• It is well-known that estimating extreme quantiles, namely, quantiles lying beyond the range of the available data, is a nontrivial problem that involves the analysis of tail behavior through the estimation of the extreme-value index. For heavy-tailed distributions, on which this paper focuses, the extreme-value index is often called the tail index and extreme quantile estimation typically in...

We introduce a bivariate distribution supported on the first quadrant with exponential, and heavy tailed Mittag–Leffer, marginal distributions. Although this distribution belongs to the class of geometric operator stable laws, it is a rather special case that does not follow their general theory. Our results include the joint density and distribution function, Laplace transform, conditional dis...

Cortical activity involves large populations of neurons, even when it is limited to functionally coherent areas. Electrophysiological recordings, on the other hand, involve comparatively small neural ensembles, even when modern-day techniques are used. Here we review results which have started to fill the gap between these two scales of inquiry, by shedding light on the statistical distribution...

To recover the topology of a manifold in presence heavy tailed or exponentially decaying noise, one must understand behavior geometric complexes whose points lie tail these noise distributions. This study advances this line inquiry, and demonstrates functional strong laws large numbers for Euler characteristic process random formed by outside an expanding ball \(\mathbb {R}^{d}\). When are draw...

This paper describes statistical analysis of web data traffic to identify the probability distribution best fitting the connection arrivals and to evaluate whether data traffic traces follow heavy-tail distributions – an indication of fractal behaviour, which is in contrast to conventional data traffic. Modelling of the fractal nature of web data traffic is used to specify classes of fractal co...

Internet traffic is composed of flows, sets of packets being transferred from one computer to another. Some visualizations for understanding the set of flows at a busy internet link are developed. These show graphically that the set of flows is dominated by a relatively few “elephants”, and a very large number of “mice”. It also becomes clear that “representative sampling” from heavy tail distr...

Stable distributions have heavy tails that are asymptotically Paretian. Accurate computations of stable densities and distribution functions are used to analyze when the Paretian tail actually appears. Implications for estimation procedures are discussed. In addition to numerically locating the mode of a general stable distribution, analytic and numeric results are given for the mode. Extensive...

− This paper is dedicated to the estimation of extreme quantiles and the tail index from heavy-tailed distributions when a covariate is recorded simultaneously with the quantity of interest. A nearest neighbor approach is used to construct our estimators. Their asymptotic normality is established under mild regularity conditions and their finite sample properties are illustrated on a simulation...

Crash data can often be characterized by over-dispersion, heavy (long) tail and many observations with the value zero. Over the last few years, a small number of researchers have started developing and applying novel and innovative multi-parameter models to analyze such data. These multi-parameter models have been proposed for overcoming the limitations of the traditional negative binomial (NB)...