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In allometric studies, the joint distribution of the log-transformed morphometric variables is typically symmetric and with heavy tails. Moreover, in the bivariate case, it is customary to explain the morphometric variation of these variables by fitting a convenient line, as for example the first principal component (PC). To account for all these peculiarities, we propose the use of multiple scaled symmetric (MSS) distributions. These distributions have the advantage to be directly defined in the PC space, the kind of symmetry involved is less restrictive than the commonly considered elliptical symmetry, the behavior of the tails can vary across PCs, and their first PC is less sensitive to outliers. In the family of MSS distributions, we also propose the multiple scaled shifted exponential normal distribution, equivalent of the multivariate shifted exponential normal distribution in the MSS framework. For the sake of parsimony, we also allow the parameter governing the leptokurtosis on each PC, in the considered MSS distributions, to be tied across PCs. From an inferential point of view, we describe an EM algorithm to estimate the parameters by maximum likelihood, we illustrate how to compute standard errors of the obtained estimates, and we give statistical tests and confidence intervals for the parameters. We use artificial and real allometric data to appreciate the advantages of the MSS distributions over well-known elliptically symmetric distributions and to compare the robustness of the line from our models with respect to the lines fitted by well-established robust and non-robust methods available in the literature. [ABSTRACT FROM AUTHOR]

The multivariate contaminated normal (MCN) distribution represents a simple heavy-tailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers (also referred to as 'bad' points herein) and automatically detect bad points. The price of these advantages is two additional parameters: proportion of good observations and degree of contamination. However, in a multivariate setting, only one proportion of good observations and only one degree of contamination may be limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution. Among its parameters, we have an orthogonal matrix Γ. In the space spanned by the vectors (principal components) of Γ, there is a proportion of good observations and a degree of contamination for each component. Moreover, each observation has a posterior probability of being good with respect to each principal component. Thanks to this probability, the method provides directional robust estimates of the parameters of the nested MN and automatic directional detection of bad points. The term 'directional' is added to specify that the method works separately for each principal component. Mixtures of MSCN distributions are also proposed, and an expectation-maximization algorithm is used for parameter estimation. Real and simulated data are considered to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails. [ABSTRACT FROM AUTHOR]

We propose the family of dimension-wise scaled normal mixtures (DSNMs) to model the joint distribution of a d -variate random variable with real-valued components. Each member of the family generalizes the multivariate normal (MN) distribution in two directions. Firstly, the DSNM has a more general type of symmetry with respect to the elliptical symmetry of the MN distribution. Secondly, the univariate marginals have similar heavy-tailed normal scale mixture distributions with (possibly) different tailedness parameters; as a consequence of practical interest, the DSNM allows for a different excess kurtosis on each dimension. These peculiarities are obtained in an MN scale mixture framework by introducing a d -variate mixing random variable with independent and similar components acting separately for each dimension. We examine several properties of DSNMs, such as the joint density function, hierarchical and stochastic representations, relations with other families of symmetric distributions, type of symmetry, marginal and conditional distributions, no correlation implying independence, moment generating function, and moments of practical interest. For illustrative purposes, we describe two members of the DSNM family obtained in the case of components of the mixing random vector being either uniform or shifted exponential; these are examples of mixing distributions that guarantee a closed-form expression for the joint density of the DSNM. For the two DSNMs analyzed in detail, we introduce parsimony by allowing the d tailedness parameters to be tied across dimensions, and describe algorithms, based on the expectation–maximization (EM) principle, to estimate the parameters by maximum likelihood. We use real data from the financial and biometrical fields to appreciate the advantages of our DSNMs over other symmetric heavy-tailed distributions available in the literature. [ABSTRACT FROM AUTHOR]