Certain bivariate distributions and random processes connected with maxima and minima

It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where alpha> 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions

Simulating High-Dimensional Multivariate Data using the bigsimr R Package

It is critical to accurately simulate data when employing Monte Carlo techniques and evaluating statistical methodology. Measurements are often correlated and high dimensional in this era of big data, such as data obtained in high-throughput biomedical experiments. Due to the computational complexity and a lack of user-friendly software available to simulate these massive multivariate constructions, researchers resort to simulation designs that posit independence or perform arbitrary data transformations. To close this gap, we developed the Bigsimr Julia package with R and Python interfaces. This paper focuses on the R interface. These packages empower high-dimensional random vector simulation with arbitrary marginal distributions and dependency via a Pearson, Spearman, or Kendall correlation matrix. bigsimr contains high-performance features, including multi-core and graphical-processing-unit-accelerated algorithms to estimate correlation and compute the nearest correlation matrix. Monte Carlo studies quantify the accuracy and scalability of our approach, up to $d=10,000$. We describe example workflows and apply to a high-dimensional data set -- RNA-sequencing data obtained from breast cancer tumor samples.Comment: 22 pages, 10 figures, https://cran.r-project.org/web/packages/bigsimr/index.htm

A note on self-decomposability of stable process subordinated to self-decomposable subordinator

We provide an example that shows that there exists a stable Lévy motion and self-decomposable subordinator , such that the corresponding subordinated process is not self-decomposable.Geometric stable law Linnik distribution Subordination Unimodality

Mixture representation of Linnik distribution revisited

Let Y[alpha] have a Linnik distribution, given by the characteristic function [psi](t) = (1 + t [alpha])-1. We extend the result of Kotz and Ostrovskii (1996) and show that Y[alpha] admits two different representations, where 0Geometric stable law Heavy tailed distribution Mittag-Leffler distribution Mixture Random summation Simulation Stable law

A note on self-decomposability of stable process subordinated to self-decomposable subordinator

We provide an example that shows that there exists a stable Lévy motion and self-decomposable subordinator , such that the corresponding subordinated process is not self-decomposable.Geometric stable law Linnik distribution Subordination Unimodality

A generalized Sibuya distribution

The Sibuya distribution arises as the distribution of the waiting time for the first success in Bernoulli trials, where the probabilities of success are inversely proportional to the number of a trial. We study a generalization that is obtained as the distribution of the excess random variable N − k given N > k, where N has the Sibuya distribution. We summarize basic facts regarding this distribution and provide several new results and characterizations, shedding more light on its origin and possible applications. In particular, we emphasize the role Sibuya distribution plays in the extreme value theory and point out its invariance property with respect to random thinning operation

Gaussian Mixture Representation of the Laplace Distribution Revisited : Bibliographical Connections and Extensions

We provide bibliographical connections and extensions of several representations of the classical Laplace distribution, discussed recently in the study of Ding and Blitzstein. Beyond presenting relation to some previous results, we also include their skew as well as multivariate versions. In particular, the distribution of det Z, where Z is an n × n matrix of iid standard normal components, is obtained for an arbitrary integer n. While the latter is a scale mixture of Gaussian distributions, the Laplace distribution is obtained only in the case n = 2. Supplementary materials for this article are available online

Esscher-transformed laplace distribution revisited

We show that the family of Esscher-transformed Laplace distributions is a subclass of asymmetric Laplace laws