Asymptotocally nonparametric tests with censored paired data

A class of asymptotically nonparametric test with contains a test proposed by Wei(1980), is considered for testing the equality of two continuous distribution funcitons when paired observations are subject to arbitrary right censorship. It is shown that under the null hypothesis each test statistic converges in distribution to the standard normal random variable. Furthermore. the Monte Carlo simulation results indicate that some tests in this class are more powerful than Wei's test. A generalization to incomplete censored paired data is also included.     

Power comparison of data depth-based nonparametric tests for testing equality of locations

In the recent years, the notion of data depth has been used in nonparametric multivariate data analysis since it gives natural ‘centre-outward’ ordering of multivariate data points with respect to the given data cloud. In the literature, various nonparametric tests are developed for testing equality of location of two multivariate distributions based on data depth. Here, we define two nonparametric tests based on two different test statistic for testing equality of locations of two multivariate distributions. In the present work, we compare the performance of these tests with the tests developed by Li and Liu [New nonparametric tests of multivariate locations and scales using data depth. Statist Sci. 2004;(1):686–696] for testing equality of locations of two multivariate distributions. Comparison in terms of power is done for multivariate symmetric and skewed distributions using simulation for three popular depth functions. Application of tests to real life data is provided. Conclusion and recommendations are also provided.     

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.     

Asymptotic test statistics for coefficients of variation

A one-sample asymptotically normal test statistic Is derived for testing the hypothesis that the coefficient of variation of a normal population is equal to a specified value. Based on this derivation, an asymptotically noraml two-sample test statistic and an asymptotically chi-square k-sample test statistic are derived for testing the hypothesis that the coefficients of variation of k ≥2 normal populations are equal. The two and k-sample test statistics allow for unequal sample sizes. Results of a simulation study which evaluate the size and power of the test statistics and compare the test statistics to earlier ones developed by McKay (1932) and Bennett (1976) are presented.     

Exact distribution-free tests for equality of several linear models

Exact tests for the equality of several linear models are developed using permutation techniques. Two cases of the linear model, characterized by either stochastic or nonstochastic predictors, are considered: the linear regression model (LRM) and the general linear model (GLM). A general class of test statistics using the volume of simplexes as the basic unit of analysis is proposed for this problem. The resulting class of statistics is shown to be a natural generalization of the multi-response permutation procedure (MRPP) test statistics which have been shown to comprise many of the statistics used in both parametric and nonparametric analysis of the standard g—sample problem. In the LRM case, exact moments of all orders are derived for the permutation distribution of any test statistic in the general class. Moment-based approximation of significance levels is shown to be computationally feasible in the simple LRM.     

Nonparametric tests for multivariate locations based on data depth

The present paper deals with the problem of testing equality of locations of two multivariate distributions using a notion of data depth. A notion of data depth has been used to measure centrality/outlyingness of a given point in a given data cloud. The paper proposes two nonparametric tests for testing equality of locations of two multivariate populations which are developed by observing the behavior of the depth versus depth plot. Simulation study reveals that the proposed tests are superior to the existing tests based on the data depth with regard to power. Illustrations with real data are provided.     

An empirical study of the type I error rate and power for some selected normal-theory and nonparametric tests of the independence of two sets of variables

Normal-theory tests of the hypothesis of no relationship among two sets of variables require assumptions of independence, hamoscedasticity, and normality. If, however, the assumption of normality is not tenable, there are few guidelines for properly using these tests. Historically, the lack of a comprehensive hypothesis-testing framework in the nonparametric case has provided few alternatives to normal-theory procedures. Fortunately, this situation has changed with the introduction of nonparametric, general linear model-based tests that can be used with existing computing packages. Multivariate-nonparametric tests due to Puri and Sen (1969, 1971, 1985) and Conover and Iman (1981) are outlined, and the results of a simulation study of the performance of three nonparametric and one normal-theory test of the hypothesis of no relationship among two sets of variables are presented. These results suggest that multivariate-nonparametric tests should be considered for a variety of data conditions. especially heavy-tailed and badly skewed data for small samples and a large number of variates.     

Generalized information matrix tests for copulas

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokhorov. The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test’s asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the nonparametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer–von Mises type tests and confirm the desired properties of the new tests in high dimensions.     

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.     

Lack-of-fit testing of a regression model with response missing at random

This paper proposes a class of lack-of-fit tests for fitting a linear regression model when some response variables are missing at random. These tests are based on a class of minimum integrated square distances between a kernel type estimator of a regression function and the parametric regression function being fitted. These tests are shown to be consistent against a large class of fixed alternatives. The corresponding test statistics are shown to have asymptotic normal distributions under null hypothesis and a class of nonparametric local alternatives. Some simulation results are also presented.     

Testing for the Equality of Two Nonparametric Regression Curves with Long Memory Errors

This article discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives for uniform design on [0,1] with long memory moving average errors. The standard deviations and the long memory parameters are possibly different for the two errors. The article adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotic null distributions. The article also shows that these tests are consistent for general alternatives and obtains their limiting distributions under a sequence of local alternatives. Since the limiting null distributions of these tests are unknown, we first conducted a Monte Carlo simulation study to obtain a few selected critical values of the proposed tests. Then based on these critical values, another Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. The article also contains a simulation study that assesses the effect of estimating the nonparametric regression function on an estimate of the long memory parameter of the errors. It is observed that the estimate based on direct observations is generally preferable over the one based on the estimated nonparametric residuals.     

Multivariate nonparametric methods in two-way balanced designs: performances and limitations in small samples

Investigations of multivariate population are pretty common in applied researches, and the two-way crossed factorial design is a common design used at the exploratory phase in industrial applications. When assumptions such as multivariate normality and covariance homogeneity are violated, the conventional wisdom is to resort to nonparametric tests for hypotheses testing. In this paper we compare the performances, and in particular the power, of some nonparametric and semi-parametric methods that have been developed in recent years. Specifically, we examined resampling methods and robust versions of classical multivariate analysis of variance (MANOVA) tests. In a simulation study, we generate data sets with different configurations of factor''s effect, number of replicates, number of response variables under null hypothesis, and number of response variables under alternative hypothesis. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the tradeoff between power and type I error, the strategic impact of increasing number of response variables, and the favourable performance of one test when the alternative is sparse. A real case study from an industrial engineering experiment in thermoformed packaging production is used to compare and illustrate the application of the various methods.     

Comparing Nonparametric Tests of Equality of Means for Randomized Block Designs

A number of nonparametric tests are compared empirically for a randomized block layout. We assess tests appropriate for when the data are not consistent with normality or when outliers invalidate traditional analysis of variance (ANOVA) tests. The objective is to assess, within this setting, tests that use ranks within blocks, the rank transform procedure that ranks the complete sample and continuous analogs of the Cochran–Mantel–Haenszel tests. The usual linear model is assumed, and our primary foci are tests of equality of means and component tests that assess linear and quadratic trends in the means. These tests include the traditional Page and Friedman tests. We conclude that the rank transform tests have competitive power and warrant greater use than is currently apparent.     

Testing heteroscedasticity in nonlinear and nonparametric regressions

This article presents new nonparametric tests for heteroscedasticity in nonlinear and nonparametric regression models. The tests have an asymptotic standard normal distribution under the null hypothesis of homoscedasticity and are robust against any form of heteroscedasticity. A Monte Carlo simulation with critical values obtained from the wild bootstrap procedure is provided to asses the finite sample performances of the tests. A real application of testing interest rate volatility functions illustrates the usefulness of the tests proposed. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Tests based on weighted rankings in complete blocks: exact distribution and monte carlo simulation

Quade (1972, 1979) proposed a family of nonparametric tests based on weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout. In this paper we give a table of the exact null distribution of these tests when the number of treatments is 3, the number of blocks is less than or equal to 14 and the block scores are linear. Moreover, a Monte Carlo study was performed to compare the powers of these tests with parametric and nonparametric competitors     

Robust testing with generalized partial linear models for longitudinal data

By approximating the nonparametric component using a regression spline in generalized partial linear models (GPLM), robust generalized estimating equations (GEE), involving bounded score function and leverage-based weighting function, can be used to estimate the regression parameters in GPLM robustly for longitudinal data or clustered data. In this paper, score test statistics are proposed for testing the regression parameters with robustness, and their asymptotic distributions under the null hypothesis and a class of local alternative hypotheses are studied. The proposed score tests reply on the estimation of a smaller model without the testing parameters involved, and perform well in the simulation studies and real data analysis conducted in this paper.     

Generalized likelihood ratio tests for the structure of semiparametric additive models

Semiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.     

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.     

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.     

Efficiency of tests based on weighted rankings for randomized blocks when the alternatives have an a prior1 ordering

Salama and Quade (1981) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects against a postulated ranking of the treatments in a complete randomized blocks layout.

These tests and others are compared with respect to asymptotic efficiency.