A non-linear congruential pseudo random number generator

A non-linear congruential pseudo random number generator is introduced. This generator does not have the lattice structure in the distribution of tuples of consecutive pseudo random numbers which appears in the case of linear congruential generators. A theorem on the period length of sequences produced by this type of generators is proved. This theorem justifies an algorithm to determine the period length. Finally a simulation problem is described where a linear congruential generator produces completely useless results whereas good results are obtained if a non-linear congruential generator of about the same period length is applied.     

Random number generators for the generalized Birnbaum–Saunders distribution

The generalized Birnbaum–Saunders distribution pertains to a class of lifetime models including both lighter and heavier tailed distributions. This model adapts well to lifetime data, even when outliers exist, and has other good theoretical properties and application perspectives. However, statistical inference tools may not exist in closed form for this model. Hence, simulation and numerical studies are needed, which require a random number generator. Three different ways to generate observations from this model are considered here. These generators are compared by utilizing a goodness-of-fit procedure as well as their effectiveness in predicting the true parameter values by using Monte Carlo simulations. This goodness-of-fit procedure may also be used as an estimation method. The quality of this estimation method is studied here. Finally, through a real data set, the generalized and classical Birnbaum–Saunders models are compared by using this estimation method.     

Correlation and variable importance in random forests

This paper is about variable selection with the random forests algorithm in presence of correlated predictors. In high-dimensional regression or classification frameworks, variable selection is a difficult task, that becomes even more challenging in the presence of highly correlated predictors. Firstly we provide a theoretical study of the permutation importance measure for an additive regression model. This allows us to describe how the correlation between predictors impacts the permutation importance. Our results motivate the use of the recursive feature elimination (RFE) algorithm for variable selection in this context. This algorithm recursively eliminates the variables using permutation importance measure as a ranking criterion. Next various simulation experiments illustrate the efficiency of the RFE algorithm for selecting a small number of variables together with a good prediction error. Finally, this selection algorithm is tested on the Landsat Satellite data from the UCI Machine Learning Repository.     

A Comparison of Two Random Number Generators for the Birnbaum–Saunders Distribution

Abstract

The Birnbaum–Saunders distribution was developed to describe fatigue failure lifetimes, however, the distribution has been shown to be applicable for a variety of situations that frequently occur in the engineering sciences. In general, the distribution can be used for situations that involve stochastic wear–out failure. The distribution does not have an exponential family structure, and it is often necessary to use simulation methods to study the properties of statistical inference procedures for this distribution. Two random number generators for the Birnbaum–Saunders distribution have appeared in the literature. The purpose of this article is to present and compare these two random number generators to determine which is more efficient. It is shown that one of these generators is a special case of the other and is simpler and more efficient to use.     

Fast Random Number Generation Using 128-Bit Multimedia Extension Registers on Pentium Class Machines

In this article, it is shown how 128-bit SSE2 multimedia extension registers, present in Pentium IV class 32-bit processors, may be used to generate random numbers at several times greater speed then when regular general purpose registers are used. In particular, a 128-bit algorithm is presented for the Marsaglia MWC1616 generator from the DIEHARD battery of random number generator tests, and its performance is compared to that of the conventional approach.     

Dependent radius marks of Laguerre tessellations: a case study

We study a particular marked three-dimensional point process sample that represents a Laguerre tessellation. It comes from a polycrystalline sample of aluminium alloy material. The ‘points’ are the cell generators while the ‘marks’ are radius marks that control the size and shape of the tessellation cells. Our statistical mark correlation analyses show that the marks of the sample are in clear and plausible spatial correlation: the marks of generators close together tend to be small and similar and the form of the correlation functions does not justify geostatistical marking. We show that a simplified modelling of tessellations by Laguerre tessellations with independent radius marks may lead to wrong results. When we started from the aluminium alloy data and generated random marks by random permutation we obtained tessellations with characteristics quite different from the original ones. We observed similar behaviour for simulated Laguerre tessellations. This fact, which seems to be natural for the given data type, makes fitting of models to empirical Laguerre tessellations quite difficult: the generator points and radius marks have to be modelled simultaneously. This may imply that the reconstruction methods are more efficient than point-process modelling if only samples of similar Laguerre tessellations are needed. We also found that literature recipes for bandwidth choice for estimating correlation functions should be used with care.     

Serial unblocked designs balanced for correlation between neighbours,II

The theory in Part I contained an error that was inferred from the output of a program, written in SAS by Eric P. Smith and David D. Morris. The program produces random BUS designs in accordance with the algorithm of Part I. The theory is here corrected by using a combinatorial argument that involves elementary number theory. The algorithm needs no change but its interpretation is now adjusted.     

Variance reduction of estimators arising from Metropolis–Hastings algorithms

The Metropolis–Hastings algorithm is one of the most basic and well-studied Markov chain Monte Carlo methods. It generates a Markov chain which has as limit distribution the target distribution by simulating observations from a different proposal distribution. A proposed value is accepted with some particular probability otherwise the previous value is repeated. As a consequence, the accepted values are repeated a positive number of times and thus any resulting ergodic mean is, in fact, a weighted average. It turns out that this weighted average is an importance sampling-type estimator with random weights. By the standard theory of importance sampling, replacement of these random weights by their (conditional) expectations leads to more efficient estimators. In this paper we study the estimator arising by replacing the random weights with certain estimators of their conditional expectations. We illustrate by simulations that it is often more efficient than the original estimator while in the case of the independence Metropolis–Hastings and for distributions with finite support we formally prove that it is even better than the “optimal” importance sampling estimator.     

Adaptive rejection sampling with fixed number of nodes

The adaptive rejection sampling (ARS) algorithm is a universal random generator for drawing samples efficiently from a univariate log-concave target probability density function (pdf). ARS generates independent samples from the target via rejection sampling with high acceptance rates. Indeed, ARS yields a sequence of proposal functions that converge toward the target pdf, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computational demanding each time it is updated. In this work, we propose a novel ARS scheme, called Cheap Adaptive Rejection Sampling (CARS), where the computational effort for drawing from the proposal remains constant, decided in advance by the user. For generating a large number of desired samples, CARS is faster than ARS.     

Matrix generators for pseudo-random vector generation

In this paper a method for the direct generation of pseudo-random vectors is considered. Thereby the n-th pseudo-random vector is recursively generated from the (n?1)-th pseudo-random vector by multiplication with a matrix. The period lengths of this generator type are examined and characterized. Furthermore it is shown that some popular pseudo-random number generators can be regarded as special cases of this method.     

A note on a universal random variate generator for integer-valued random variables

A universal generator for integer-valued square-integrable random variables is introduced. The generator relies on a rejection technique based on a generalization of the inversion formula for integer-valued random variables. This approach allows to create a dominating probability function, whose evaluation solely involves two integrals depending on the characteristic function of the random variable to be generated. The proposal gives rise to a simple algorithm which may be implemented in a few code lines and which may show good performance when the classical families of distributions—such as the Poisson and the Binomial—are considered. In addition, applications to the Poisson-Tweedie and the Luria-Delbrück distributions are provided.     

On the Deng-Lin random number generators and related methods

We study the structure and point out weaknesses of recently proposed random number generators based on special types of linear recurrences with small coefficients, which allow fast implementations. Our theoretical analysis is complemented by the results of simple empirical statistical tests that the generators fail decisively. Directions for improvement and alternative generators are also pointed out.     

Implementation of fortran random number generators on computers with one's complement arithmetic †

Random number generators can be programmed in FORTRAN using several methods on 1's complement machines. We have presented several methods for the SRU 1107–1108 and CDC 6000 and CYBER series, the two most popular that utilize 1's complement arithmetic. We believe the logical mask method to be the most efficient technique based on execution time on both machines. This method requires only a little more than one-half the time of the signed residue method on the SRU 1107–1108 and a little more than two thirds the time of the unsigned residue method on the CDC 6000–CYBER. A fast composite generator in FORTRAN uses the logical mask merhod combined with the FLD function to implement the table approach of MacLaren and Marsaglia (1965). This generator proves to be more than twice as fast as a previous one for the SRU 1108, Marsaglia and Bray (1968)     

Fitting asymmetric bimodal data with selected distributions

ABSTRACT

This article discusses two asymmetrization methods, Azzalini's representation and beta generation, to generate asymmetric bimodal models including two novel beta-generated models. The practical utility of these models is assessed with nine data sets from different fields of applied sciences. Besides this tutorial assessment, some methodological contributions are made: a random number generator for the asymmetric Rathie–Swamee model is developed (generators for the other models are already known and briefly described) and a new likelihood ratio test of unimodality is compared via simulations with other available tests. Several tools have been used to quantify and test for bimodality and assess goodness of fit including Bayesian information criterion, measures of agreement with the empirical distribution and the Kolmogorov–Smirnoff test. In the nine case studies, the results favoured models derived from Azzalini's asymmetrization, but no single model provided a best fit across the applications considered. In only two cases the normal mixture was selected as best model. Parameter estimation has been done by likelihood maximization. Numerical optimization must be performed with care since local optima are often present. We concluded that the models considered are flexible enough to fit different bimodal shapes and that the tools studied should be used with care and attention to detail.     

Sequential Monte Carlo for counting vertex covers in general graphs

In this paper we describe a sequential importance sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting SIS algorithm shows polynomial complexity for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show that the resulting SIS algorithm enjoys excellent practical performance in comparison with existing methods. In particular the method is compared with cachet—an exact model counter, and the state of the art SampleSearch, which is based on Belief Networks and importance sampling.     

A New Test of Randomness for Lehmer Generators Based on the Manhattan Distance Between Pairs of Consecutive Random Numbers

This article considers the Marsaglia effect by proposing a new test of randomness for Lehmer random number generators. Our test is based on the Manhattan distance criterion between consecutive pairs of random numbers rather than the usually adopted Euclidian distance. We derive the theoretical distribution functions for the Manhattan distance for both overlapping (two dimensional) as well as non-overlapping cases. Extensive goodness-of-fit testing as well as empirical experimentation provides ample proof of the merits of the proposed criterion.     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.     

Simulation of Constrained Variables in Engineering Risk Analyses

The problem of sampling random variables with overlapping pdfs subject to inequality constraints is addressed. Often, the values of physical variables in an engineering model are interrelated. This mutual dependence imposes inequality constraints on the random variables representing these parameters. Ignoring the interdependencies and sampling the variables independently can lead to inconsistency/bias. We propose an algorithm to generate samples of constrained random variables that are characterized by typical continuous probability distributions and are subject to different kinds of inequality constraints. The sampling procedure is illustrated for various representative cases and one realistic application to simulation of structural natural frequencies.     

A moment-matching Ferguson & Klass algorithm

Completely random measures (CRM) represent the key building block of a wide variety of popular stochastic models and play a pivotal role in modern Bayesian Nonparametrics. The popular Ferguson & Klass representation of CRMs as a random series with decreasing jumps can immediately be turned into an algorithm for sampling realizations of CRMs or more elaborate models involving transformed CRMs. However, concrete implementation requires to truncate the random series at some threshold resulting in an approximation error. The goal of this paper is to quantify the quality of the approximation by a moment-matching criterion, which consists in evaluating a measure of discrepancy between actual moments and moments based on the simulation output. Seen as a function of the truncation level, the methodology can be used to determine the truncation level needed to reach a certain level of precision. The resulting moment-matching Ferguson & Klass algorithm is then implemented and illustrated on several popular Bayesian nonparametric models.