlo results.     

Linear rank tests for censored survival data

We describe a class of rank test procedures for the two-sample problem with right censored survival data. The class of tests is directly generalized from the linear rank tests by assigning each observation a rank according to its corresponding Wilcoxon scores. It allows a flexible choice of score functions, in particular, those powerful against scale differences between the two survival distributions. Monte Carlo simulations have shown that some members of this class have great power in detecting crossing-curve alternatives (alternatives where underlying survival curves cross over). The class also contains tests essentially equivalent to the Gehan-Wilcoxon and the logrank tests.     

A class of nonparametric tests for location and scale parameters

A class of nonparametric two-sample tests for testing identity of distributions versus alternatives containing both location and scale parameters is proposed and some properties are derived. A recursion formula for the exact distribution under the hypothesis is presented and, the asymptotic distribution is given under both the hypothesis and a contiguous sequence of alternatives. Some asymptotic optimality properties are deduced for particular tests of the class and finally, the asymptotic efficiency is found.     

Some linear and nonlinear rank tests for competing risks models

Using the methods of asymptotic decision theory asymptotically optimal for translation and scale families as well as for certian nonparmetric families. Moreover, two new classes of nonlinear rank tests are introduced. These tests are designed for detecting either “ omnibus alternatives ” or “ one sided alternatives of trend ”. Under the null hypothesis of randomness all tests are distribution - free. The asymptotic distributions of the test statistics are derived under contiguous alternatives.     

Augmenting lehmann alternatives for improving asymptotic efficiency of rank tests

Locally most powerful tests for augmented simple Lehmann alternatives are obtained. These tests turn out to be linearcombinations of the Savage and Mann-Whitney-Wilcoxon test criteria. We study their performance in terms of the asymptotic efficiency relative to their parametric competitors against location and scale alternatives. For small sample sizes, critical points of a couple of test procedures are given.     

Saddlepoint p-values for group of linear rank tests

Many nonparametric tests in one sample problem, matched pairs, and competingrisks under censoring have the same underlying permutation distribution. This article proposes a saddlepoint approximation to the exact p-values of these tests instead of the asymptotic approximations. The performance of the saddlepoint approximation is assessed by using simulation studies that show the superiority of the saddlepoint methods over the asymptotic approximations in several settings. The use of the saddlepoint to approximate the p-values of class of two sample tests under complete randomized design is also discussed.     

Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review

The paper reviews recent contributions to the statistical inference methods, tests and estimates, based on the generalized median of Oja. Multivariate analogues of sign and rank concepts, affine invariant one-sample and two-sample sign tests and rank tests, affine equivariant median and Hodges–Lehmann-type estimates are reviewed and discussed. Some comparisons are made to other generalizations. The theory is illustrated by two examples.     

The chernoff efficiency of the wilcoxon rank test

Exact limiting Chernoff efficiencies of the Wilcoxon rank test are derived using Hoadley's results. Efficiency curves are derived for the two-sample Wilcoxon rank test relative to the two-sample t test for normal shift alternatives when the null hypothesis is that of common normality. The comparisons with Bahadur efficiency and small sample Hodges-Lehmann efficiency-are also made.     

A bivariate signed rank test for two sample location problem

An affine-invariant signed rank test for the difference in location between two symmetric populations is proposed. The proposed test statistic is compared with Hotelling's T2 test statistic, Mardia's(1967)test statistic, Peters-Randles(1991) test statistic and Wilcoxon's rank sum test statistic using a Monte Carlo Study. It performs better than Mardia's test statistic under almost all populations considered. Under the bivariate normal distribution, it performs better than other test statistics compared for small differences in location between two populations except Hotelling's T2. It performs better than all statistics, including Hotelling's T , for sample size 15 when samples are drawn from Pearson type.