Band 23, Heft 2

We give an explicit error bound between the invariant density of an elliptic reflected diffusion in a smooth compact domain and the kernel estimator built on the symmetric Euler scheme introduced in [3].

Generating pseudorandom numbers is a prerequisite for many areas including Monte Carlo simulation and randomized algorithms. The performance of pseudorandom number generators (PRNGs) depends on the quality of the generated random sequences. They must be generated quickly and have good statistical properties. Several statistical test suites have been developed to evaluate a single stream of random numbers such as those from the TestU01 library, the DIEHARD test suite, the tests from the SPRNG package, and a set of tests designed to evaluate bit sequences developed at NIST. This paper presents a new pseudorandom number generation scheme that produces pseudorandom sequences with good statistical properties via a scrambling procedure motivated by cryptographic transformations. We will specifically apply this to a popular set of PRNGs called the Linear Congruential generators (LGCs). The scrambling technique is based on a simplified version of a Feistel network. The proposed method seeks to improve the quality of the LCGs output stream. We show that this Feistel-inspired scrambling technique breaks up the regularities that are known to exist in LCGs. The Feistel-inspired scrambling technique is modular, and can be applied to any 64-bit PRNG, and so we believe that it can serve as an inexpensive model for a scrambler that can be used with most PRNGs via post-processing.

A probabilistic collocation based polynomial chaos expansion method is developed for simulation of particle transport in porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure. The flow is modeled in a two-dimensional domain with mixed Dirichlet–Neumann boundary conditions. The relevant Karhunen–Loève expansion is constructed by a special randomized singular value decomposition (SVD) of the correlation matrix which makes possible to treat problems of high dimension. The simulation results are compared against a direct Monte Carlo calculation of different Eulerian and Lagrangian statistical characteristics of the solutions. As a byproduct, we suggest an approach to solve an inverse problem of recovering the variance of the log-conductivity.

Count outcomes are often modelled using the Poisson regression. However, this model imposes a strict mean-variance relationship that is unappealing in many contexts. Several studies in the life sciences result in count outcomes with excessive amounts of zeros. The presence of the excess zeros introduces extra dispersion in the data which cannot be accounted for by the traditional Poisson regression. The zero-inflated Poisson (ZIP) and zero-inflated negative binomial models are popular alternative. The zero-inflated models comprise two key components; a logistic part which models the zeros, and a Poisson component to handle the positive counts. Both components allow the inclusion of covariates. Civettini and Hines [3] investigated misspecification effects in the zero-inflated negative binomial regression models. Long,Preisser, Herring and Golin [10] proposed a so-called marginalized zero-inflated Poisson (MZIP) model that allows direct marginal interpretation for fixed effect estimates to overcome the often sub-population specific interpretation of the traditional zero-inflated models. In this research, the effects of misspecification of components of the MZIP regression model are investigated through a comprehensive simulation study. Two different incorrect specifications of the components of an MZIP model were considered, namely ‘Omission’ and ‘Misspecification’. Bias, standard error (precision) of estimates and mean square error (MSE) are computed while varying the sample size. Type I error rates are also evaluated for the misspecified models. Results of a Monte Carlo simulation are reported. It was observed that omissions in both parts of the models lead to biases in the estimated parameters. The intercept parameters were the most severely affected. Furthermore, in all the types of omissions, parameters in the zero-inflated part of the models were much affected compared to the Poisson part in terms of both bias and MSE. Generally, bias and MSE decrease as sample sizes increase for all parameters. It was also found that misspecification can either increase, preserve or decrease the type I error rates depending on the sample size.

A Monte Carlo method for solving the multi-dimensional optimal stopping problem is considered. Consistent estimators for a general jump-diffusion are pointed out. It is shown that the variance of estimators is inverse proportional to the number of points in each layer of the mesh.

Bivariate counts are collected in many sectors of research but the analysis of such data is often challenging because each series of counts may exhibit different levels and types of dispersion. This paper addresses this problem by proposing a flexible bivariate COM-Poisson model that may handle any combination of over-, equi- and under-dispersion at any levels. In this paper, the bivariate COM-Poisson is developed via Archimedean copulas. The Generalized Quasi-Likelihood (GQL) approach is used to estimate the unknown mean parameters in the copula-based bivariate COM-Poisson model while the dependence parameter is estimated using the copula likelihood. We further introduce a Monte Carlo experiment to generate bivariate COM-Poisson data under different dispersion levels. The performance of the GQL approach is assessed on the simulated data. The model is applied to analyze real-life epileptic seizures data.