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.     

Quasi-random numbers for copula models

The present work addresses the question how sampling algorithms for commonly applied copula models can be adapted to account for quasi-random numbers. Besides sampling methods such as the conditional distribution method (based on a one-to-one transformation), it is also shown that typically faster sampling methods (based on stochastic representations) can be used to improve upon classical Monte Carlo methods when pseudo-random number generators are replaced by quasi-random number generators. This opens the door to quasi-random numbers for models well beyond independent margins or the multivariate normal distribution. Detailed examples (in the context of finance and insurance), illustrations and simulations are given and software has been developed and provided in the R packages copula and qrng.     

Random number generators for microcomputers

This paper presents the results of comparative tests of several random number generators to generate uniform [0,1] pseudo-random numbers. Tests were run on two different kinds of microcomputers with similar results. The computers used were a Tandy Model 4D (an 8 bit machine) and a Zenith model 150 (a 16 bit IBM compatible machine operating under MS DOS 2.11 and 3.1) withouta math coprocessor chip. All programming was done in FORTRAN using the Microsoft FORTRAN compilers available for the two machines (two MS DOS compilers were used; Microsoft's FORTRAN 77, versions 3.2 and 4.0).     

Statistical simulations on the BBC microcomputer: significance tests of the pseudo-random number generator

Statistical tests of significance are carried out on the feedback shift register pseudo-random number generator employed on the BBC microcomputer. The tests are based on the practicalities of using a microcomputer in simulations for statistical education. The results indicate that the generator is not universally acceptable in this role.     

Cross-correlation between simultaneously generated sequences of pseudo-random uniform deviates

Systematic patterns are revealed when sequences of pseudo-random uniform deviates are generated from multiplicative congruential generators. If the initial seeds, x ₀ and y ₀, for two such sequences are related by y ₀ = (n ₁/n ₂)x ₀, where n ₁ and n ₂ are relatively prime, positive integers, then an approximate argument suggests that the asymptotic correlation coefficient between corresponding members of the two sequences is (n ₁ n ₂)^–1. This unsettling phenomenon is discussed in the context of related, existing literature.     

Some criteria for projection pursuit

The idea of searching for orthogonal projections, from a multidimensional space into a linear subspace, as an aid to detecting non-linear structure has been named exploratory projection pursuit.Most approaches are tied to the idea of searching for interesting projections. Typically, an interesting projection is one where the distribution of the projected data differs from the normal distribution. In this paper we define two projection indices which are aimed specifically at finding projections that best show grouped structure in the plane, if this exists in the multi-dimensional space. These involve a numerical optimization problem which is tackled in two stages, the projection and the pursuit; the first is based on a procedure to generate pseudo-random rotation matrices in the sense of the grand tour by D. Asimov (1985), and the second is a local numerical optimization procedure. One artificial and one real example illustrate the performance of the suggested indices.     

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.     

On the autocorrelation structure of inversive congruential pseudorandom number sequences

The inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators with their well known deficiencies. In the present paper inversive congruential generators with power of two modulus are considered. Known favourable results on the period length and the statistical independence properties of the generated sequences are summarized. Besides that the autocorrelation structure of inversive congruential sequences is studied. The main result reveals a remarkable autocorrelation property of the inversive congruential method.     

Quadratic reflected BSDEs and related obstacle problems for PDEs

Abstract

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.     

Generating multivariate continuous data via the notion of nearest neighbors

Taylor and Thompson [15] introduced a clever algorithm for simulating multivariate continuous data sets that resemble the original data. Their approach is predicated upon determining a few nearest neighbors of a given row of data through a statistical distance measure, and subsequently combining the observations by stochastic multipliers that are drawn from a uniform distribution to generate simulated data that essentially maintain the original data trends. The newly drawn values are assumed to come from the same underlying hypothetical process that governs the mechanism of how the data are formed. This technique is appealing in that no density estimation is required. We believe that this data-based simulation method has substantial potential in multivariate data generation due to the local nature of the generation scheme, which does not have strict specification requirements as in most other algorithms. In this work, we provide two R routines: one has a built-in simulator for finding the optimal number of nearest neighbors for any given data set, and the other generates pseudo-random data using this optimal number.     

On simulation and properties of the stable law

The stable distribution, in its many parametrizations, is central to many stochastic processes. Many random variables that occur in the study of Lévy processes are related to it. Good progress has been made recently for simulating various quantities related to the stable law. In this note, we survey exact random variate generators for these distributions. Many distributional identities are also reviewed.     

Some new construction of circular weakly balanced repeated measurements designs in periods of two different sizes

Abstract

Balanced repeated measurements designs (RMDs) balance out the residual effects. Williams Latin square designs work as minimal combinatorial balanced as well as variance balanced for RMDs for p (period sizes) = v (number of treatments). If minimal balanced RMDs cannot be constructed for the situations where p must be less than v then weakly balanced RMDs should be preferred. In this article, some generators are developed to generate circular weakly balanced RMDs in periods of two different sizes. To obtain the proposed designs, some construction procedures are also described for some of the cases where we could not develop generators.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

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.     

BSDEs driven by time-changed Lévy noises with non-Lipschitz generators

This paper deals with a class of backward stochastic differential equations (BSDEs for short) driven by time-changed Lévy noises. The existence and uniqueness of L^p(p ? 2) solutions for this kind of BSDEs with non-Lipschitz generators are obtained, which extend the corresponding results of Di Nunno and Sjursen (2014) [Stochastic Process. Appl. 124(4):1679-1709]. Furthermore, representation theorem for generators as well as converse comparison theorem for this kind of BSDEs are also studied.     

Sampling nested Archimedean copulas

We give algorithms for sampling from non-exchangeable Archimedean copulas created by the nesting of Archimedean copula generators, where in the most general algorithm the generators may be nested to an arbitrary depth. These algorithms are based on mixture representations of these copulas using Laplace transforms. While in principle the approach applies to all nested Archimedean copulas, in practice the approach is restricted to certain cases where we are able to sample distributions with given Laplace transforms. Precise instructions are given for the case when all generators are taken from the Gumbel parametric family or the Clayton family; the Gumbel case in particular proves very easy to simulate.     

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.     

Quasi-random resampling for the bootstrap

Quasi-random sequences are known to give efficient numerical integration rules in many Bayesian statistical problems where the posterior distribution can be transformed into periodic functions on then-dimensional hypercube. From this idea we develop a quasi-random approach to the generation of resamples used for Monte Carlo approximations to bootstrap estimates of bias, variance and distribution functions. We demonstrate a major difference between quasi-random bootstrap resamples, which are generated by deterministic algorithms and have no true randomness, and the usual pseudo-random bootstrap resamples generated by the classical bootstrap approach. Various quasi-random approaches are considered and are shown via a simulation study to result in approximants that are competitive in terms of efficiency when compared with other bootstrap Monte Carlo procedures such as balanced and antithetic resampling.     

Construction of repeated measurements designs strongly balanced for residual effects

Abstract

Repeated Measurements Designs have been widely used in agriculture, animal husbandry, education, biology, botany and engineering. Balanced or strongly balanced repeated measurements designs are useful to balance out the residual effects. In this article, some new generators and construction procedures are proposed to obtain circular strongly balanced repeated measurements designs in periods of (a) equal sizes, (b) two different sizes, and (c) three different sizes.     

A computer program for the generation of random variables from any discrete distribution

This paper provides specific directions for the preparation of a discrete pseudo-random number generator computer program using basic machine instructions. The scheme is a table look-up first suggested by Marsaglia. It is applicable to any discrete probability distribution. The general procedure is described for probabilities expressed as fractions in a number system of arbitrary base β. A brief example is given using the decimal system. Flow diagrams accompany the directions which will enable an experienced programmer to write the program for any computer system with only modest storage requirements. Results of chi-square tests performed on samples from specific binomial, Poisson, negative binomial, and hypergeornetric distributions generated using this procedure are given