Heavy-tailed Phenomena and Tail Index Inference

This thesis focuses on the analysis of heavy-tailed distributions, which are widely applied to model phenomena in many disciplines. The definition of heavy tails based on the theory of regular variation highlights the importance of the tail index, which indicates the existence of moments and characterises the rate at which the tail decays. Two new approaches to make inference for the tail index are proposed. The first approach employs a regression technique and constructs an estimator of the tail index. It exploits the fact that the behaviour of the characteristic function near the origin reflects the behaviour of the distribution function at infinity. The main advantage of this approach is that it utilises all observations to constitute each point in the regression, not just extreme values. Moreover, the approach does not rely on prior information on the starting point of the tail behaviour of the underlying distribution and shows excellent performance in a wide range of cases: Pareto distributions, heavy-tailed distributions with a non-constant slowly varying factor, and composite distributions with heavy tails. The second approach is motivated by the asymptotic properties of a special moment statistic, the so-called partition function. This statistic considers blocks of data and is generally used in the context of multifractality. Due to the interplay between the weak law of large numbers and the generalised central limit theorem, the asymptotic behaviour of the partition function is strongly affected by the existence of moments even for weakly dependent samples. Via a quantity, the scaling function, a graphical method to identify the existence of heavy tails is proposed. Moreover, the plot of the scaling function allows one to make inference for the underlying distribution: with infinite variance, finite variance with tail index larger than two, or all moments finite. Furthermore, since the tail index is reflected at the breakpoint of the plot of the scaling function, this gives the possibility to estimate the tail index. Both these two approaches use the entire distribution, not just the tail, to analyse the tail behaviour. This sheds a new light on the analysis of heavy-tailed distributions. At the end of this thesis, these two approaches are used to detect power laws in empirical data sets from a variety of fields and contribute to the debate on whether city sizes are better approximated by a power law or a log-normal distribution.