Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

On some nonparametric tests for profile analysis of several multivariate samples

For profile analysis of independent samples from several multivariate populations, a nonparametric analog of the hypothesis of parallelism of population profiles is formulated. A class of asymptotically distribution-free statistics is offered to test this hypothesis. These are based on generalized U statistics and are in some sense modifications of statistics offered previously by one of the authors for testing the homogeneity hypothesis. Consistency of these statistics is established for suitable alternatives and also asymptotic power is investigated.     

Testing the equality of several covariance matrices with fewer observations than the dimension

For normally distributed data from the k populations with m×m covariance matrices Σ₁,…,Σ_k, we test the hypothesis H:Σ₁=?=Σ_k vs the alternative A≠H when the number of observations N_i, i=1,…,k from each population are less than or equal to the dimension m, N_i≤m, i=1,…,k. Two tests are proposed and compared with two other tests proposed in the literature. These tests, however, do not require that N_i≤m, and thus can be used in all situations, including when the likelihood ratio test is available. The asymptotic distributions of the test statistics are given, and the power compared by simulations with other test statistics proposed in the literature. The proposed tests perform well and better in several cases than the other two tests available in the literature.     

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S, can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.     

Rank test for heteroscedastic functional data

In this paper, we consider (mid-)rank based inferences for testing hypotheses in a fully nonparametric marginal model for heteroscedastic functional data that contain a large number of within subject measurements from possibly only a limited number of subjects. The effects of several crossed factors and their interactions with time are considered. The results are obtained by establishing asymptotic equivalence between the rank statistics and their asymptotic rank transforms. The inference holds under the assumption ofα-mixing without moment assumptions. As a result, the proposed tests are applicable to data from heavy-tailed or skewed distributions, including both continuous and ordered categorical responses. Simulation results and a real application confirm that the (mid-)rank procedures provide both robustness and increased power over the methods based on original observations for non-normally distributed data.     

Nonparametric lack-of-fit tests for parametric mean-regression models with censored data

We developed two kernel smoothing based tests of a parametric mean-regression model against a nonparametric alternative when the response variable is right-censored. The new test statistics are inspired by the synthetic data and the weighted least squares approaches for estimating the parameters of a (non)linear regression model under censoring. The asymptotic critical values of our tests are given by the quantiles of the standard normal law. The tests are consistent against fixed alternatives, local Pitman alternatives and uniformly over alternatives in Hölder classes of functions of known regularity.     

Density testing in a contaminated sample

We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z₁,Z₂,… which are observed under additional noise with density ψ. The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Z_i. In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g=f*ψ instead on the initial density of interest f.     

Model checking in errors-in-variables regression

This paper discusses a class of minimum distance tests for fitting a parametric regression model to a class of regression functions in the errors-in-variables model. These tests are based on certain minimized distances between a nonparametric regression function estimator and a deconvolution kernel estimator of the conditional expectation of the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics under the null hypothesis and that of the corresponding minimum distance estimators. We also prove the consistency of the proposed tests against a fixed alternative and obtain the asymptotic distributions for general local alternatives. Simulation studies show that the testing procedures are quite satisfactory in the preservation of the finite sample level and in terms of a power comparison.     

Nonparametric rank-based tests of bivariate extreme-value dependence

A new class of tests of extreme-value dependence for bivariate copulas is proposed. It is based on the process comparing the empirical copula with a natural nonparametric rank-based estimator of the unknown copula under extreme-value dependence. A multiplier technique is used to compute approximate p-values for several candidate test statistics. Extensive Monte Carlo experiments were carried out to compare the resulting procedures with the tests of extreme-value dependence recently studied in Ben Ghorbal et al. (2009) [1] and Kojadinovic and Yan (2010) [19]. The finite-sample performance study of the tests is complemented by local power calculations.     

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).     

Two inequalities with an application

Summary An inequality used in Brown and Cohen (1974,Ann. Statist., 2(5), 963–976) and Bhattacharya (1978,Ann. Inst. Statist. Math., A,30, 407–414) is generalized and another useful inequality derived from it. An application of the latter which provides a more elegant approach to and an improvement over a result in Shinozaki (1978,Commun. Statist. Theor. Meth., A,7, 1421–1432), is also presented.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

Power comparisons of two-sided tests of equality of two covariance matrices based on six criteria

Power studies of tests of equality of covariance matrices of twop-variate normal populations Σ₁=Σ₂ against two-sided alternatives have been made based on the following six criteria: 1) Roy's largest root, 2) Hotelling's trace, 3) Pillai's trace, 4) Wilks' criterion, 5) Roy's largest-smallest roots and 6) modified likelihood ratio. A general theorem has been proved establishing the local unbiasedness conditions connecting the two critical values for tests 1) to 5). Extensive unbiased power tabulations have been made forp=2, for various values ofn ₁,n ₂, λ₁ and λ₂ wheren _i is the df of the SP matrix from theith sample and λ _i is theith latent root of Σ₁Σ ₂ ⁻¹ (i=1,2). Further, comparisons of powers of tests 1) to 5) have been made with those of the modified likelihood ratio after obtaining the exact distribution of the latter forn ₂=2n ₁ andp=2. Equal tail areas approach has also been used further to compute powers of tests 1) to 4) forp=2 for studying the bias. Again, a separate study has been made to compare the powers of the largest-smallest roots test with its three biased approximate approaches as well as the largest root. Since the largest root test was observed to have some advantage over the others, critical values were also obtained for this test in the unbiased as well as equal tail areas case forp=3. This research was supported by David Ross Grant from Purdue University. S. Sylvia Chu is now with Northwestern University.     

Testing independence in nonparametric regression

We propose a new test for independence of error and covariate in a nonparametric regression model. The test statistic is based on a kernel estimator for the L₂-distance between the conditional distribution and the unconditional distribution of the covariates. In contrast to tests so far available in literature, the test can be applied in the important case of multivariate covariates. It can also be adjusted for models with heteroscedastic variance. Asymptotic normality of the test statistic is shown. Simulation results and a real data example are presented.     

Signed-rank tests for location in the symmetric independent component model

The so-called independent component (IC) model states that the observed p-vector X is generated via X=ΛZ+μ, where μ is a p-vector, Λ is a full-rank matrix, and the centered random vector Z has independent marginals. We consider the problem of testing the null hypothesis H₀:μ=0 on the basis of i.i.d. observations X₁,…,X_n generated by the symmetric version of the IC model above (for which all ICs have a symmetric distribution about the origin). In the spirit of [M. Hallin, D. Paindaveine, Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks, Annals of Statistics, 30 (2002), 1103-1133], we develop nonparametric (signed-rank) tests, which are valid without any moment assumption and are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at given densities. Our tests are measurable with respect to the marginal signed ranks computed in the collection of null residuals , where is a suitable estimate of Λ. Provided that is affine-equivariant, the proposed tests, unlike the standard marginal signed-rank tests developed in [M.L. Puri, P.K. Sen, Nonparametric Methods in Multivariate Analysis, Wiley & Sons, New York, 1971] or any of their obvious generalizations, are affine-invariant. Local powers and asymptotic relative efficiencies (AREs) with respect to Hotelling’s T² test are derived. Quite remarkably, when Gaussian scores are used, these AREs are always greater than or equal to one, with equality in the multinormal model only. Finite-sample efficiencies and robustness properties are investigated through a Monte Carlo study.     

Dual divergence estimators and tests: Robustness results

The class of dual ?-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.     

Some tests for the covariance matrix with fewer observations than the dimension under non-normality

This article analyzes whether some existing tests for the p×p covariance matrix Σ of the N independent identically distributed observation vectors work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Σ is proportional to an identity matrix I_p; (2) the covariance matrix Σ is an identity matrix I_p; and (3) the covariance matrix is a diagonal matrix. It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether N≤p or N≥p, but (N,p)→∞, and N/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N,p).