Sull'approccio dei cumulanti multivariati per la stima di parametri di processi subordinati

The purpose of a statistical model is to guide analysts of every scientific field through the journey of understanding raw data and to help decision makers in their role of deduction of practical information from this data analysis. In such a framework, the issue of estimating models and the need for fast efficient methods for doing so have become increasingly relevant. This thesis explores a new approach to estimate the parameters of a multivariate statistical model by using its cumulants, which are, in turn, approximated by k-statistics. A number of multivariate Lévy processes and their cumulants will be resented, together with a brief overview of their application in finance. The estimation approach is then tested for a specific distribution, the bivariate Normal Inverse Gaussian (NIG), with simulated and real-life data. The R package kStatistics is used to obtain unbiased estimators for the cumulants, i.e. to calculate the k-statistics. The cumulants of a probability distribution are a sequence of quantities analogous to the moments and equally useful for the characterization of the distribution. For example these measures can determine mean, variance, skewness and kurtosis. Compared with the moments, cumulants have more interesting properties, such as the additivity and the scale-invariance, that make them more flexible to use. They can be estimated without bias through k-statistics and from this estimation the parameters of the underlying model can be retrieved. A classical tool to recover k-statistics relies on augmented symmetric function in the data points. This method is quite involved and, thus, time consuming if done directly, in particular in the multivariate case. Nevertheless, some combinatorics algorithms are available that lead to a faster computation of k-statistics. Those methods were developed into a package, which is available in Maple and in R. Having such a package at hand, one could clearly try to fit a statistical model using the method of cumulants. The final procedure is analogous to the method of moments: the theoretical cumulants are calculated as a function of the parameters and then they are equated to the k-statistics retrieved from the data to get an estimation of the parameters. We will focus our analysis on multivariate subordinated Lévy processes not only because of the many financial applications but also because their cumulants can be written in a closed form as functions of the parameters. The aim of this thesis is to compare the method of cumulants with the most popular likelihood estimation, since it is well-known that the latter method slows down dramatically in higher dimension. Therefore we will pit the methods against each other and asses which of the two performs better in a multivariate scenario.