The multivariate tail-inflated normal distribution and its application in finance

Abstract

The research objective of this paper is to handle situations where the empirical distribution of multivariate real-valued data is elliptical and with heavy tails. Many statistical models already exist that accommodate these peculiarities. This paper enriches this branch of literature by introducing the multivariate tail-inflated normal (MTIN) distribution, an elliptical heavy tails generalization of the multivariate normal (MN). The MTIN belongs to the family of MN scale mixtures by choosing a convenient continuous uniform as mixing distribution. Moreover, it has a closed-form for the probability density function characterized by only one additional ‘inflation’ parameter, with respect to the nested MN, governing the tail-weight. The moment generating function, and the first four moments, are also derived; interestingly, the latter always exist and the excess kurtosis can assume any positive value. The method of moments and maximum likelihood (ML) are considered for estimation. As concerns the latter, a direct approach, as well as a variant of the EM algorithm (namely, the ECME algorithm), are illustrated. Furthermore, a way to approximate covariance matrix of the ML estimator is suggested and the existence of the ML estimates is evaluated. Since the inflation parameter is estimated from the data, robust estimates of the mean vector of the nested MN distribution are automatically obtained by down-weighting. Simulations are performed to compare the estimation methods/algorithms, to investigate the ability of AIC and BIC to select among a set of candidate elliptical models, and to evaluate the robustness of these candidate methods when data are skewed. The findings are the following: ML is better than MM, direct ML is suggested for low dimensions, while the ECME algorithm is to be preferred when the number of variables is higher, AIC and BIC work comparably in selecting the true underlying model, and the MTIN outperforms the competing models in terms of robustness toward skew data. For illustrative purposes, the MTIN distribution is finally fitted to multivariate financial data and compared with other well-established multivariate elliptical distributions. The analysis shows how the proposed model represents a valid alternative to the considered competitors in terms of AIC and BIC, but also in reproducing the higher empirical kurtosis which is common in the financial context.