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).     

Multivariate linear rank statistics for profile analysis

For some general multivariate linear models, linear rank statistics are used in conjunction with Roy's Union-Intersection Principle to develop some tests for inference on the parameter (vector) when they are subject to certain linear constraints. More powerful tests are designed by incorporating the a priori information on these constraints. Profile analysis is an important application of this type of hypothesis testing problem; it consists of a set of hypothesis testing problem for the p responses q-sample model, where it is a priori assumed that the response-sample interactions are null.     

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.     

Rates of convergence of quadratic rank statistics

The rates of convergence of the distribution function of quadratic rank statistics to the X²-distribution under hypothesis and near alternatives are investigated. The considered quadratic rank statistics are used for testing the multivariate hypothesis of randomness. The method suggested by Jure?ková [7] is applied.     

A novel trace test for the mean parameters in a multivariate growth curve model

A trace test for the mean parameters of the growth curve model is proposed. It is constructed using the restricted maximum likelihood followed by an estimated likelihood ratio approach. The statistic reduces to the Lawley-Hotelling trace test for the Multivariate Analysis of Variance (MANOVA) models. Our test statistic is, therefore, a natural extension of the classical trace test to GMANOVA models. We show that the distribution of the test under the null hypothesis does not depend on the unknown covariance matrix Σ. We also show that the distributions under the null and alternative hypotheses can be represented as sums of weighted central and non-central chi-square random variables, respectively. Under the null hypothesis, the Satterthwaite approximation is used to get an approximate critical point. A novel Satterthwaite type approximation is proposed to obtain an approximate power. A simulation study is performed to evaluate the performance of our proposed test and numerical examples are provided as illustrations.     

Wilcoxon‐type rank‐sum statistics for selecting the best population: Some advances

In this article, we introduce three new nonparametric procedures for testing the equality of two lifetime distributions. The proposed testing processes are based on appropriately modified Wilcoxon‐type rank‐sum statistics. The exact null distribution of these statistics is studied and closed formulae for the corresponding exact probability of correct selection of the best population are derived for the class of Lehmann alternatives. A detailed numerical study is carried out to elucidate the performance of the proposed testing schemes. For illustration purposes, a real data application is presented in some detail.     

A Berry-Esséen Theorem for Serial Rank Statistics

Berry-Esséen bounds of the optimal O(n-1/2) order are obtained, under the null hypothesis of randomness, for serial linear rank statistics, of the form a1 (Rt)a2(Rt-k). Such statistics play an essential role in distribution-free methods for time-series analysis, where they provide nonparametric analogues to classical (Gaussian) correlogram-based methods. Berry-Esséen inequalities are established under mild conditions on the score-generating functions, allowing for normal (van der Waerden) scores. They extend to the serial case the earlier result of Does (1982, Ann. Probab., 10, 982-991) on (nonserial) linear rank statistics, and to the context of nonparametric rank-based statistics the parametric results of Taniguchi (1991, Higher Order Asymptotics for Time Series Analysis, Springer, New York) on quadratic forms of Gaussian stationary processes.     

Simultaneous control of false positives and false negatives in multiple hypotheses testing

Multiple hypotheses testing is concerned with appropriately controlling the rate of false positives, false negatives or both when testing several hypotheses simultaneously. Nowadays, the common approach to testing multiple hypotheses calls for controlling the expected proportion of falsely rejected null hypotheses referred to as the false discovery rate (FDR) or suitable measures based on the positive false discovery rate (pFDR). In this paper, we consider the problem of determining levels that both false positives and false negatives can be controlled simultaneously. As our risk function, we use the expected value of the maximum between the proportions of false positives and false negatives, with the expectation being taken conditional on the event that at least one hypothesis is rejected and one is accepted, referred to as hybrid error rate (HER). We then develop, based on HER, an analog of p-value termed as h-value to test the individual hypotheses. The use of the new procedure is illustrated using the well-known public data set by Golub et al. [Molecular classification of cancer: class discovery and class prediction by gene expression monitoring, Science 386 (1999) 531-537] with Affymetrix arrays of patients with acute lymphoic leukemia and acute myeloid leukemia.     

A note on testing hypotheses for stationary processes in the frequency domain

In a recent paper, Eichler (2008) [11] considered a class of non- and semiparametric hypotheses in multivariate stationary processes, which are characterized by a functional of the spectral density matrix. The corresponding statistics are obtained using kernel estimates for the spectral distribution and are asymptotically normally distributed under the null hypothesis and local alternatives. In this paper, we derive the asymptotic properties of these test statistics under fixed alternatives. In particular, we also show weak convergence but with a different rate compared to the null hypothesis. We also discuss potential statistical applications of the asymptotic theory by means of a small simulation study.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

Regression type tests for parametric hypotheses based on optimally selected subsets of the order statistics

Through use of a regression framework, a general technique is developed for determining test procedures based on subsets of the order statistics for both simple and composite parametric null hypotheses. Under both the null hypothesis and sequences of local alternatives these procedures are asymptotically equivalent in distribution to the generalized likelihood ratio statistic based on the corresponding order statistics. A simple, approximate method for selecting quantiles for such tests, which endows the corresponding test statistics with optimal power properties, is also given.     

Nonparametric Methods for Checking the Validity of Prior Order Information

A large number of statistical procedures have been proposed in the literature to explicitly utilize available information about the ordering of treatment effects at increasing treatment levels. These procedures are generally more efficient than those ignoring the order information. However, when the assumed order information is incorrect, order restricted procedures are inferior and, strictly speaking, invalid. Just as any statistical model needs to be validated by data, order information to be used in a statistical analysis should also be justified by data first. A common statistical format for checking the validity of order information is to test the null hypothesis of the ordering representing the order information. Parametric tests for ordered null hypotheses have been extensively studied in the literature. These tests are not suitable for data with nonnormal or unknown underlying distributions. The objective of this study is to develop a general distribution-free testing theory for ordered null hypotheses based on rank order statistics and score generating functions. Sufficient and necessary conditions for the consistency of the proposed general tests are rigorously established.     

A sharpening of the remainder term in the higher-dimensional central limit theorem for multilinear rank statistics

A new approach to the asymptotic normality of the multivariate linear rank statistics is provided along with the Berry-Esséen and the Prohorov distance estimates for the remainder term in the convergence to normality.     

Semi-aligned rank tests

Summary The distribution-free test based on semi-aligned rankings for no treatment effects in a two-way layout, with unequal number of replications in each cell is considered. The asymptotic χ-square distribution of the test statistic under the null hypothesis is derived. The Pitman asymptotic relative efficiency of the test (i) based on semi-aligned rankings with respect to the test (ii) based on within-block rankings, is shown to be larger than one as the number of blocks tends to infinity. Also the asymptotic properties of linear rank statistics (i) and (ii) are investigated and the asymptotic relative efficiency of the test (i) with respect to the test (ii) is again shown to be larger than one.     

Asymptotically efficient two-sample rank tests for modal directions on spheres

A general class of optimal and distribution-free rank tests for the two-sample modal directions problem on (hyper-) spheres is proposed, along with an asymptotic distribution theory for such spherical rank tests. The asymptotic optimality of the spherical rank tests in terms of power-equivalence to the spherical likelihood ratio tests is studied, while the spherical Wilcoxon rank test, an important case for the class of spherical rank tests, is further investigated. A data set is reanalyzed and some errors made in previous studies are corrected. On the usual sphere, a lower bound on the asymptotic Pitman relative efficiency relative to Hotelling’s T²-type test is established, and a new distribution for which the spherical Wilcoxon rank test is optimal is also introduced.     

Rank Tests Based on Exceeding Observations

Rank tests based on the maximum number of exceeding observations for several standard nonparametric hypotheses are proposed. An approach to constructing nonparametric rank tests via metrics on the permutation group is used. The test statistics are based on a metric induced by Chebyshev's norm.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.     

Enhanced one-sided confidence regions for a multivariate location parameter

If a one-sided test for a multivariate location parameter is inverted, the resulting confidence region may have an unpleasant shape. In particular, if the null and alternative hypothesis are both composite and complementary, the confidence region usually does not resemble the alternative parameter region in shape, but rather a reflected version of the null parameter region.We illustrate this effect and show one possibility of obtaining confidence regions for the location parameter that are smaller and have a more suitable shape for the type of problems investigated. This method is based on the closed testing principle applied to a family of nested hypotheses.     

Hájek-Šidák approach to the asymptotic distribution of multivariate rank order statistics

Let X_α = (X_1α,…, X_pα), 1 ≤ α ≤ N_ν, ν ≥ 1 be N_ν independent observations from a density function f(x) where x ∈ R^p, the p-dimensional real space. Let R_νjα denote the rank of X_jα in the ordered array of X_j1 ,…, X_jNν; 1 ≤ j ≤ p and consider the multivariate rank order statistics

$\text{T}{}_{\mathrm{vj}}\text{=}\text{∑}\text{α = 1}\text{N}{}_{\mathrm{v}}\text{C}{}_{\mathrm{y\alpha }}\text{a}{}_{\mathrm{vj}}\text{(R}{}_{\mathrm{vj\alpha }}\text{),}$

where the constants, c_να, 1 ≤ α ≤ N_ν satisfy the Noether condition and the scores, a_νj(α), 1 ≤ j ≤ p, 1 ≤ α ≤ N_ν converge as ν → ∞, for each j, in quadratic mean to a nonconstant, square integrable function π_j(u), 0 < u < 1. Under the hypothesis of randomness, the joint asymptotic conditional and uncoditional normality of the statistics T_νj, 1 ≤ j ≤ p is established. Further, under mild conditions on the underlying density functions and assuming contiguous location shift alternatives, the joint asymptotic normality of these statistics is also established.     

Local efficiency of a Cramér-von Mises test of independence

Deheuvels proposed a rank test of independence based on a Cramér-von Mises functional of the empirical copula process. Using a general result on the asymptotic distribution of this process under sequences of contiguous alternatives, the local power curve of Deheuvels’ test is computed in the bivariate case and compared to that of competing procedures based on linear rank statistics. The Gil-Pelaez inversion formula is used to make additional comparisons in terms of a natural extension of Pitman's measure of asymptotic relative efficiency.