A note on the asymptotic distribution of mardia's measure of multivariate kurtosis

Weak convergence results are used to investigate asymptotic properties of Mardia's measure of multivariate kurtosis in the context of assessing multivariate normality.     

A MATLAB package for multivariate normality test

A MATLAB package testing for multivariate normality (TMVN) is implemented as an interactive and graphical tool to examine multivariate normality (MVN). Monte Carlo simulation studies have failed to find a uniformly most powerful MVN test, which requires a rather extensive statistical inference procedure. TMVN contains several competitive MVN tests and provides a flexible and extensive testing environment for univariate or multivariate data analyses. Simulated results provide information of which test may possess more power for the selected non-MVN alternatives. Fisher's Iris data are used to show how TMVN can be used in practice.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

A new test for multivariate normality by combining extreme and nonextreme BHEP tests

In this article, we propose a new multiple test procedure for assessing multivariate normality, which combines BHEP (Baringhaus–Henze–Epps–Pulley) tests by considering extreme and nonextreme choices of the tuning parameter in the definition of the BHEP test statistic. Monte Carlo power comparisons indicate that the new test presents a reasonable power against a wide range of alternative distributions, showing itself to be competitive against the most recommended procedures for testing a multivariate hypothesis of normality. We further illustrate the use of the new test for the Fisher Iris dataset.     

Applications of empirical characteristic functions in some multivariate problems

In this note we extend univariate tests for normality and symmetry based on empirical characteristic functions to the multivariate case.     

A power study of goodness-of-fit tests for multivariate normality implemented in R

Multivariate statistical analysis procedures often require data to be multivariate normally distributed. Many tests have been developed to verify if a sample could indeed have come from a normally distributed population. These tests do not all share the same sensitivity for detecting departures from normality, and thus a choice of test is of central importance. This study investigates through simulated data the power of those tests for multivariate normality implemented in the statistic software R and pits them against the variant of testing each marginal distribution for normality. The results of testing two-dimensional data at a level of significance α=5% showed that almost one-third of those tests implemented in R do not have a type I error below this. Other tests outperformed the naive variant in terms of power even when the marginals were not normally distributed. Even though no test was consistently better than all alternatives with every alternative distribution, the energy-statistic test always showed relatively good power across all tested sample sizes.     

Test for normality based on two new estimators of entropy

In this article, two new consistent estimators are introduced of Shannon's entropy that compares root of mean-square error with other estimators. Then we define new tests for normality based on these new estimators. Finally, by simulation, the powers of the proposed tests are compared under different alternatives with other entropy tests for normality.     

Tests for multinormality with applications to time series

Making use of a characterization of multivariate normality by Hermitian polynomials, we propose a multivariate normality test. The approach is then applied to time series analysis by constructing a test for Gaussianity of a stationary univariate series. Simulation study shows that the proposed test has reasonable power and outperforms other tests available in the literature when the innovation series of the time series is symmetric, but non-Gaussian.     

Normality testing: two new tests using L-moments

Establishing that there is no compelling evidence that some population is not normally distributed is fundamental to many statistical inferences, and numerous approaches to testing the null hypothesis of normality have been proposed. Fundamentally, the power of a test depends on which specific deviation from normality may be presented in a distribution. Knowledge of the potential nature of deviation from normality should reasonably guide the researcher's selection of testing for non-normality. In most settings, little is known aside from the data available for analysis, so that selection of a test based on general applicability is typically necessary. This research proposes and reports the power of two new tests of normality. One of the new tests is a version of the R-test that uses the L-moments, respectively, L-skewness and L-kurtosis and the other test is based on normalizing transformations of L-skewness and L-kurtosis. Both tests have high power relative to alternatives. The test based on normalized transformations, in particular, shows consistently high power and outperforms other normality tests against a variety of distributions.     

A Measure of Multivariate Kurtosis with Principal Components

Srivastava (1984 Srivastava , M. S. ( 1984 ). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality . Statist. Probab. Lett. 2 ( 5 ): 263 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) defined a measure of multivariate kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Some new results are obtained by generalizing Srivastava's theorem to an asymptotic expansion up to higher order. Finally, two numerical examples are presented.     

A new functional statistic for multivariate normality

The test statistics of assessing multivariate normality based on Roy’s union-intersection principle (Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1953) are generalizations of univariate normality, and are formed as the optimal value of a nonlinear multivariate function. Due to the difficulty of solving multivariate optimization problems, researchers have proposed various approximations. However, this paper shows that the (nearly) global solution contrarily results in unsatisfactory power performance in Monte Carlo simulations. Thus, instead of searching for a true optimal solution, this study proposes a functional statistic constructed by the q% quantile of the objective function values. A comparative Monte Carlo analysis shows that the proposed method is superior to two highly recommended tests when detecting widely-selected alternatives that characterize the various properties of multivariate normality.     

Estimation and hypothesis testing in multivariate linear regression models under non normality

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.     

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.     

Asymptotic normality of estimators for parameters of a multivariate skew-normal distribution

In this paper, asymptotic normality is established for the parameters of the multivariate skew-normal distribution under two parametrizations. Also, an analytic expression and an asymptotic normal law are derived for the skewness vector of the skew-normal distribution. The estimates are derived using the method of moments. Convergence to the asymptotic distributions is examined both computationally and in a simulation experiment.     

Applications of a new power normal family

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson-Hilferty power transformation setting.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.     

An efficient method for the estimation of multivariate moving averge models

Durbin's (1959) efficient method for the estimation of univariate moving average models is generalized to the vector case. Strong consistency and asymptotic normality of the estimator is proved. A simulation experiment is performed to illustrate the behaviour of the method in finite samples.     

Asymptotic Normality for Regression Function Estimate Under Truncation and α-Mixing Conditions

In this article we establish pointwise asymptotic normality of nonparametric kernel estimator of regression function for a left truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimation of the covariable's density is considered. As a by-product, we obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and truncated distributions under dependence, which is interesting independently. Finite sample behavior of the estimator of the regression function is investigated as well.     

Adaptive plotting position and test of normality

The quantile–quantile plot is widely used to check normality. The plot depends on the plotting positions. Many commonly used plotting positions do not depend on the sample values. We propose an adaptive plotting position that depends on the relative distances of the two neighbouring sample values. The correlation coefficient obtained from the adaptive plotting position is used to test normality. The test using the adaptive plotting position is better than the Shapiro–Wilk W test for small samples and has larger power than Hazen's and Blom's plotting positions for symmetric alternatives with shorter tail than normal and skewed alternatives when n is 20 or larger. The Brown–Hettmansperger T* test is designed for detecting bad tail behaviour, so it does not have power for symmetric alternatives with shorter tail than normal, but it is generally better than the other tests when β₂ is greater than 3.25.