We also evaluate the existence of the ML estimates. The membership to the family of MNSMs allows us a simple computation of the maximum likelihood (ML) estimates of the parameters via the expectation-maximization (EM) algorithm advantageously, the M-step is computationally simplified by closed-form updates of all the parameters. The first four moments exist and the excess kurtosis can assume any positive value. The probability density function of the MSEN has a simple closed-form characterized by only one additional parameter, with respect to the nested MN, governing the tail weight. The MSEN belongs to the family of MN scale mixtures (MNSMs) by choosing a convenient shifted exponential as mixing distribution. To account for these peculiarities, we introduce the multivariate shifted exponential normal (MSEN) distribution, an elliptical heavy-tailed generalization of the multivariate normal (MN). We fit the MSEN distribution to multivariate allometric data where we show its usefulness also in comparison with other well-established multivariate elliptical distributions.Ībstract = "In allometric studies, the joint distribution of the log-transformed morphometric variables is typically elliptical and with heavy tails. Since the parameter governing the tail weight is estimated from the data, robust estimates of the mean vector of the nested MN distribution are automatically obtained by downweighting we show this aspect theoretically but also by means of a simulation study. In allometric studies, the joint distribution of the log-transformed morphometric variables is typically elliptical and with heavy tails.
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