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This article proposes the elliptical multivariate leptokurtic‐normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram–Charlier expansion of the multivariate normal (MN) distribution and has a closed‐form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model‐based clustering. An EM algorithm is presented to obtain estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions. The Canadian Journal of Statistics 45: 95–119; 2017 © 2016 Statistical Society of Canada

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The reliability of risk measures for financial portfolios crucially rests on the availability of sound representations of the involved random variables. The trade-off between adherence to reality and specification parsimony can find a fitting balance in a technique that "adjust" the moments of a density function by making use of its associated orthogonal polynomials. This approach rests on the Gram-Charlier expansion of a Gaussian law which, allowing for leptokurtosis to an appreciable extent, makes the resulting random variable a tail-sensitive density function. In this paper we determine the density of sums of leptokurtic normal variables duly adjusted for excess kurtosis by means of their Gram-Charlier expansions based on Hermite polynomials. The resultant density can be effectively used to represent a portfolio return and as such proves suitable for computing some risk measures such as Value at Risk and expected short fall. An application to a portfolio of financial returns is used to provide evidence of the effectiveness of the proposed approach.

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In this paper a novel partitioned inversion formula is obtained in terms of the orthogonal complements of off-diagonal blocks, with the emblematic matrix of unit-root econometrics emerging as the leading diagonal block of the inverse. The result paves the way to a straightforward derivation of a key result of vector autoregressive econometrics.

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