Nicola Loperfido

Generalized skew-elliptical distributions and their quadratic forms

This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-t, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.

Quadratic forms of skew-normal random vectors

The sum of squares and products matrix has a Wishart distribution, when the rows of the corresponding data matrix are i.i.d. according to a skew-normal distribution centered at the origin. Applications include robustness of the t-test, time series and spatial statistics.

Statistical implications of selectively reported inferential results

Researchers sometimes report only the largest estimate, the most significant test statistic and the most shifted confidence interval. The following result quantifies the statistical implications of this behavior, when the choice is restricted to two inferential procedures: the minimum and maximum of two standardized random variables whose distribution is jointly normal is skew-normal. More generally, the distribution of the minimum and maximum of two random variables whose distribution is bivariate normal centered at the origin is a mixture with equal weights of scaled skew-normal distributions.

Skewed distributions in finance and actuarial science: a review

That the returns on financial assets and insurance claims are not well described by the multivariate normal distribution is generally acknowledged in the literature. This paper presents a review of the use of the skew-normal distribution and its extensions in finance and actuarial science, highlighting known results as well as potential directions for future research. When skewness and kurtosis are present in asset returns, the skew-normal and skew-Student distributions are natural candidates in both theoretical and empirical work. Their parameterization is parsimonious and they are mathematically tractable. In finance, the distributions are interpretable in terms of the efficient markets hypothesis. Furthermore, they lead to theoretical results that are useful for portfolio selection and asset pricing. In actuarial science, the presence of skewness and kurtosis in insurance claims data is the main motivation for using the skew-normal distribution and its extensions. The skew-normal has been used in studies on risk measurement and capital allocation, which are two important research fields in actuarial science. Empirical studies consider the skew-normal distribution because of its flexibility, interpretability, and tractability. This paper comprises four main sections: an overview of skew-normal distributions; a review of skewness in finance, including asset pricing, portfolio selection, time series modeling, and a review of its applications in insurance, in which the use of alternative distribution functions is widespread. The final section summarizes some of the challenges associated with the use of skew-elliptical distributions and points out some directions for future research.

A multivariate skew-garch model

Empirical research on European stock markets has shown that they behave differently according to the performance of the leading financial market identified as the US market. A positive sign is viewed as good news in the international financial markets, a negative sign means, conversely, bad news. As a result, we assume that European stock market returns are affected by endogenous and exogenous shocks. The former raise in the market itself, the latter come from the US market, because of its most influential role in the world. Under standard assumptions, the distribution of the European market index returns conditionally on the sign of the one-day lagged US return is skew-normal. The resulting model is denoted Skew-GARCH. We study the properties of this new model and illustrate its application to time-series data from three European financial markets.

Some Relationships Between Skew-Normal Distributions and Order Statistics from Exchangeable Normal Random Vectors

The article explores the relationship between distributions of order statistics from random vectors with exchangeable normal distributions and several skewed generalizations of the normal distribution. In particular, we show that the order statistics of exchangeable normal observations have closed skew-normal distributions, and that the corresponding density function is log-concave when the order statistic is extreme. Special attention is given to the bivariate case, which is related to the univariate skew-normal distribution. The applications discussed focus on the lifetimes of coherent systems.

Modelling multivariate skewness in financial returns: a SGARCH approach

Skewness of financial time series is a relevant topic, due to its implications for portfolio theory and for statistical inference. In the univariate case, its default measure is the third cumulant of the standardized random variable. It can be generalized to the third multivariate cumulant that is a matrix containing all centered moments of order three which can be obtained from a random vector. The present paper examines some properties of the third cumulant under the assumptions of the multivariate SGARCH model introduced by De Luca, Genton, and Loperfido [2006. A multivariate skew-GARCH model. Advances in Econometrics 20: 33–57]. In the first place, it allows for parsimonious modelling of multivariate skewness. In the second place, all its elements are either null or negative, consistently with previous empirical and theoretical findings. A numerical example with financial returns of France, Spain and Netherlands illustrates the theoretical results in the paper.

Maximum likelihood estimation of correlation between maximal oxygen consumption and the 6-min walk test in patients with chronic heart failure

Maximal oxygen consumption (VO2max) is the standard measurement used to quantify cardiovascular functional capacity and aerobic fitness. Unfortunately, it is a costly, impractical and labour-intensive measure to obtain. The 6-min walk test (6MWT) also assesses cardiopulmonary function, but in contrast to the VO2max test, it is inexpensive and can be performed almost anywhere. Various medical studies have addressed the correlation between VO2max and 6MWT in patients with chronic heart failure. Of particular interest, from a medical point of view, is the conditional correlation between the two measures given the individuals height, weight, age and gender. In this paper, we have calculated the maximum likelihood estimate of the conditional correlation in patients with chronic heart failure under the assumption of skew normality. Data were recorded from 98 patients in the Operative Unit of Thoracic Surgery in Bari, Italy. The estimated conditional correlation was found to be much smaller than estimated marginal correlations reported in the medical literature.

Third Cumulant for Multivariate Aggregate Claim Models

The third 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. Two important special cases are considered. In the first 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. It is discussed how these results that deal only with dependence in the claim sizes can be used to obtain a formula for the third cumulant for more complex aggregate models of multivariate claims in which the dependence is also in the aggregating variables.