A collection of robust Mahalanobis distances for multivariate outlier detection is proposed, based on the notion of shrinkage. Robust intensity and scaling factors are optimally estimated to define the shrinkage. Some properties are investigated, such as affine equivariance and breakdown value. The performance of the proposal is illustrated through the comparison to other techniques from the literature, in a simulation study and with a real dataset. The behavior when the underlying distribution is heavy-tailed or skewed, shows the appropriateness of the method when we deviate from the common assumption of normality. The resulting high true positive rates and low false positive rates in the vast majority of cases, as well as the significantly smaller computation time show the advantages of our proposal.

This work presents a nonparametric statistical test, S-maup, to measure the sensitivity of a spatially intensive variable to the effects of the Modifiable Areal Unit Problem (MAUP). To the best of our knowledge, S-maup is the first statistic of its type and focuses on determining how much the distribution of the variable, at its highest level of spatial disaggregation, will change when it is spatially aggregated. Through a computational experiment, we obtain the basis for the design of the statistical test under the null hypothesis of non-sensitivity to MAUP. We performed an exhaustive simulation study for approaching the empirical distribution of the statistical test, obtaining its critical values, and computing its power and size. The results indicate that, in general, both the statistical size and power improve with increasing sample size. Finally, for illustrative purposes, an empirical application is made using the Mincer equation in South Africa, where starting from 206 municipalities, the S-maup statistic is used to find the maximum level of spatial aggregation that avoids the negative consequences of the MAUP.

In economics, insurance and finance, value at risk (VaR) is a widely used measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, time horizon, and probability α $\alpha$ <!--mathContainer--> <!--Loading Mathjax--> , the 100α% $100\alpha \%$ <!--mathContainer--> <!--Loading Mathjax--> VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is α $\alpha$ <!--mathContainer--> <!--Loading Mathjax--> . That is to say, it is a quantile of the distribution of the losses, which has both good analytic properties and easy interpretation as a risk measure. However, its extension to the multivariate framework is not unique because a unique definition of multivariate quantile does not exist. In the current literature, the multivariate quantiles are related to a specific partial order considered in Rⁿ ${\mathbb{R}}^n$ <!--mathContainer--> <!--Loading Mathjax--> , or to a property of the univariate quantile that is desirable to be extended to Rⁿ ${\mathbb{R}}^n$ <!--mathContainer--> <!--Loading Mathjax--> . In this work, we introduce a multivariate value at risk as a vector-valued directional risk measure, based on a directional multivariate quantile, which has recently been introduced in the literature. The directional approach allows the manager to consider external information or risk preferences in her/his analysis. We derive some properties of the risk measure and we compare the univariate VaR over the marginals with the components of the directional multivariate VaR. We also analyze the relationship between some families of copulas, for which it is possible to obtain closed forms of the multivariate VaR that we propose. Finally, comparisons with other alternative multivariate VaR given in the literature, are provided in terms of robustness.

This article presents a novel nonparametric approach to generate synthetic data using copulas, which are functions that explain the dependency structure of the real data. The proposed method addresses several challenges faced by existing synthetic data generation techniques, such as the preservation of complex multivariate structures presented in real data. By using all the information from real data and verifying that the generated synthetic data follows the same behavior as the real data under homogeneity tests, our method is a significant improvement over existing techniques. Our method is easy to implement and interpret, making it a valuable tool for solving class imbalance problems in machine learning models, improving the generalization capabilities of deep learning models, and anonymizing information in finance and healthcare domains, among other applications.

Both cell-wise and case-wise outliers may appear in a real data set at the same time. Few methods have been developed in order to deal with both types of outliers when formulating a regression model. In this work, a robust estimator is proposed based on a three-step method named 3S-regression, which uses the comedian as a highly robust scatter estimate. An intensive simulation study is conducted in order to evaluate the performance of the proposed comedian 3S-regression estimator in the presence of cell-wise and case-wise outliers. In addition, a comparison of this estimator with recently developed robust methods is carried out. The proposed method is also extended to the model with continuous and dummy covariates. Finally, a real data set is analyzed for illustration in order to show potential applications.