We introduce a scale invariance property for Poisson point processes and use this property to define a series representation for a correlated bivariate gamma process. This approach is quite general and can be used to define other types of multidimensional Lévy processes with given marginals. Some important special cases are bivariate G-processes, bivariate variance gamma processes and multivari...

We propose a new method for estimating intrinsic dimension of a dataset derived by applying the principle of maximum likelihood to the distances between close neighbors. We derive the estimator by a Poisson process approximation, assess its bias and variance theoretically and by simulations, and apply it to a number of simulated and real datasets. We also show it has the best overall performanc...

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, non-asymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galto...

In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven by infinite variance α-stable Lévy motion. We show that the solution is regularly varying with index α. An important step in the proof is the study of a Poisson number of products of independent random variables with regularly varying tail. The study of these products dese...

A considerable number of equivalent formulas defining conditional value-at-risk and expected shortfall are gathered together. Then we present a simple method to bound the conditional value-at-risk of compound Poisson loss distributions under incomplete information about its severity distribution, which is assumed to have a known finite range, mean, and variance. This important class of nonnorma...

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families which include, for example, Poisson regression, binomial regression, and gamma regressi...

Several models and/or several variable combinations could be used to predict the diameter distribution of a stand. Typically, a fi xed model and a fi xed variable combination is used in all conditions. The calibration procedure, however, makes it possible to choose the measurement combination from among many possibilities, although the model used is fi xed. In this study, the usefulness of util...

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance funct...

Statistical process control (SPC) is a significant method to monitor processes and ensure quality. Control charts are the most important tools in SPC. As production parts become more complex, there need design using complex distributions. One of number nonconformities C-chart, which uses Poisson distribution as quality characteristic distribution. However, fit count data, equality mean variance...

In this study, we derive algorithms for estimating mixed β-divergences. Such cost functions are useful for Nonnegative Matrix and Tensor Factorization models with a compound Poisson observation model. Compound Poisson is a particular Tweedie model, an important special case of exponential dispersion models characterized by the fact that the variance is proportional to a power function of the me...