A new dimension to nonparametric tests

It is demonstrated that a statistician can take a preliminary peek- at the data in a special vay in order to select an appropriate nonparametric test statistic without spoiling the distribution-free property. This process (adaptive selection and then testing) generally improves zhe paver over a vide range of underlying distributions and is the nev -dimension to nonparametric tests. Several illustrations are given using standard nonparametric techniques involving the usual one-sample, tvo-sample, k-sample, and independence testing situations. More recent (and not standard) applications to regression and classification problems are then considered. Finally one 'difficulty concerning this adapting process is noted, but a corrective action is suggested.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

Small sample distribution for a nonparametric test for trend

The exact distribution of a nonparametric test statistic for ordered alternatives, the _rank ² statistic, is computed for small sample sizes. The exact distribution is compared to an approximation.     

On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

A survey of nonparametric tests for scale

A surrey of nonparametric tests for scale is presented. Some parametric procedures are also presented for the purpose of comparison. Suggestions on which procedures to use in various situations are given. Both 2-sample and c-sample tests are included     

Two sample nonparametric tests based on subsamples

The paper introduces a general class of nonparametric tests for the two-sample location problem based on subsamples. Includ- ed in this class is the Mann-Whitney (or the Wilcoxon rank sum) test. General formulas for the Pitman efficacy for different methods of subsampling are derived. A small sample power simu- lation compares the performance of members of this class     

Stochastic equivalence of ranking methods

The asymptotic behavior of linear rank statistics for comparing the locations of two populations, where the observations are ranked jointly with other populations, is considered. Under certain conditions, the asymptotic behavior of these statistics does not depend on which other populations are included in the ranking. In particular, the difference of a pair of these statistics, with the same score function, but based on two different rankings, converges to zero in probability under Pitman alternatives and Chernoff-Savage conditions on the scores and underlying distributions.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

A nonparametric test to compare survival distributions with covariate adjustment

Summary.  The analysis of covariance is a technique that is used to improve the power of a k -sample test by adjusting for concomitant variables. If the end point is the time of survival, and some observations are right censored, the score statistic from the Cox proportional hazards model is the method that is most commonly used to test the equality of conditional hazard functions. In many situations, however, the proportional hazards model assumptions are not satisfied. Specifically, the relative risk function is not time invariant or represented as a log-linear function of the covariates. We propose an asymptotically valid k -sample test statistic to compare conditional hazard functions which does not require the assumption of proportional hazards, a parametric specification of the relative risk function or randomization of group assignment. Simulation results indicate that the performance of this statistic is satisfactory. The methodology is demonstrated on a data set in prostate cancer.     

A nonparametric statistical procedure for the detection of marine pollution

This paper is devoted to the estimation of the derivative of the regression function in fixed-design nonparametric regression. We establish the almost sure convergence as well as the asymptotic normality of our estimate. We also provide concentration inequalities which are useful for small sample sizes. Numerical experiments on simulated data show that our nonparametric statistical procedure performs very well. We also illustrate our approach on high-frequency environmental data for the study of marine pollution.     

A class of distribution-free tests for the two-sample slippage problem

A class of distribution-free tests for the two-sample slippage problem, when the random variables take only nonnegative values, is proposed. These tests are consistent and unbiased against the general slippage alternative. Recurrence relations for generating small sample significance points are given. The tests have been compared with the Savage test, the Wilcoxon test and the appropriate locally most powerful rank test by considering Pitman asymptotic relative efficiencies for several alternative hypotheses. Some of these tests exhibit considerable robustness in terms of efficiency for the various alternative hypotheses which are considered.     

A nonparametric test for proving noninferiority in clinical trials with ordered categorical data

We consider the problem of proving noninferiority when the comparison is based on ordered categorical data. We apply a rank test based on the Wilcoxon–Mann–Whitney effect where the asymptotic variance is estimated consistently under the alternative and a small‐sample approximation is given. We give the associated 100(1?α)% confidence interval and propose a formula for sample size determination. Finally, we illustrate the procedure and possible choices of the noninferiority margin using data from a clinical trial. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparisons of the asymptotic efficiencies of two sample tests for discrete distributions

The asymptotic efficiencies are computed for several popular two sample rank tests when the underlying distributions are Poisson, binomial, discrete uniform, and negative binomial The rank tests examined include the Mann-Whitney test, the van der Waerden test, and the median test. Three methods for handling ties are discussed and compared. The computed asymptotic efficiencies apply also to the k-sample extensions of the above tests, such as the Kruskal-Wallis test, etc.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Some tests of whether several samples are from identical populations

We observe s Independent samples, from unknown continuous distributions. The problem is to test the hypothesis that all the distributions are identical. The distribution of the numbers of observations from s-1 of the samples which fall in cells whose Boundaries are selected order statistics of the remaining sample, the number of cells increasing gradually with the sample sizes, is investigated. It is shown that under the null hypothesis and nearDy alternatives, as the sample sizes Increase these numbers of observations can be considered to be slightly rounded off normal random variables, the amount rounded off decreasing as sample sizes increase. Using these results, various tests of the hypothesis can be constructed and analyzed.     

Robust and nonparametric subset selection procedures

Subset selection procedures based on ranks have been investigated by a number of authors previously. Their methods are based on ranking the samples from all the populations jointly. However, as was pointed out by Rizvi and Woodworth (1970), the procedures they proposed cannot control the probability of a correct selection over the entire parameter space. In this paper, we propose a subset selection procedure based on pairwise rather than joint ranking of the samples. It is shown that this procedure controls the probability of a correct selection over the entire parameter space. It is also shown that the Pitman efficiency of this nonparametric procedure relative to the multivariate t procedure of Gupta (1956, 1965) is the same as the Pitman efficiency of the Mann-Whitney-Wilcoxon test relative to the t-test.     

On the sample breakdown robustness of some nonparametric tests

In this paper, relying on the sample breakdown points, we investigate the sample breakdown properties of some nonparametric tests. It is shown that the sample breakdown points of the sign test asymptotically dominate those of the Wilcoxon test for one–sided hypotheses, However, the different conclusion is derived in the case of testing some shrinking neighborhood hypotheses. The breakdown behaviors of the Kolmogorov test and X²–test are also explored. These studies unify or refine some existing breakdown analyses of tests.