In this paper we study the problem of testing the null hypothesis that errors from k independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the k residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data.
This paper studies goodness of fit tests and specification tests for an extension of the log-GARCH model which is stable by scaling. A Lagrange-Multiplier test is derived for testing the null assumption of extended log-GARCH against more general formulations including the Exponential GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended log-GARCH. Simulations illustrating the theoretical results and an application to real financial data are proposed.
We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted $L^2$-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the ne…
We generalize a recent class of tests for univariate normality that are based on the empirical moment generating function to the multivariate setting, thus obtaining a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for multinormality. The test statistics are suitably weighted $L^2$-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We study the finite-sample behavior of the new tests, compare the criteria with alternative e…
In the classical two-sample problem, the conventional approach for testing distributions equality is based on the difference between the two marginal empirical distribution functions, whereas a test for independence is based on the contrast between the bivariate and the product of the marginal empirical distribution functions. In this article we consider the problem of testing independence and distributions equality when the observer is "colour blind" so he cannot distinguish the distribution which has generated each of the two measurements. Within a nonparametric framework, we propose an empi…
In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study if this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert--Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the tests statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour…
We propose a robust test for the equality of the covariance structures in two functional samples. The test statistic has a chi-square asymptotic distribution with a known number of degrees of freedom, which depends on the level of dimension reduction needed to represent the data. Detailed analysis of the asymptotic properties is developed. Finite sample performance is examined by a simulation study and an application to egg-laying curves of fruit flies.
Several procedures have been recently proposed to test the simplifying assumption for conditional copulas. Instead of considering pointwise conditioning events, we study the constancy of the conditional dependence structure when some covariates belong to general borelian conditioning subsets. Several test statistics based on the equality of conditional Kendall's tau are introduced, and we derive their asymptotic distributions under the null. When such conditioning events are not fixed ex ante, we propose a data-driven procedure to recursively build such relevant subsets. It is based on decisio…
We present a general approach to constructing permutation tests that are both exact for the null hypothesis of equality of distributions and asymptotically correct for testing equality of parameters of distributions while allowing the distributions themselves to differ. These robust permutation tests transform a given test statistic by a consistent estimator of its limiting distribution function before enumerating its permutation distribution. This transformation, known as prepivoting, aligns the unconditional limiting distribution for the test statistic with the probability limit of its permu…
This paper considers the problem of testing the equality of two unspecified distributions. The classical omnibus tests such as the Kolmogorov-Smirnov and Cramèr-von Mises are known to suffer from low power against essentially all but location-scale alternatives. We propose a new two-sample test that modifies the Neyman's smooth test and extend it to the multivariate case based on the idea of projection pursue. The asymptotic null property of the test and its power against local alternatives are studied. The multiplier bootstrap method is employed to compute the critical value of the multivaria…
The Savage-Dickey density ratio is a specific expression of the Bayes factor when testing a precise (equality constrained) hypothesis against an unrestricted alternative. The expression greatly simplifies the computation of the Bayes factor at the cost of assuming a specific form of the prior under the precise hypothesis as a function of the unrestricted prior. A generalization was proposed by Verdinelli and Wasserman (1995) such that the priors can be freely specified under both hypotheses while keeping the computational advantage. This paper presents an extension of this generalization when …
In the context of data that lie on the simplex, we investigate use of empirical and exponential empirical likelihood, and Hotelling and James statistics, to test the null hypothesis of equal population means based on two independent samples. We perform an extensive numerical study using data simulated from various distributions on the simplex. The results, taken together with practical considerations regarding implementation, support the use of bootstrap-calibrated James statistic.
This paper considers a semiparametric generalized autoregressive conditional heteroskedasticity (S-GARCH) model. For this model, we first estimate the time-varying long run component for unconditional variance by the kernel estimator, and then estimate the non-time-varying parameters in GARCH-type short run component by the quasi maximum likelihood estimator (QMLE). We show that the QMLE is asymptotically normal with the parametric convergence rate. Next, we construct a Lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking, and obtain their asymptot…
The Gini correlation plays an important role in measuring dependence of random variables with heavy tailed distributions, whose properties are a mixture of Pearson's and Spearman's correlations. Due to the structure of this dependence measure, there are two Gini correlations between each pair of random variables, which are not equal in general. Both the Gini correlation and the equality of the two Gini correlations play important roles in Economics. In the literature, there are limited papers focusing on the inference of the Gini correlations and their equality testing. In this paper, we devel…
Several interesting models for contingency tables are defined by a system of equality and inequality constraints on a suitable set of marginal log-linear parameters. After reviewing the most common difficulties which are intrinsic to order restricted testing problems, we propose two new families of testing procedures, based on similar attempts appeared in the econometric literature, in order to increase the probability of detecting several relevant violations of the supposed order relations. One set of procedures is based on the decomposition of the log-likelihood ratio when testing the given …
We consider the hypothesis testing problem that two vertices $i$ and $j$ of a generalized random dot product graph have the same latent positions, possibly up to scaling. Special cases of this hypothesis test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors, or testing whether two vertices in a popularity adjusted block model have the same community assignment. We propose several test statistics based on the empirical Mahalanobis distances between the $i$th and $j$th rows of either the adjac…
Estimation of the covariance matrix of asset returns is crucial to portfolio construction. As suggested by economic theories, the correlation structure among assets differs between emerging markets and developed countries. It is therefore imperative to make rigorous statistical inference on correlation matrix equality between the two groups of countries. However, if the traditional vector-valued approach is undertaken, such inference is either infeasible due to limited number of countries comparing to the relatively abundant assets, or invalid due to the violations of temporal independence ass…
In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of $k$th powers of residuals, CUSUM processes and self-normalized partial sum processes. The $k$th power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the $k$th moment $μ_k$ of the innovation sequence. If $μ_k=0$, then the correction term is 0 and, thus, the $k$th power partial sum process converges weakly to the same Gaussian process as does the $k$th power partial sum of the…
Recently Liu and Wang derived the likelihood ratio test (LRT) statistic and its asymptotic distribution for testing equality of two multinomial distributions vs. the alternative that the second distribution is larger in terms of increasing convex order (ICX). ICX is less restrictive than stochastic order and is a notion that has found applications in insurance and actuarial science. In this paper we propose a new test for ICX. The new test has several advantages over the LRT and over any test procedure that depends on asymptotic theory for implementation. The advantages include the following: …
The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical $χ^2$ statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the nonparametric maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution st…
We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form…
We consider the problem of testing equality of functions $f_j:[0,1]\to \mathbb{R}$ for $j=1,2,...,J$ the basis of $J$ independent samples from possibly different distributions under the assumption that the functions are monotone. We provide a uniform approach that covers testing equality of monotone regression curves, equality of monotone densities and equality of monotone hazards in the random censorship model. Two test statistics are proposed based on $L_1$-distances. We show that both statistics are asymptotically normal and we provide bootstrap implementations, which are shown to have crit…
This article presents a homogeneity test for testing the equality of several high-dimensional covariance matrices for stationary processes with ignoring the assumption of normality. We give the asymptotic distribution of the proposed test. The simulation illustrates that the proposed test has perfect performance. Moreover, the power of the test can approach any high probability uniformly on a set of covariance matrices.
Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate population covariance matrices, or testing the independence between sub-groups of a multivariate random vector. This paper is concerned with the properties of a large-dimensional Fisher matrix when the dimension of the population is proportionally large compared to the sample size. Most of existing works on Fisher matrices deal with a particular Fisher matr…
In this paper, we propose two new tests for testing the equality of the covariance functions of several functional populations, namely a quasi GPF test and a quasi $F_{\max}$ test. The asymptotic random expressions of the two tests under the null hypothesis are derived. We show that the asymptotic null distribution of the quasi GPF test is a chi-squared-type mixture whose distribution can be well approximated by a simple scaled chi-squared distribution. We also adopt a random permutation method for approximating the null distributions of the quasi GPF and $F_{\max}$ tests. The random permutati…
In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for testing homoscedasticity of covariance matrix. In this generalization of Bartlett's test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covariance structures.
Texas Tech University（德州理工大学）；North Dakota State University（北达科他州立大学）；New Jersey Institute of Technology（新泽西理工学院）
Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with SGD, PSGD forces its iterative values into the constrained parameter space via projection. From a statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfy some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD-based estimate. As a byproduct, we pr…
In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problem in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by \citet{BaiS96E} and \cite{ChenQ10T}, a test statistic is introduced and the asymptomatic distributions under the null hypothesis as well as the alternative hypothesis are given. In addition, it is compared with a test statistic recently proposed by \cite{SrivastavaK13Ta}. It is shown that our test statist…
This paper concerns a method of testing equality of distribution of random convex compact sets and the way how to use the test to distinguish between two realisations of general random sets. The family of metrics on the space of distributions of random convex compact sets is constructed using the theory of N-distances and characteristic functions of random convex compact sets. Further, the approximation of the metrics through its finite dimensional counterparts is proposed, which lead to a new statistical test for testing equality in distribution of two random convex compact sets. Then, it is …
Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This paper proposes permutation tests for equality of parameters that are estimated at root-$n$ or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymp…
Methods of hypotheses testing for equality of extrinsic antimeans on compact manifolds are unveiled in this paper. The two and multiple sample problem for antimeans on compact manifolds is addressed for large samples via asymptotic distributions, as well as for small samples using nonparametric bootstrap. An example of face differentiation using 3D VW antimean projective shape analysis for data extracted from digital camera images is also given.
There has been an increasing interest in testing the equality of large Pearson's correlation matrices. However, in many applications it is more important to test the equality of large rank-based correlation matrices since they are more robust to outliers and nonlinearity. Unlike the Pearson's case, testing the equality of large rank-based statistics has not been well explored and requires us to develop new methods and theory. In this paper, we provide a framework for testing the equality of two large U-statistic based correlation matrices, which include the rank-based correlation matrices as s…
In this paper, we investigate the testing problem that the spectral density matrices of several, not necessarily independent, stationary processes are equal. Based on an $L_2$-type test statistic, we propose a new nonparametric approach, where the critical values of the tests are calculated with the help of randomization methods. We analyze asymptotic exactness and consistency of these randomization tests and show in simulation studies that the new procedures posses very good size and power characteristics.
In this paper, we generalize the metric-based permutation test for the equality of covariance operators proposed by Pigoli et al. (2014) to the case of multiple samples of functional data. To this end, the non-parametric combination methodology of Pesarin and Salmaso (2010) is used to combine all the pairwise comparisons between samples into a global test. Different combining functions and permutation strategies are reviewed and analyzed in detail. The resulting test allows to make inference on the equality of the covariance operators of multiple groups and, if there is evidence to reject the …
We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on fully functional tests for the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomes some common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can …
Bayesian supervised predictive classifiers, hypothesis testing, and parametric estimation under Partition Exchangeability are implemented. The two classifiers presented are the marginal classifier (that assumes test data is i.i.d.) next to a more computationally costly but accurate simultaneous classifier (that finds a labelling for the entire test dataset at once based on simultanous use of all the test data to predict each label). We also provide the Maximum Likelihood Estimation (MLE) of the only underlying parameter of the partition exchangeability generative model as well as hypothesis te…
In this paper, we propose a new test for the equality of several covariance functions for functional data. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function, hoping to obtain a more powerful test than some existing tests for the same testing problem. The asymptotic random expression of this test statistic under the null hypothesis is obtained. To approximate the null distribution of the proposed test statistic, we describe a parametric bootstrap method and a non…
The problem of comparing the entire second order structure of two functional processes is considered and a $L^2$-type statistic for testing equality of the corresponding spectral density operators is investigated. The test statistic evaluates, over all frequencies, the Hilbert-Schmidt distance between the two estimated spectral density operators. Under certain assumptions, the limiting distribution under the null hypothesis is derived. A novel frequency domain bootstrap method is introduced, which leads to a more accurate approximation of the distribution of the test statistic under the null t…
Nonparametric tests for equality of multivariate distributions are frequently desired in research. It is commonly required that test-procedures based on relatively small samples of vectors accurately control the corresponding Type I Error (TIE) rates. Often, in the multivariate testing, extensions of null-distribution-free univariate methods, e.g., Kolmogorov-Smirnov and Cramer-von Mises type schemes, are not exact, since their null distributions depend on underlying data distributions. The present paper extends the density-based empirical likelihood technique in order to nonparametrically app…
We propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared $t$-statistics rather than a direct summation of those components. More importantly, to d…
