If one ranks cities by population, the rank of a city is inversely related to its size, a well-documented phenomenon known as Zipf's Law. Further, the growth rate of a city's population is uncorrelated with its size, another well-known characteristic known as Gibrat's Law. In this paper, I show...

We calculate the probability of returns exceeding a multiple of Expected Shortfall for fat-tailed portfolios. Our results show that, taken in isolation, the Basel 3 FRTB Market Risk capital requirements are insufficient to prevent a large number of insolvencies. The impact of applying a cutoff...

This paper introduces a two-piece generalized distributional framework based on some minimum requirements for the probability density function used as the basis for the two-way split. These requirements are symmetricity, unimodality and continuity. The basic characteristics of the two-piece...

We study the tail distributions of multi-day index returns across a variety of asset classes. Fitting power laws to the tail distributions, we find tail indices in the range [2-4] for all underlyings, for returns up to 250 days. We also find that the power laws can not be statistically ruled out...

We study the uniform convergence rates of a nonparametric estimator for a probability density function and its derivatives when the density has a known pole. Such situation arises in some structural microeconometric models, e.g. in auction, labor, and consumer search. Existing uniform...

Topological data analysis provides a new perspective on many problems in the domain of complex systems. Here, we establish the dependency of the mean value of functional $p$-norms of 'persistence landscapes' on a uniform scaling of the underlying multivariate distribution. Furthermore, we...

The matrix that transforms the response variable in a regression to its predicted value is commonly referred to as the hat matrix. The trace of the hat matrix is a standard metric for calculating degrees of freedom. The two prominent theoretical frameworks for studying hat matrices to calculate...