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.     

Nonparametric repeated significance tests for one-way anova with adoptation to multiple comparisons

Based on two-sample rank order statistics, a repeated significance testing procedure for a multi-sample location problem is considered. The asymptotic distribution theory of the proposed tests is given under the null hypothesis as well as under local alternatives. A Bahadur efficiency result of the repeated significance test relative to the terminal test based solely on the target sample size is presented. In the adaptation of the proposed tests to multiple comparisons, an asymptotically equivalent test statistic in terms of the rank estimators of the location parameters is derived from which the Scheffé method of multiple comparisons can be obtained in a convinient way.     

Simple nonparametric tests for a known standard survival based on censored data

It is often of interest in survival analysis to test whether the distribution of lifetimes from which the sample under study was derived is the same as a reference distribution. The latter can be specified on the basis of previous studies or on subject matter considerations. In this paper several tests are developed for the above hypothesis, suitable for right-censored observations. The tests are based on modifications of Moses' one-sample limits of some classical two-sample rank tests. The asymptotic distributions of the test statistics are derived, consistency is established for alternatives which are stochastically ordered with respect to the null, and Pitman asymptotic efficiencies are calculated relative to competing tests. Simulated power comparisons are reported. An example is given with data on the survival times of lung cancer patients.     

Empirical likelihood test for equality of two distributions using distance of characteristic functions

In this paper, we investigate the problem of testing for the equality of two distributions. We employ a two-sample Jackknife Empirical Likelihood (JEL) approach to construct a test statistic whose limiting distribution is Chi-square distribution with degree of freedom 1, no matter what the data dimension (fixed) is. A variety of synthetic data experiments demonstrate that our JEL test statistic performs very well, with a very neat asymptotic distribution under the null hypothesis. Furthermore, we apply the test procedure to a real dataset to obtain competitive results.     

Asymptotic spacings theory with applications to the two-sample problem

The asymptotic distribution theory of test statistics which are functions of spacings is studied here. Distribution theory under appropriate close alternatives is also derived and used to find the locally most powerful spacing tests. For the two-sample problem, which is to test if two independent samples are from the same population, test statistics which are based on “spacing-frequencies” (i.e., the numbers of observations of one sample which fall in between the spacings made by the other sample) are utilized. The general asymptotic distribution theory of such statistics is studied both under the null hypothesis and under a sequence of close alternatives.     

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.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient 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 offers 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 rank test for bivariate location and scale problem for elliptically symmetric populations ∗

A nonparametric rank test for the two-sample location and scale bivariate problem is proposed. The asymptotic distribution of the statistic of the test is derived under the null hypothesis and under a class of contiguous alternatives. The asymptotic relative efficiency is given and a simulation study gives the performance of the test and some competitors.     

Nonparametric tests for right-censored data with biased sampling

Testing the equality of two survival distributions can be difficult in a prevalent cohort study when non random sampling of subjects is involved. Due to the biased sampling scheme, independent censoring assumption is often violated. Although the issues about biased inference caused by length-biased sampling have been widely recognized in statistical, epidemiological and economical literature, there is no satisfactory solution for efficient two-sample testing. We propose an asymptotic most efficient nonparametric test by properly adjusting for length-biased sampling. The test statistic is derived from a full likelihood function, and can be generalized from the two-sample test to a k-sample test. The asymptotic properties of the test statistic under the null hypothesis are derived using its asymptotic independent and identically distributed representation. We conduct extensive Monte Carlo simulations to evaluate the performance of the proposed test statistics and compare them with the conditional test and the standard logrank test for different biased sampling schemes and right-censoring mechanisms. For length-biased data, empirical studies demonstrated that the proposed test is substantially more powerful than the existing methods. For general left-truncated data, the proposed test is robust, still maintains accurate control of type I error rate, and is also more powerful than the existing methods, if the truncation patterns and right-censoring patterns are the same between the groups. We illustrate the methods using two real data examples.     

Power Comparison of Multivariate Wilcoxon-type Tests Based on the Jurečková–Kalina’s Ranks of Distances

A multivariate two-sample testing problem is one of the most important topics in nonparametric statistics. One of the multivariate two-sample testing problems based on the Jure?ková–Kalina ranks of distance is discussed in this article. Further, a multivariate Wilcoxon-type test is proposed for testing the equality of two continuous distribution functions. Simulations are used to investigate the power of this test for the two-sided alternative with various population distributions. The results show that the proposed test statistic is more suitable than various existing statistics for testing a shift in the locationt and location-scale parameters.     

Weighted Mean Survival Test Statistics: a Class of Distance Tests for Censored Survival Data

A class of test statistics is introduced which is sensitive against the alternative of stochastic ordering in the two-sample censored data problem. The test statistics for evaluating a cumulative weighted difference in survival distributions are developed while taking into account the imbalances in base-line covariates between two groups. This procedure can be used to test the null hypothesis of no treatment effect, especially when base-line hazards cross and prognostic covariates need to be adjusted. The statistics are semiparametric, not rank based, and can be written as integrated weighted differences in estimated survival functions, where these survival estimates are adjusted for covariate imbalances. The asymptotic distribution theory of the tests is developed, yielding test procedures that are shown to be consistent under a fixed alternative. The choice of weight function is discussed and relies on stability and interpretability considerations. An example taken from a clinical trial for acquired immune deficiency syndrome is presented.     

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.     

How Important Is the Effect of Rounding in Goodness-of-Fit

Abstract

In the area of goodness-of-fit there is a clear distinction between the problem of testing the fit of a continuous distribution and that of testing a discrete distribution. In all continuous problems the data is recorded with a limited number of decimals, so in theory one could say that the problem is always of a discrete nature, but it is a common practice to ignore discretization and proceed as if the data is continuous. It is therefore an interesting question whether in a given problem of test of fit, the “limited resolution” in the observed recorded values may be or may be not of concern, if the analysis done ignores this implied discretization. In this article, we address the problem of testing the fit of a continuous distribution with data recorded with a limited resolution. A measure for the degree of discretization is proposed which involves the size of the rounding interval, the dispersion in the underlying distribution and the sample size. This measure is shown to be a key characteristic which allows comparison, in different problems, of the amount of discretization involved. Some asymptotic results are given for the distribution of the EDF (empirical distribution function) statistics that explicitly depend on the above mentioned measure of degree of discretization. The results obtained are illustrated with some simulations for testing normality when the parameters are known and also when they are unknown. The asymptotic distributions are shown to be an accurate approximation for the true finite n distribution obtained by Monte Carlo. A real example from image analysis is also discussed. The conclusion drawn is that in the cases where the value of the measure for the degree of discretization is not “large”, the practice of ignoring discreteness is of no concern. However, when this value is “large”, the effect of ignoring discreteness leads to an exceded number of rejections of the distribution tested, as compared to what would be the number of rejections if no rounding is taking into account. The error made in the number of rejections might be huge.     

Goodness of fit and homogeneity tests on the basis of N-distances

In this paper we propose an application of N-distance theory [Klebanov, L.B., 2005. N-distances and their applications. Karolinum, Prague] for testing simple hypotheses of goodness of fit and homogeneity. The asymptotic null distribution of test statistics is established and coincides with the distribution of infinite quadratic form of independent standard normal random variables. A construction of multivariate free-of-distribution homogeneity test is considered. The power of proposed criteria is compared with classical tests using Monte-Carlo simulations.     

Goodness-of-fit tests for general linear models with covariates missed at random

In this paper, we consider a model checking problem for general linear models with randomly missing covariates. Two types of score type tests with inverse probability weight, which is estimated by parameter and nonparameter methods respectively, are proposed to this goodness of fit problem. The asymptotic properties of the test statistics are developed under the null and local alternative hypothesis. Simulation study is carried out to present the performance of the sizes and powers of the tests. We illustrate the proposed method with a data set on monozygotic twins.     

Function-based hypothesis testing in censored two-sample location-scale models

Function-based hypothesis testing in two-sample location-scale models has been addressed for uncensored data using the empirical characteristic function. A test of adequacy in censored two-sample location-scale models is lacking, however. A plug-in empirical likelihood approach is used to introduce a test statistic, which, asymptotically, is not distribution free. Hence for practical situations bootstrap is necessary for performing the test. A multiplier bootstrap and a model appropriate resampling procedure are given to approximate critical values from the null asymptotic distribution. Although minimum distance estimators of the location and scale are deployed for the plug-in, any consistent estimators can be used. Numerical studies are carried out that validate the proposed testing method, and real example illustrations are given.

     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.     

Bootstrap and permutation rank tests for proportional hazards under right censoring

We address the testing problem of proportional hazards in the two-sample survival setting allowing right censoring, i.e., we check whether the famous Cox model is underlying. Although there are many test proposals for this problem, only a few papers suggest how to improve the performance for small sample sizes. In this paper, we do exactly this by carrying out our test as a permutation as well as a wild bootstrap test. The asymptotic properties of our test, namely asymptotic exactness under the null and consistency, can be transferred to both resampling versions. Various simulations for small sample sizes reveal an actual improvement of the empirical size and a reasonable power performance when using the resampling versions. Moreover, the resampling tests perform better than the existing tests of Gill and Schumacher and Grambsch and Therneau . The tests’ practical applicability is illustrated by discussing real data examples.

     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.