Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality

Multivariate tree-indexed Markov processes are discussed with applications. A Galton-Watson super-critical branching process is used to model the random tree-indexed process. Martingale estimating functions are used as a basic framework to discuss asymptotic properties and optimality of estimators and tests. The limit distributions of the estimators turn out to be mixtures of normals rather than normal. Also, the non-null limit distributions of standard test statistics such as Wald, Rao’s score, and likelihood ratio statistics are shown to have mixtures of non-central chi-square distributions. The models discussed in this paper belong to the local asymptotic mixed normal family. Consequently, non-standard limit results are obtained.     

Goodness-of-fit tests for copulas

This paper defines two distribution free goodness-of-fit test statistics for copulas. It states their asymptotic distributions under some composite parametric assumptions in an independent identically distributed framework. A short simulation study is provided to assess their power performances.     

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed.Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels.For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable.As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.     

A generalized Hollander-Proschan type test for NBUE alternatives

In this note we develop a family of test statistics for testing exponentiality against NBUE alternatives. The asymptotic distribution of the test statistics is derived. The test statistics are shown to be asymptotically normal and consistent. This family of test statistics includes the test proposed by Hollander and Proschan (1975) as a special case. Efficiency studies have also been done.     

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.     

High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus

We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy.     

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.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

Variance component testing in semiparametric mixed models

It is of considerable interest to test for heteroscedasticity in statistical studies. In this paper, we investigate such a problem under the framework of a semiparametric mixed model. A score test is proposed for the hypothesis that all the variance components are zero. We establish the asymptotic property of the test, and examine its performance in a simulation study. The test is illustrated with the analysis of a longitudinal study of measurements of serum creatinine.     

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.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Properties of spatial cross-periodograms using fixed-domain asymptotics

Cross-periodograms can be used to study a multivariate spatial process observed on a lattice. For spatial data, it is often appropriate to study asymptotic properties of statistical procedures under fixed-domain asymptotics in which the number of observations increases in a fixed region while shrinking distances between neighboring observations. Using fixed-domain asymptotics, we prove relative asymptotic unbiasedness and relative consistency of a smoothed cross-periodogram after appropriate filtering of the data. In addition, we show that smoothed cross-periodograms are asymptotically normal when the process is stationary multivariate Gaussian with appropriate assumptions on high-frequency behavior of the spectral density.     

Asymptotic statistical equivalence for scalar ergodic diffusions

For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam's deficiency between statistical experiments is considered under long-time asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed. The authors acknowledge the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH     

A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.     

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.     

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.     

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.     

A lack-of-fit test in Tobit errors-in-variables regression models

The problem of fitting a parametric model in Tobit errors-in-variables regression models is discussed in this paper. The proposed test is based on the supremum of the Khmaladze type transformation of a certain partial sum process of calibrated residuals. This framework covers the usual error-free Tobit model as a special case. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Consistency against some fixed alternatives and asymptotic power under some local nonparametric alternatives of this test are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test.     

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.     

ON SOME TESTS FOR ELLIPTICAL SYMMETRY

In this paper, some test statistics of Kolmogorov type and Cramer-von Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis and any fixed alternative. The asymptotic properties of Bootstrap approximation are investigated. Furthermore, for computational reasons, an approximation for the statistics, based on number theoretic method, is suggested.