Search

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]

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. [ABSTRACT FROM AUTHOR]

Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models. [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]

The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector θ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of θ. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution. • A way to make more flexible the tail-behavior of the Laplace distribution. • The influence of observations with large scaled absolute distances is reduced. • Good models to describe the stylized facts of the cryptocurrency return distribution. [ABSTRACT FROM AUTHOR]

• We propose a robust method for simultaneous unsupervised classification and dimensionality reduction for high-dimensional data. • The proposed approach is effective for identifying mild outliers in high-dimensional unsupervised classification problems. • The proportion of mild outliers is learned and so does not need to be pre-specified. The contaminated Gaussian distribution represents a simple heavy-tailed elliptical generalization of the Gaussian distribution; unlike the often-considered t -distribution, it also allows for automatic detection of mild outlying or "bad" points in the same way that observations are typically assigned to the groups in the finite mixture model context. Starting from this distribution, we propose the contaminated factor analysis model as a method for dimensionality reduction and detection of bad points in higher dimensions. A mixture of contaminated Gaussian factor analyzers (MCGFA) model follows therefrom, and extends the recently proposed mixture of contaminated Gaussian distributions to high-dimensional data. We introduce a family of 32 parsimonious models formed by introducing constraints on the covariance and contamination structures of the general MCGFA model. We outline a variant of the expectation-maximization algorithm for parameter estimation. Various implementation issues are discussed, and the novel family of models is compared to well-established approaches on both simulated and real data. [ABSTRACT FROM AUTHOR]