Mini Symposium on Perspective Research Directions in Complex Stochastic Structures

Budapest, March 19, 2012

Contact and venue

Expected time span: 10:00 - 15:00

Kjersti Aas, Norwegian Computing Center, Oslo

Title: Pair-copula constructions - even more flexible than copulas
Abstract: A copula is a multivariate distribution with standard uniform marginal distributions. While the literature on copulas is substantial, most of the research is still limited to the bivariate case. However, recently hierarchical copula-based structures have been proposed as an alternative to the standard copula methodology. One of the most promising of these structures is the pair-copula construction (PCC). Pair-copula constructions are also called regular vines (R-vines). The modeling scheme is based on a decomposition of a multivariate density into a cascade of pair copulae, applied on original variables and on their conditional and unconditional distribution functions. Each pair copula can be chosen arbitrarily and the full model exhibit complex dependence patterns such as asymmetry and tail dependence. In this talk I will give an introduction to pair-copula constructions and apply the methodology to a 19- dimensional financial data set.

Ágnes Backhausz, Eötvös Loránd University, Budapest

Title: A random graph model based on interactions of three vertices.
Abstract: (in PDF) Random graphs evolving by some “preferential attachment” rule are inevitable in modelling real-world networks [1]. There is a vast number of publications inventing and studying different models of that kind, but in most of them the dynamics is only driven by vertex-vertex interactions. However, one can easily find networks (that is, objects equipped with links) in economy or other areas where simultaneous interactions can take place among three or even more vertices, and those interactions determine the evolution of the process.
We consider a random graph model evolving in discrete time-steps that is based on 3-interactions among vertices. Triangles, edges and vertices have different weights; objects with larger weight are more likely to participate in future interactions and to increase their weights. Thus it is also a “preferential attachment” model.
We prove the scale free property of the model, that is, the ratio of weights of weight w tends to xw almost surely as the number of vertices goes to infinity, where xw is polinomially decreasing as w \to \infty. Techniques of discrete parameter martingales are applied in the proofs.
The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TAMOP 4.2.1./B-09/KMR-2010-0003.

Alexander Bulinski, Lomonosov Moscow State University

Title: Statistical methods for multidimensional genetic data analysis
Abstract: We concentrate on the new modifications of multifactor dimensionality reduction, logic regression, random forests and stochastic gradient boosting. Theorems justifying application of these methods are established. The roles of certain combinations of single nucleotide polymorphisms and non-genetic risk factors are examined to assess the risks of complex diseases such as cardiovascular ones.

Probal Chaudhuri, Indian Statistical Institute, Calcutta

Title: Comparison of Multivariate Distributions Using Quantile-Quantile Plots and Related Tests
Abstract: The univariate quantile-quantile (Q-Q) plot is a well-known graphical tool for examining whether two data sets are generated from the same distribution or not. It is also used to determine how well a specified probability distribution fits a given sample. We develop and study a multivariate version of Q-Q plot based on spatial quantiles. The usefulness of the proposed graphical device is illustrated on different real and simulated data, some of which have fairly large dimensions. We also develop certain statistical tests that are related to the proposed multivariate Q-Q plots and study their asymptotic properties. The performance of those tests are compared with that of some other well-known tests for multivariate distributions available in the literature. This is joint work with Subhra S. Dhar (Cambridge University) and Biman Chakraborty (University of Birmingham).

László Martinek, Eötvös Loránd University, Budapest

Title: Estimation of Claim Numbers in Automobile Insurance
Abstract: The use of bonus-malus systems in compulsory liability automobile insurance is a worldwide applied method for premium pricing. If certain assumptions hold, like the conditional Poisson distribution of the policyholders claim number, then an interesting task is to evaluate the so called claims frequency of the individuals. Here we introduce 3 techniques, two is based on the bonus-malus class, and the third based on claims history. The article is devoted to choose the method, which fits to the frequency parameters the best for certain input parameters. For measuring the goodness-of-fit we will use scores, similar to better known divergence measures. The detailed method is also suitable to compare bonus-malus systems in the sense that how much information they contain about drivers.
The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP 4.2.1/B-09/KMR-2010-0003.

Axel Munk, University of Göttingen


Fabrizio Ruggeri, Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche, Milano

Title: Bayesian Estimation in Stochastic Predator-Prey Models
Abstract: Parameter estimation for the functional response of predator-prey systems is a critical methodological problem in population ecology. Population dynamics is described by a system of stochastic differential equations in which behavioural stochasticities are represented by noise terms affecting each population as well as their interaction. In our work we are interested in methods for model parameter estimation based on time series of field data.

Michael Sorensen, University of Copenhagen, Department of Mathematical Sciences

Title: Martingale estimating functions for diffusions with jumps
Abstract: For discrete time observations of a stochastic differential equation driven not only by a Wiener process, but also by a Lévy process with jumps, the likelihood function is not explicitly known and extremely difficult to calculate numerically. Therefore alternatives like estimating functions are even more useful for jump diffusions than for classical diffusions. We present a broad and flexible class of diffusions with jumps for which explicit optimal martingale estimating functions are available. Generalizing earlier work with Mathieu Kessler on ordinary diffusion processes, these estimating functions are based on eigenfunctions of the generator. A simple condition on the compensator of the jump measure (the Levy measure) is shown to ensure that explicit optimal martingale estimating functions can be found. We illustrate the general theory by concrete examples.

Charles Taylor, University of Leeds

Title: Regression models on the torus
Abstract: Regression with a circular response is a topic of current interest. Firstly, we consider a nonparametric estimator, in which simple adaptations of a weight function enable a unified formulation for both real-line and circular predictors. Secondly, we consider various parametric models for the circular-circular case, in which we propose robust fitting procedures, as well as potential diagnostic tools to identify influential observations. Real data examples are used to illustrate the methods.

Rasmus Plenge Waagepetersen, University of Aalborg

Title: Optimal estimation of the intensity function of a spatial point process
Abstract: Maximum likelihood estimation for Cox and cluster point processes can be cumbersome in practice due to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation and reduces to the likelihood score in case of a Poisson process. The Fredholm equation must be solved numerically and it turns out that the approximated optimal estimating function is equivalent to a quasi-likelihood for binary spatial data.

The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP 4.2.1/B-09/KMR-2010-0003.