Multivariate goodness-of-fit tests based on kernel density estimators

The paper is devoted to multivariate goodness-of-fit ests based on kernel density estimators. Both simple and composite null hypotheses are investigated. The test statistic is considered in the form of maximum of the normalized deviation of the estimate from its expected value. The produced comparative Monte Carlo power study shows that the proposed test is a powerful competitor to the existing classical criteria for testing goodness of fit against a specific type of an alternative hypothesis. An analytical way to establish the asymptotic distribution of the test statistic is discussed, using the approximation results for the probabilities of high excursions of differentiable Gaussian random fields

Combinando testes de Mardia e BHEP na avaliação duma hipótese multivariada de normalidade

https://thekeep.eiu.edu/den_1998_feb/1015/thumbnail.jp

Are You All Normal? It Depends!

The assumption of normality has underlain much of the development of statistics, including spatial statistics, and many tests have been proposed. In this work, we focus on the multivariate setting and we first provide a synopsis of the recent advances in multivariate normality tests for i.i.d. data, with emphasis on the skewness and kurtosis approaches. We show through simulation studies that some of these tests cannot be used directly for testing normality of spatial data, since the multivariate sample skewness and kurtosis measures, such as the Mardia's measures, deviate from their theoretical values under Gaussianity due to dependence, and some related tests exhibit inflated type I error, especially when the spatial dependence gets stronger. We review briefly the few existing tests under dependence (time or space), and then propose a new multivariate normality test for spatial data by accounting for the spatial dependence of the observations in the test statistic. The new test aggregates univariate Jarque-Bera (JB) statistics, which combine skewness and kurtosis measures, for individual variables. The asymptotic variances of sample skewness and kurtosis for standardized observations are derived given the dependence structure of the spatial data. Consistent estimators of the asymptotic variances are then constructed for finite samples. The test statistic is easy to compute, without any smoothing involved, and it is asymptotically $\chi^2_{2p}$ under normality, where $p$ is the number of variables. The new test has a good control of the type I error and a high empirical power, especially for large sample sizes

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schr\"odinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure.Comment: 29 pages, 1 figure, 7 table

Tests for multivariate normality—a critical review with emphasis on weighted L2L^2-statistics

This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted L$^{2}$-statistics. Since weighted L$^{2}$-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt

Tests for multivariate normality -- a critical review with emphasis on weighted L2L^2-statistics

This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted $L^2$-statistics. Since weighted $L^2$-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

We study a novel class of affine invariant and consistent tests for normality in any dimension in an i.i.d.‐setting. The tests are based on a characterization of the standard d‐variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrödinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood‐of‐model validation procedure

A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function

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 L2-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 existing procedures, and apply the new procedure to a data set of monthly log returns.Ministerio de Economía y Competitividad (MINECO). Españ