An efficient nonparametric test for bivariate two-sample location problem

We propose a new test for the two-sample bivariate location problem. The proposed test statistic has a U-statistic representation with a degenerate kernel. The limiting distribution is found for the proposed test statistic. The power of the test is compared using Monte Carlo simulation to the tests of Blumen [I. Blumen, A new bivariate sign-test for location, Journal of the American Statistical Association 53 (1958) 448–456], Mardia [K.V. Mardia, A non-parametric test for the bivariate two-sample location problem, Journal of the Royal Statistical Society, Series B 29 (1967) 320–342], Peters and Randles [D. Peters, R.H. Randles, A bivariate signed-rank test for the two-sample location problem, Journal of the Royal Statistical Society, Series B 53 (1991) 493–504], LaRocque, Tardif and van Eeden [D. LaRocque, S. Tardif, C. van Eeden, An affine-invariant generalization of the Wilcoxon signed-rank test for the bivariate location problem, Australian and New Zealand Journal of Statistics 45 (2003) 153–165], and Baringhaus and Franz [L. Baringhaus, C. Franz, On a new multivariate two-sample test, Journal of Multivariate Analysis 88 (2004) 190–206]. Under the bivariate normal and bivariate t distributions the proposed test was more powerful than the competitors for almost every change in location. Under the other distributions the proposed test reached the desired power of one at a faster rate than the other tests in the simulation study. Application of the test is presented using bivariate data from a synthetic and a real-life data set.     

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.     

Adaptive nonparametric tests for the two-sample scale problem under symmetry

The present paper considers two-sample tests for scale problem under symmetry without any assumption regarding the equality of medians. Two adaptive procedures are proposed—one is probabilistic while the other is deterministic. The proposed probabilistic approach is shown, by simulation studies, to maintain its significance level for various symmetric distributions and is found to be superior to the other existing competitors in terms of both robustness of size and power. Both the adaptive procedures are illustrated by using a real data. Some relevant asymptotic properties are also discussed.     

The extended two-sample problem: nonparametric case

The classical two-sample problem is extended here to the case where the distribution functions of the observable random variables are specified functions of unknown distribution functions and the null hypotheses to be tested or the parameters to be estimated relate to these unknown distributions. Various properties of the proposed rank tests and derived estimates are studied.     

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 statistical software procedure for exact parametric and nonparametric likelihood-ratio tests for two-sample comparisons

Two-sample comparisons belonging to basic class of statistical inference are extensively applied in practice. There is a rich statistical literature regarding different parametric methods to address these problems. In this context, most of the powerful techniques are assumed to be based on normally distributed populations. In practice, the alternative distributions of compared samples are commonly unknown. In this case, one can propose a combined test based on the following decision rules: (a) the likelihood-ratio test (LRT) for equality of two normal populations and (b) the Shapiro–Wilk (S-W) test for normality. The rules (a) and (b) can be merged by, e.g., using the Bonferroni correction technique to offer the correct comparison of the samples distribution. Alternatively, we propose the exact density-based empirical likelihood (DBEL) ratio test. We develop the tsc package as the first R package available to perform the two-sample comparisons using the exact test procedures: the LRT; the LRT combined with the S-W test; as well as the newly developed DBEL ratio test. We demonstrate Monte Carlo (MC) results and a real data example to show an efficiency and excellent applicability of the developed procedure.     

Distribution-free tests for parallelism and concurrence in two-sample regression problem

This paper suggests some distribution-free methods for testing hypothesis of parallelism and concurrence of two linear regressions. We assume that the independent variable x is equally spaced. The proposed procedures are compared with nonparametric competitors and the normal theory t-test.     

A two-sample test for location

A new test that is based on U-statistics is oroposed for the two-sample local, on problem. This test is sensitive to heavy-tailed distributions.     

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.     

Asymptotically optimal rank tests for the two-sample problem with randomly censored data

Using the methods of asymptotic decision theory asymptotically optimal rank tests are constructed in the two-sample testing problem for translation families and positive scale families under the assumption of equal censoring in both samples. The resulting tests have a simple form extending the known tests for un-censored data in a natural way. Relations to a recent proposal by Albers and Akritas are discussed.     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Bounds on efficiencies for some two-sample nonparametric statistics

Bounds on the Pitman efficiency for two-sample scale and loca-tion statistics are presented along with densities for which these bounds are sharp.     

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.     

Rank tests in the two-sample scale problem with unequal and unknown locations

The two-sample scale problem is studied in the case of unequal and unknown location parameters. The method proposed is based on the idea of Moses (1963) and it is distribution-free. The two samples are separated into random subgroups of the same sizek. It is proposed to choosek=4 and to apply the Wilconxon test or the Savage test to the ranges or sample variances of the subgroups. The asymptotic power functions of the tests are compared. For small and moderate sample sizes simulations are carried out. Relations to some other procedures, especially to the method of Compagnone and Denker (1996) are briefly discussed.     

Combination of distribution-free tests for the general two-sample problem with application to the social sciences

ABSTRACT

The problem of detecting any differences between the distributions of two populations is addressed within the non parametric permutation framework of combined tests. Combined testing has been very useful to address the location, the scale, and the location/scale problems. The aim of the paper is to see whether combined testing is useful also for the general two-sample problem. The framework of combined testing for the general two-sample problem is presented and some tests are proposed. These tests are valid even when a non random sample of units is randomized into two groups. Type 1 error rate and power characteristics of the new tests are investigated and compared to former tests. It is shown that the new tests compare favorably with the former ones. An application to a very important socioeconomic problem is discussed.     

Robust algorithms for economic designing of a nonparametric control chart for abrupt shift in location

The existing statistical process control procedures typically rely on the fundamental assumption of a parametric distribution of the quality characteristic. However, when there is a lack of knowledge about the underlying distribution (as full knowledge is not available in practice), the performance of these parametric charts is very likely to be heavily degraded. Motivated by this problem, a one-sided nonparametric monitoring procedure using the single sample sign statistic is proposed for detecting a shift in the location parameter of a continuous distribution. An economic model of the control chart is developed to optimize the sample size, sampling interval, and control limits. Three data-dependent estimation approaches for the unknown parameter are evaluated and discussed. Simulation results exhibit that our proposed procedure generally performs well under a great variety of continuous distributions and hence it is recommended as an alternative scheme especially when the knowledge of the underlying distribution is imperfect. Furthermore, beneficial recommendations of estimation approach selection are provided for practical implementation of the control chart.     

Robust nonparametric estimation for spatial regression

In this paper, we investigate a nonparametric robust estimation for spatial regression. More precisely, given a strictly stationary random field _Zi=(_Xi,_{Yi)i∈NNN≥1}$Z_{\mathbf{i}}=(X_{\mathbf{i}},Y_{\mathbf{i}}{)}_{\mathbf{i}\in {\mathbb{N}}^{N}N\ge 1}$, we consider a family of robust nonparametric estimators for a regression function based on the kernel method. Under some general mixing assumptions, the almost complete consistency and the asymptotic normality of these estimators are obtained. A robust procedure to select the smoothing parameter adapted to the spatial data is also discussed.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.