Likelihood ratio tests for multivariate normality

This paper presents some powerful omnibus tests for multivariate normality based on the likelihood ratio and the characterizations of the multivariate normal distribution. The power of the proposed tests is studied against various alternatives via Monte Carlo simulations. Simulation studies show our tests compare well with other powerful tests including multivariate versions of the Shapiro–Wilk test and the Anderson–Darling test.     

Invariant tests for multivariate normality: a critical review

d -dimensional random vector X is some nondegenerate d-variate normal distribution, on the basis of i.i.d. copies X ₁, ..., X _x of X. Particular emphasis is given to progress that has been achieved during the last decade. Furthermore, we stress the typical diagnostic pitfall connected with purportedly ‘directed’ procedures, such as tests based on measures of multivariate skewness. Received: April 30, 2001; revised version: October 30, 2001     

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.     

New tests for multivariate normality based on Small's and Srivastava's graphical methods

In this paper, we propose a new measure of fit which can be used in the case of quantile–quantile plots. This measure, when applied to Small's and Srivastava's graphical methods provides two new tests for assessing multivariate normality. For different sample sizes and numbers of variables, the critical values of these tests were evaluated via simulations. The power of the new tests and its comparison with some other tests for multivariate normality are presented herein.     

Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S^-1 (x-[Xbar]) = c_i centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x² (M-1) and x² (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.     

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.     

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.     

A comparison of uniformity tests

Problems of goodness-of-fit to a given distribution can usually be reduced to test uniformity. The uniform distribution appears due to natural random events or due to the application of methods for transforming samples from any other distribution to the samples with values uniformly distributed in the interval (0, 1). Thus, one can solve the problem of testing if a sample comes from a given distribution by testing whether its transformed sample is distributed according to the uniform distribution. For this reason, the methods of testing for goodness-of-fit to a uniform distribution have been widely investigated. In this paper, a comparative power analysis of a selected set of statistics is performed in order to give suggestions on which one to use for testing uniformity against the families of alternatives proposed by Stephens [Stephens, M.A., 1974, EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730–737.]. Definition and some relevant features of the considered test statistics are given in section 1. Implemented numerical processes to calculate percentage points of every considered statistic are described in section 2. Finally, a Monte Carlo simulation experiment has been carried out to fulfill the mentioned target of this paper.     

Power studies of tests for uniformity,II

Results from a power study of six statistics for testing that a sample is from a uniform distribution on the unit interval (0,1) are reported. The test statistics are all well-known and each of them was originally proposed because they should have high power against some alternative distributions. The tests considered are the Pearson probability product test, the Neyman smooth test, the Sukhatme test, the Durbin-Kolmogorov test, the Kuiper test, and the Sherman test. Results are given for each of these tests against each of four classes of alternatives. Also, the most powerful test against each member of the first three alternatives is obtained, and the powers of these tests are given for the same sample sizes as for the six general "omnibus" test statistics. These values constitute a "power envelope" against which all tests can be compared. The Neyman smooth tests with 2nd and 4th degree polynomials are found to have good power and are recommended as general tests for uniformity.     

Robust multivariate transformations to normality: Constructed variables and likelihood ratio tests

The assumption of multivariate normality provides the customary powerful and convenient ways of analysing multivariate data: if the data are not normal, the analysis may often be simplified by an appropriate transformation. In this context, the most widely used test is the likelihood ratio, which requires the maximum likelihood estimate of the transformation parameter for each variable. Given that this estimate can only be found numerically, when the number of variables is large (> 20) it is impossible or infeasible to compute the test. In this paper we introduce alternative tests which do not require the maximum likelihood estimate of the transformation parameters and prove algebraically their relationships. We also give insights both using theoretical arguments and a robust simulation study, based on the forward search algorithm, about the distribution of the tests previously introduced.     

Efficiency of ranked set sampling in tests for normality

In this article, we consider the ranked set sampling (RSS) and investigate seven tests for normality under RSS. Each test is described and then power of each test is obtained by Monte Carlo simulations under various alternatives. Finally, the powers of the tests based on RSS are compared with the powers of the tests based on the simple random sampling and the results are discussed.     

Comparisons of various types of normality tests

Normality tests can be classified into tests based on chi-squared, moments, empirical distribution, spacings, regression and correlation and other special tests. This paper studies and compares the power of eight selected normality tests: the Shapiro–Wilk test, the Kolmogorov–Smirnov test, the Lilliefors test, the Cramer–von Mises test, the Anderson–Darling test, the D'Agostino–Pearson test, the Jarque–Bera test and chi-squared test. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from alternative distributions that follow symmetric short-tailed, symmetric long-tailed and asymmetric distributions. Our simulation results show that for symmetric short-tailed distributions, D'Agostino and Shapiro–Wilk tests have better power. For symmetric long-tailed distributions, the power of Jarque–Bera and D'Agostino tests is quite comparable with the Shapiro–Wilk test. As for asymmetric distributions, the Shapiro–Wilk test is the most powerful test followed by the Anderson–Darling test.     

Asymptotic power of tests of normality under local alternatives

Seven tests of univariate normality are studied in view of their asymptotic power under local alternatives. The procedures under consideration are either based on the empirical skewness and/or kurtosis, including the popular Jarque-Bera statistic, as well as Cramér-von Mises, Anderson-Darling and Kolmogorov-Smirnov functionals of an empirical process with estimated parameters. The large-sample behavior of these test statistics under contiguous sequences is obtained; this allows for the computation of their associated local power curves and of their asymptotic relative efficiency in the light of a measure proposed by Berg and Quessy (2009). Comparisons are made under four classes of local alternatives, including those used by Thadewald and Büning (2007) in a recent Monte-Carlo power study. These theoretical results are related to empirical ones and many recommendations are formulated.     

A re-appraisal of tests for normality

Tests for normality can be divided into two groups - those based upon a function of the empirical distribution function and those based upon a function of the original observations. The latter group of statistics test spherical symmetry and not necessarily normality. If the distribution is completely specified then the first group can be used to test for ‘spherical’ normality. However, if the distribution is incompletely specified and F‘‘x_i - x’/s’ is used these test statistics also test sphericity rather than normality. A Monte Carlo study was conducted for the completely specified case, to investigate the sensitivity of the distance tests to departures from normality when the alternative distributions are non-normal spherically symmetric laws. A “new” test statistic is proposed for testing a completely specified normal distribution     

New invariant and consistent chi-squared type goodness-of-fit tests for multivariate normality and a related comparative simulation study

ABSTRACT

New invariant and consistent goodness-of-fit tests for multivariate normality are introduced. Tests are based on the Karhunen–Loève transformation of a multidimensional sample from a population. A comparison of simulated powers of tests and other well-known tests with respect to some alternatives is given. The simulation study demonstrates that power of the proposed McCull test almost does not depend on the number of grouping cells. The test shows an advantage over other chi-squared type tests. However, averaged over all of the simulated conditions examined in this article, the Anderson–Darling type and the Cramer–von Mises type tests seem to be the best.     

Some tests for a poisson process and their powers with respect to trend in the intensity

Recently many authors have worked on Wei bull process in the area of modelling and analysis. Much less work is done in the area of testing of hypothesis. In this article, some tests for testing the Poisson process against a class of Wei bull process based on the conditional distribution of observations given the sufficient statistic, are proposed. The percentage points of the distributions of the proposed test statistics are simulated. The powers of the tests under alternatives are computed by Monte Carlo method. It is seen that the suggested tests perform well for decreasing intensities.     

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 tests detecting difference in means and variances simultaneously under normality

In this paper, we propose several tests for detecting difference in means and variances simultaneously between two populations under normality. First of all, we propose a likelihood ratio test. Then we obtain an expression of the likelihood ratio statistic by a product of two functions of random quantities, which can be used to test the two individual partial hypotheses for differences in means and variances. With those individual partial tests, we propose a union-intersection test. Also we consider two optimal tests by combining the p-values of the two individual partial tests. For obtaining null distributions, we apply the permutation principle with the Monte Carlo approach. Then we compare efficiency among the proposed tests with well-known ones through a simulation study. Finally, we discuss some interesting features related to the simultaneous tests and resampling methods as concluding remarks.     

Empirical behavior of some tests for normality

The behavior of a range of tests assessing the normality of a sequence of independent and identically distributed random variables is investigated.An examination of the empirical significance level of the tests is undertaken for different sample sizes. The empirical power associated with these tests is also calculated under some alternative distributions.     

Critical values of some nonparametric statistics used for tests of total stochastic independence

The problem of testing for total independence of the variates of a stochastic p(≧3) component vector using rank correlation statistics is considered. Two distribution free statistics are considered, one based on the determinant of the matrix of rank correlation statistics, the second on their sum of squares. Tables of critical values are given for p=3,4 for the cases when (a) ranks, and (b) exponential scores are used to replace the ordered observations within each variate. Some approximations to the critical values are proposed and evaluated.