MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness

Skewness plays a relevant role in several multivariate statistical techniques. Sometimes it is used to recover data features, as in cluster analysis. In other circumstances, skewness impairs the performances of statistical methods, as in the Hotelling's one-sample test. In both cases, there is the need to check the symmetry of the underlying distribution, either by visual inspection or by formal testing. The R packages MaxSkew and MultiSkew address these issues by measuring, testing and removing skewness from multivariate data. Skewness is assessed by the third multivariate cumulant and its functions. The hypothesis of symmetry is tested either nonparametrically, with the bootstrap, or parametrically, under the normality assumption. Skewness is removed or at least alleviated by projecting the data onto appropriate linear subspaces. Usages of MaxSkew and MultiSkew are illustrated with the Iris dataset

Sampling Distribution of the Gini Index from a Skew Normal

The Gini concentration ratio has been extensively used in the study of "inequality" of distributions. According to Chakrabarthy (1982), "the Lorenz curve and the Gini index have remained the most popular and powerful tool in the analyses of size distribution of income". A major statistical limitation of Gini Concentration Ratio is the absence and the intractability of appropriate sampling distribution (Hart, 1971, Nigard Sandstrom, 1981) We find that the exact sampling distribution of the Gini Concentration Index for a sample from a Skew Normal population has an Extended Skew Normal distribution. This result can be easily particularized to the normal case.Skew Normal, L-statistics, Small Sample, Gini Concentration Ratio, Exact Distribution.

On the exact sampling distribution of L-statistics

This paper shows that linear functions of order statistics (L-statistics) based on random samples have a Fundamental Skew distribution (Arellano-Valle Genton, 2003). The paper also examines the exact distribution of L-statistics when the sampled population is Skew Normal. Exact distributions of L-statistics from normal samples easily follows as special case.

Statistical Analysis of the Correlation between Italian and U.S. Stock Returns

An estimator of the correlation between Italian and U.S. Stock Returns is introduced. The properties of the estimator are invariant with respect to a wide class of GARCH models. The empirical evidence shows the existence of a positive correlation between Italian an U. S. stock returns.GARCH models, Invariance, Stock Returns.

Linear transformations to symmetry

Abstract We obtain random vectors with null third-order cumulants by projecting the data onto appropriate subspaces. Statistical applications include, but are not limited to, the robustification of Hotelling's T 2 test against nonnormality. Our approach only requires the existence of the third-order moments and leads to normal transformed variables when the parent distribution belongs to well-known classes of sample selection models

Canonical correlations and nonlinear dependencies

Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones

Third Cumulant for Multivariate Aggregate Claim Models

The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of multivariate claims. Two important special cases are considered. In the rst one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results

Bayesian inference for skew-symmetric distributions

Skew-symmetric distributions are a popular family of flexible distributions that conveniently model non-normal features such as skewness, kurtosis and multimodality. Unfortunately, their frequentist inference poses several difficulties, which may be adequately addressed by means of a Bayesian approach. This paper reviews the main prior distributions proposed for the parameters of skew-symmetric distributions, with special emphasis on the skew-normal and the skew-t distributions which are the most prominent skew-symmetric models. The paper focuses on the univariate case in the absence of covariates, but more general models are also discussed