F. Marta L. Di Lascio

Truncation invariant copulas and a testing procedure

ABSTRACT The class of bivariate copulas that are invariant under truncation with respect to one variable is considered. A simulation algorithm for the members of the class and a novel construction method are presented. Moreover, inspired by a stochastic interpretation of the members of such a class, a procedure is suggested to check whether the dependence structure of a given data set is truncation invariant. The overall performance of the procedure has been illustrated on both simulated and real data.

Copula–based clustering methods

We review some recent clustering methods based on copulas. Specifically, in the dissimilarity–based clustering framework, we describe and compare methods based on concordance or tail-dependence concept. An illustration is hence provided by using a time series dataset formed by the constituent data of the S&P 500 observed during the financial crisis of 2007-2008. Next, in the likelihood–based clustering framework, we present and discuss a clustering algorithm based on copula and called CoClust. Here, an application to the gene expression profiles of human tumour cell lines is provided to describe the methodology. Finally, a comparison between the two different approaches is performed through a case study on environmental data.

A copula-based clustering algorithm to analyse EU country diets

The aim of the paper is to suggest a novel clustering technique to explore the changes of the food diet in 40 European countries in accordance with common European policies and guidelines on healthy diets and lifestyles. The proposed clustering algorithm is based on copulas and it is called CoClust. The CoClust algorithm is able to find clusters according to the multivariate dependence structure of the data generating process. The database analysed contains information on the proportions of calories from 16 food aggregates in 40 European countries observed over 40 years by the Food and Agriculture Organization of the United Nations (FAO). The findings suggest that European country diets are changing, individually or as a group, but not in a unique direction. Central and Eastern European countries are becoming unhealthier, while the tendency followed by the majority of the remaining countries is to integrate the common European guidelines on healthy, balanced, and diversified diets in their national policies.

Percutaneous real-time sonoelastography as a non-invasive tool for the characterization of solid focal liver lesions: A prospective study

BACKGROUND Real-time sonoelastography is currently used for the characterization of superficial solid lesions such as thyroid and breast masses. This study evaluates the usefulness of percutaneous sonoelastography for the characterization of solid focal liver lesions. METHODS 30 out of 43 patients with 38 known liver lesions were included in a prospective, diagnostic study. Qualitative analysis (pattern of deformation, elasticity type of liver tumour) and semi-quantitative measurements (strain ratio, hardness percentage, histogram) were evaluated. Sensitivity, specificity, positive and negative predictive values were calculated and the area under the receiver operating characteristics curve was constructed. RESULTS Patterns A and C-D are specific of benign lesions and metastases respectively. The patterns for haemangiomas, focal nodular hyperplasia and metastases were significantly different to each other in terms of strain ratio, hardness percentage and histogram (p<0.05). A statistically significant difference (p<0.001) was observed between the median values of the 3 measured parameters for benign (1.02; 12%; 47) and malignant lesions (1.66; 65%; 20.5) respectively. The area under the receiver operating characteristics curve values for strain ratio, hardness percentage and histogram were 0.88, 0.89, and 0.86 respectively for cut-off values of 1.2, 45, and 30. CONCLUSIONS By percutaneous sonoelastography it is possible to differentiate benign versus malignant focal liver lesions, metastases in particular, with good diagnostic performance.

A Clustering Approach and a Rule of Thumb for Risk Aggregation

The problem of establishing reliable estimates or bounds for the (T)VaR of a joint risk portfolio is a relevant subject in connection with the computation of total economic capital in the Basel regulatory framework for the finance sector as well as with the Solvency regulations for the insurance sector. In the computation of total economic capital, a financial institution faces a considerable amount of model uncertainty related to the estimation of the interdependence amongst the marginal risks. In this paper, we propose to apply a clustering procedure in order to partition a risk portfolio into independent subgroups of positively dependent risks. Based on available data, the portfolio partition so obtained can be statistically validated and allows for a reduction of capital and the corresponding model uncertainty. We illustrate the proposed methodology in a simulation study and two case studies considering an Operational and a Market Risk portfolio. A rule of thumb stems from the various examples proposed: in a mathematical model where the risk portfolio is split into independent subsets with comonotonic dependence within, the smallest VaR-based capital estimate (at the high regulatory probability levels typically used) is produced by assuming that the infinite-mean risks are comonotonic and the finite-mean risks are independent. The largest VaR estimate is instead generated by obtaining the maximum number of independent infinite-mean sums.

A Test for Truncation Invariant Dependence

A test is proposed to check whether a random sample comes from a truncation invariant copula C, that is, if C is the copula of a pair (U, V) of random variables uniformly distributed on [0, 1], then C is also the copula of the conditional distribution function of \((U,V\mid U\le \alpha )\) for every \(\alpha \in (0,1]\). The asymptotic normality of the test statistics is shown. Moreover, a procedure is described to simplify the approximation of the asymptotic variance of the test. Its performance is investigated in a simulation study.