Measures Of Multivariate Kurtosis Research Articles

On a Vector-Valued Measure of Multivariate Skewness

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

Multivariate normality test based on kurtosis with two-step monotone missing data

This paper deals with a sample measure of multivariate kurtosis, which is used as a test statistic in multivariate normality testing problems. We define a new multivariate sample kurtosis measure to provide a multivariate normality test for data with a two-step monotone missing structure. Furthermore, we derive its expectation and variance using a perturbation method. To evaluate the accuracy of a normal approximation, we conducted a Monte Carlo simulation for certain parameters. Finally, we present a numerical example to illustrate the proposed procedure.

Some remarks on Koziol’s kurtosis

James Koziol, in his 1987 and 1989 papers, proposed to use the sums of either the squared fourth-order cumulants or moments as test statistics for multivariate normality. His proposals are by far less popular than Mardia’s measure of multivariate kurtosis, that is the fourth moment of the Mahalanobis distance of a random vector from its mean. We investigate some properties of Koziol’s measures of multivariate kurtosis which motivate their use in statistical practice. Firstly, we show some of their connections with Mahalanobis angles. Secondly, we use inequalities to highlight their connections with other measures of multivariate skewness and kurtosis. Thirdly, we obtain their analytical formulae for some well-known multivariate statistical models. Simple examples illustrate the interpretation of Koziol’s measures of multivariate kurtosis and detect a wrong statement about them which appeared in the statistical literature. We suggest that Mardia’s and Koziol’s measures of kurtosis should be used together to detect interesting data structures.

Asymptotic distributions for the sample measures of multivariate kurtosis

Asymptotic distributions for the sample measures of multivariate kurtosis

A new kurtosis matrix, with statistical applications

The number of fourth-order moments which can be obtained from a random vector rapidly increases with the vector's dimension. Scalar measures of multivariate kurtosis may not satisfactorily capture the fourth-order structure, and matrix measures of multivariate kurtosis are called for. In this paper, we propose a kurtosis matrix derived from the dominant eigenpair of the fourth standardized moment. We show that it is the best symmetric, positive semidefinite Kronecker square root approximation to the fourth standardized moment. Additional properties are derived for realizations from GARCH and reversible random processes. Statistical applications include independent component analysis and projection pursuit. The star product of matrices highlights the connection between the proposed kurtosis matrix and other kurtosis matrices which appeared in the statistical literature. A simulation study assesses the practical relevance of theoretical results in the paper.

A measure of multivariate kurtosis for the identification of the dynamics of a [formula omitted]-dimensional market

This paper investigates the common intuition suggesting that during crises the shape of the financial market clearly differentiates from that of random walk processes. In this sense, it challenges the traditional analysis of the nature of financial markets implicit in the most popular models. For this, a geometric approach is proposed in order to define the patterns of change of the market and a measure of multivariate kurtosis is used in order to test deviations from multinormality. The statistical difficulties of this approach are discussed and a new solution is proposed to the consideration of a large space of variables in an accurate measurement of the dynamics of the market. The emergence of crises can be measured in this framework, using all the available information about the returns of the stocks under consideration and not only a single index representing the market.

Wybrane własności kurtozy wektora losowego

The paper is a continuation of a discussion of the kurtosis of a random vector. Multivariate kurtosis is defined as the fourth central moment divided by the square of the variance of a random vector. This term is built on the definition of the power of a random vector proposed by J. Tatar. The paper presents selected, essential properties of multivariate kurtosis - among other things the invariance property under a number of affine transformations. Besides that, the relation between kurtosis of the random vector and its skewness is fixed. In view of these properties, the fourth central moment divided by the square of the variance of a random vector may be regarded as a satisfactory measure of multivariate kurtosis.

On the Asymptotic Normality of Test Statistics Using Song's Kurtosis

In this article, we consider the multivariate normality test based on the sample measure of multivariate kurtosis defined by Song (2001). We calculate moments of Song's kurtosis when covariance matrix is known and unknown. In addition, we derive expectations and variances of the multivariate sample kurtosis under normality and propose three standardized test statistics using these expectations and variances. When is known, we also propose a test statistic whose approximate accuracy for multivariate normal distribution is improved rather than the standardized statistics by normalizing transformation. In order to evaluate accuracy of proposed test statistics, the numerical results by Monte Carlo simulation for some selected values of parameters are presented.

A new multivariate kurtosis and its asymptotic distribution

In this paper, we propose a new definition for multivariate kurtosis based on the two measures of multivariate kurtosis defined by Mardia (1970) and Srivastava (1984), respectively. Under normality, the expectation and the variance for the new multivariate sample measure of kurtosis are given exactly. We also give the third moment for the sample measure of new multivariate kurtosis. After that standardized statistics and normalizing transformation statistic for the sample measure of a new multivariate kurtosis are derived by using these results. Finally, in order to evaluate accuracy of these statistics, we present the numerical results by Monte Carlo simulation for some selected values of parameters.

A Measure of Multivariate Kurtosis with Principal Components

Srivastava (1984) 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 note on scale mixtures of skew normal distribution

Moments of scale mixtures of skew normal distribution and their quadratic forms are derived using the simple stochastic relationship between skew normal distribution and scale mixtures of skew normal distribution. An application to time series is discussed. It is shown that the mean, covariance, and correlation structure of the sample autocovariance function for a particular class of time series depend on a measure of multivariate kurtosis, but not on a shape parameter. Unlike the multivariate normal distribution and the elliptical distributions with Muirhead kurtosis parameter 0, the covariance and the correlation of the sample autocovariance function for a particular class of time series with underlying scale mixtures of normal distribution are not always negative. We show this using an example of the generalized t distribution. As a by-product, we derived Muirhead kurtosis for some scale mixtures of normal distribution.

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β 2 , p and Song's measure S ( f ) . In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t -distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.

Mardia's Multivariate Kurtosis with Missing Data

Mardia's measure of multivariate kurtosis has been implemented in many statistical packages commonly used by social scientists. It provides important information on whether a commonly used multivariate procedure is appropriate for inference. Many statistical packages also have options for missing data. However, there is no procedure for applying Mardia's kurtosis to a data set with missing values. In this article Mardia's kurtosis is extended to a dataset with missing values. Based on the complete data counterpart, the behavior of a standardized version of the extended sample kurtosis can be described by the standard normal distribution. Analytical and Monte-Carlo studies imply that the proposed distribution description is as good as its complete data counterpart when missing variables are either missing completely at random or missing at random when observed marginals do not sit in a cluster with a restricted range. Application of this procedure is illustrated with a real data set.

Moments of random vectors with skew t distribution and their quadratic forms

Moments of skew t random vectors and their quadratic forms are derived. It is shown that the moments of the sample autocovariance function and of the sample variogram estimator depend on a measure of multivariate kurtosis, but not on a skewness parameter.

Correlation of spatial climate/weather maps and the advantages of using the Mahalanobis metric in predictions

he skill in predicting spatially varying weather/climate maps depends on the definition of the measure of similarity between the maps. Under the justifiable approximation that the anomaly maps are distributed multinormally, it is shown analytically that the choice of weighting metric, used in defining the anomaly correlation between spatial maps, can change the resulting probability distribution of the correlation coefficient. The estimate of the numbers of degrees of freedom based on the variance of the correlation distribution can vary from unity up to the number of grid points depending on the choice of weighting metric. The (pseudo-) inverse of the sample covariance matrix acts as a special choice for the metric in that it gives a correlation distribution which has minimal kurtosis and maximum dimension. Minimal kurtosis suggests that the average predictive skill might be improved due to the rarer occurrence of troublesome outlier patterns far from the mean state. Maximum dimension has a disadvantage for analogue prediction schemes in that it gives the minimum number of analogue states. This metric also has an advantage in that it allows one to powerfully test the null hypothesis of multinormality by examining the second and third moments of the correlation coefficient which were introduced by Mardia as invariant measures of multivariate kurtosis and skewness. For these reasons, it is suggested that this metric could be usefully employed in the prediction of weather/climate and in fingerprinting anthropogenic climate change. The ideas are illustrated using the bivariate example of the observed monthly mean sea-level pressures at Darwin and Tahitifrom 1866—1995.

On Mardia’s kurtosis test for multivariate normality

Let be independent identically distributed random(d-vectors with mean μ and nonsingular covariance matrix ∑ such that . We show that Mardia’s measure of multivariate kurtosis satisfies with σ2 depending on the distribution of X 1. As a consequence we obtain an approximation to the power function of a commonly proposed test for multivariate normality based on d2,d . Moreover, this test is consistent if, and only if. Examples include normal mixtures and elliptically symmetric distributions.

Probability Plots for Assessing Multivariate Normality

A method of assessing multivariate normality by decomposition of measures of multivariate skewness and kurtosis into orthogonal components is described. The components can readily be inspected analytically, as well as graphically with probability plots. Examination of the individual components can often reveal departures from multivariate normality that may be masked in their underlying measures of skewness and kurtosis. There exist many techniques for assessing multivariate normality; see, for example, extensive reviews by Gnanadesikan (1977), Cox and Small (1978), Mardia (1980) and Koziol (1986). One graphical method devolves from the 'radius and angles' approach, which is based on the well-known transformation of Euclidean to spherical coordinates. Gnanadesikan (1977) in particular describes various informal tests for multivariate normal- ity with graphical procedures for examining the radii and angles following spherical coordinate transformation; Small (1978), Koziol (1982, 1983), Royston (1983), Szkutnik (1987) and Quiroz and Dudley (1991) among others have considered more formal procedures related to this approach. A different paradigm for assessing multivariate normality is based on the attributes of multivariate moments. In a series of seminal papers (Mardia, 1970, 1974, 1975; Mardia & Foster, 1983; Mardia & Kanazawa, 1983), Mardia introduced affine invariant measures of multivariate skewness and kurtosis, and comprehensively examined their properties under various circumstances. From a different perspective, Koziol (1986, 1987) considered measures of multivariate skewness and kurtosis suggested from the notion of Neyman's smooth tests; upon deriving smooth tests for assessing multivariate normality, he further found that the smooth test for multivariate skewness coincides with Mardia's measure of multivariate skewness, but the smooth test for multivariate kurtosis differed somewhat from Mardia's measure of multivariate kurtosis. Koziol noted that the smooth tests may be decomposed into individual components, which are distributed as independent %2 random variables, each with one degree of freedom. See also Mardia (1987) and Mardia and Kent (1991) for a unifying treatment devolving from Rao's score tests. The purpose of this note is to describe graphical techniques for depicting the components, and to demonstrate that examination of the individual components can be a valuable tool for assessing multivariate normality. For completeness, the smooth tests and their components are briefly reviewed in Section 2; examples of assessing multivariate normality with the components are found in

Some new tests for multivariate normality

A family of statistics is presented that can be used for testing goodness of fit to a parametric family. These statistics include Mardia's measure of multivariate kurtosis and Moore and Stubblebine's test for multivariate normality. The asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and, in the case of multivariate normality, the distribution is shown to be independent of the “true” parameter. A class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations.

Mardia's coefficient of kurtosis in elliptical populations

Mardia (1970) defined a measure of multivariate kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Some new results on elliptical distributions are used to extend Mardia's results to samples from an elliptical distribution. These results provide a method for testing hypotheses on the kurtosis parameter of elliptical distributions. An appendix provides extensions of Kendall and Stuart's (1977) standard errors of bivariate moments to the third and fourth order.

A Note on Measures of Multivariate Kurtosis

AbstractWe propose a measure of multivariate kurtosis suggested from Mardia's measure of multivariate skewness b1,p, and examine its relationship both to Mardia's measure of multivariate kurtosis b2,p, and to a smooth test of multivariate kurtosis Ǔ42.