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.     

New families of estimators and test statistics in log-linear models

In this paper we consider categorical data that are distributed according to a multinomial, product-multinomial or Poisson distribution whose expected values follow a log-linear model and we study the inference problem of hypothesis testing in a log-linear model setting. The family of test statistics considered is based on the family of ?-divergence measures. The unknown parameters in the log-linear model under consideration are also estimated using ?-divergence measures: Minimum ?-divergence estimators. A simulation study is included to find test statistics that offer an attractive alternative to the Pearson chi-square and likelihood-ratio test statistics.     

The likelihood ratio tests for the dimensionality of regression coefficients

In this paper we give a unified derivation of the likelihood ratio (LR) statistics for testing the hypothesis on the dimensionality of regression coefficients under a usual MANOVA model. We also derive the LR statistics under a general MANOVA model and study their asymptotic null and nonnull distributions. Further it is shown that the test statistic used by Bartlett [4] for testing the hypothesis that the last p?k canonical correlations are all zero is the LR statistic.     

K-sample subsampling in general spaces: The case of independent time series

The problem of subsampling in two-sample and K-sample settings is addressed where both the data and the statistics of interest take values in general spaces. We focus on the case where each sample is a stationary time series, and construct subsampling confidence intervals and hypothesis tests with asymptotic validity. Some examples are also given, and the problem of optimal block size choice is discussed.     

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.     

Nonparametric tests for homogeneity of scale against ordered alternatives

Summary In this paper the nonparametric several sample scale problem is considered and some tests are proposed for the hypothesis of homogeneity versus ordered alternatives. These tests are based on statistics that are weighted linear combinations of Sugiura (1965,Osaka J. Math.,2, 385–426) type statistics proposed for testing homogeneity of scale against the omnibus alternative. For each class of test statistics suggested, the member with maximum Pitman efficiency is identified. The optimal statistics are compared with their parametric and nonparametric competitors.     

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.     

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.     

An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T² distribution up to n^-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

Multivariate nonparametric tests in a randomized complete block design

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example.     

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.     

Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.     

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).     

Estimation of location and scale parameters for the Burr XII distribution using generalized order statistics

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution.     

Characterizations of symmetric distributions based on Rényi entropy

It is proved that the equality of Rényi entropies of upper and lower order statistics as well as upper and lower k-records is a characteristic property of symmetric distributions. Also, for Farlie-Gumbel-Morgenstern (FGM) family, it is shown that under some conditions the equality of entropies of concomitants of upper and lower order statistics as well as concomitants of upper and lower record values is a characteristic property for the uniform distribution.     

Estimation of the mean vector of a multivariate normal distribution: subspace hypothesis

This paper considers the estimation of the mean vector θ of a p-variate normal distribution with unknown covariance matrix Σ when it is suspected that for a p×r known matrix B the hypothesis θ=Bη, η∈R^r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis θ=Bη holds (iii) the preliminary test estimator (PTE), (iv) the James-Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis θ=Bη is true.     

Order restricted inference for sequential k-out-of-n systems

Sequential order statistics have been introduced to model sequential k-out-of-n systems which, as an extension of k-out-of-n systems, allow the failure of some components of the system to influence the remaining ones. Based on an independent sample of vectors of sequential order statistics, the maximum likelihood estimators of the model parameters of a sequential k-out-of-n system are derived under order restrictions. Special attention is paid to the simultaneous maximum likelihood estimation of the model parameters and the distribution parameters for a flexible location-scale family. Furthermore, order restricted hypothesis tests are considered for making the decision whether the usual k-out-of-n model or the general sequential k-out-of-n model is appropriate for a given data.     

On Hadamard differentiability in k-sample semiparametric models—with applications to the assessment of structural relationships

Semiparametric models to describe the functional relationship between k groups of observations are broadly applied in statistical analysis, ranging from nonparametric ANOVA to proportional hazard (ph) rate models in survival analysis. In this paper we deal with the empirical assessment of the validity of such a model, which will be denoted as a “structural relationship model”. To this end Hadamard differentiability of a suitable goodness-of-fit measure in the k-sample case is proved. This yields asymptotic limit laws which are applied to construct tests for various semiparametric models, including the Cox ph model. Two types of asymptotics are obtained, first when the hypothesis of the semiparametric model under investigation holds true, and second for the case when a fixed alternative is present. The latter result can be used to validate the presence of a semiparametric model instead of simply checking the null hypothesis “the model holds true”. Finally, various bootstrap approximations are numerically investigated and a data example is analyzed.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.