A Measure of Dependence Between Two Compositions

We consider the problem of describing the correlation between two compositions. Using a bicompositional Dirichlet distribution, we calculate a joint correlation coefficient, based on the concept of information gain, between two compositions. Numerical values of the joint correlation coefficient are calculated for compositions of two and three components, respectively. We also present an estimator of the joint correlation coefficient for a sample from a bicompositional Dirichlet distribution. Two confidence intervals are presented and we examine their empirical confidence coefficients using a Monte Carlo study. Finally, we apply the estimator to a data set analysing the joint correlation between the 1967 and 1997, and the 1977 and 1997 compositions of the government gross domestic product for the 50 states of the USA and the District of Columbia.     

On the hyper-dirichlet type 1 and hyper-liouville distributions

This paper concerns the characterization of a new family of multivariate beta distribution functions - the hyper-Dirichlet type 1 distribution. This family describes the joint density function of the terminal variates of an arbitrary tree constructed from finite sequences of probability vectors having independent Dirichlet type 1 distributions. Expressions for the general properties of the hyper-Dirichlet type 1 distribution are presented. In addition, the hyper-Liouville distribution is described and its properties are discussed as well as a generalization of the Liouville integral identity.     

On the effect of inducted negative correlation rate for beta acceptance–rejection algorithms

One of the variance reduction methods in simulation experiments is negative correlation induction, and in particular the use of the antithetic variates. The simultaneous use of antithetic variates and an acceptance–rejection method has been studied in some papers, where the inducted negative correlation has been calculated. In this study, the factors affecting the inducted negative correlation rate are addressed. To do this, the beta distribution is first selected to generate negatively correlated random variates using the acceptance–rejection method. The effects of both the efficiency of the acceptance–rejection method and the initial negative correlation rate on the inducted negative correlation are explored. Results show that both factors have significant effects; therefore, a combination of both can lead to algorithms better able to generate negative correlations.     

Generating random points in a polytope

An algorithm is presented for generating pseudorandom variates distributed uniformly over an arbitrary convex polytope in a Euclidean space of arbitrary dimension. Many commonly used methods for generating uniform variates require that the polytope be expressed as the solution set of a system of linear inequalities; the algorithm presented here requires instead that the polytope be presented in terms of a finite generating set, typically the set of its vertices. Included in the algorithm are procedures for identifying all faces of the polytope and for decomposing the polytope into simplices.     

The distribution of the product of two independent generalized trapezoidal random variables

ABSTRACT

In this article, we derive the probability density function (pdf) of the product of two independent generalized trapezoidal random variables having different supports, in closed form, by considering all possible cases. We also show that the results for the product of two triangular and uniform random variables follow as special cases of our main result. As an illustration, we obtain pdf of product for a suitably constrained set of parameters and plot some graphs using MATLAB, which express variation in pdf with change in different parameters of the generalized trapezoidal distribution.     

A revisit of the distribution of linear combinations of Dirichlet components

Provost and Cheong (2000 Provost, S. B., and Y.-H. Cheong. 2000. On the distribution of linear combinations of the components of a Dirichlet random vector. The Canadian Journal of Statistics 28 (2):417–25.[Crossref], [Web of Science ®] , [Google Scholar]) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. The c-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when its c-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results.     

Bayesian density estimation using bernstein polynomials

We propose a Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval, say [0,1]. To this aim, we use a prior based on Bemstein polynomials. This corresponds to expressing the density of the data as a mixture of given beta densities, with random weights and a random number of components. The density estimate is then obtained as the corresponding predictive density function. Comparison with classical and Bayesian kernel estimates is provided. The proposed procedure is illustrated in an example; an MCMC algorithm for approximating the estimate is also discussed.     

Moments of the product and ratio of two correlated chi-square variables

The exact probability density function of a bivariate chi-square distribution with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution.      

A simple approach for generating correlated binary variates ∗

Correlated binary data arise frequently in medical as well as other scientific disciplines; and statistical methods, such as generalized estimating equation (GEE), have been widely used for their analysis. The need for simulating correlated binary variates arises for evaluating small sample properties of the GEE estimators when modeling such data. Also, one might generate such data to simulate and study biological phenomena such as tooth decay or periodontal disease. This article introduces a simple method for generating pairs of correlated binary data. A simple algorithm is also provided for generating an arbitrary dimensional random vector of non-negatively correlated binary variates. The method relies on the idea that correlations among the random variables arise as a result of their sharing some common components that induce such correlations. It then uses some properties of the binary variates to represent each variate in terms of these common components in addition to its own elements. Unlike most previous approaches that require solving nonlinear equations or use some distributional properties of other random variables, this method uses only some properties of the binary variate. As no intermediate random variables are required for generating the binary variates, the proposed method is shown to be faster than the other methods. To verify this claim, we compare the computational efficiency of the proposed method with those of other procedures.     

A generalization of the adaptive rejection sampling algorithm

Rejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.     

A note on Markov chain Monte Carlo sweep strategies

Markov chain Monte Carlo (MCMC) routines have become a fundamental means for generating random variates from distributions otherwise difficult to sample. The Hastings sampler, which includes the Gibbs and Metropolis samplers as special cases, is the most popular MCMC method. A number of implementations are available for running these MCMC routines varying in the order through which the components or blocks of the random vector of interest X are cycled or visited. The two most common implementations are the deterministic sweep strategy, whereby the components or blocks of X are updated successively and in a fixed order, and the random sweep strategy, whereby the coordinates or blocks of X are updated in a randomly determined order. In this article, we present a general representation for MCMC updating schemes showing that the deterministic scan is a special case of the random scan. We also discuss decision criteria for choosing a sweep strategy.     

Multivariate generalized linear mixed models with?semi-nonparametric and smooth nonparametric random effects densities

We extend the family of multivariate generalized linear mixed models to include random effects that are generated by smooth densities. We consider two such families of densities, the so-called semi-nonparametric (SNP) and smooth nonparametric (SMNP) densities. Maximum likelihood estimation, under either the SNP or the SMNP densities, is carried out using a Monte Carlo EM algorithm. This algorithm uses rejection sampling and automatically increases the MC sample size as it approaches convergence. In a simulation study we investigate the performance of these two densities in capturing the true underlying shape of the random effects distribution. We also examine the implications of misspecification of the random effects distribution on the estimation of the fixed effects and their standard errors. The impact of the assumed random effects density on the estimation of the random effects themselves is investigated in a simulation study and also in an application to a real data set.     

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.     

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.     

On the distribution of linear combinations of the components of a Dirichlet random vector

On making use of a result of Imhof, an integral representation of the distribution function of linear combinations of the components of a Dirichlet random vector is obtained. In fact, the distributions of several statistics such as Moran and Geary's indices, the Cliff‐Ord statistic for spatial correlation, the sample coefficient of determination, F‐ratios and the sample autocorrelation coefficient can be similarly determined. Linear combinations of the components of Dirichlet random vectors also turn out to be a key component in a decomposition of quadratic forms in spherically symmetric random vectors. An application involving the sample spectrum associated with series generated by ARMA processes is discussed.     

One sided acceptance sampling by variables: the case of the laplace distribution

The theory of acceptance sampling by variables is well known when the underlying distribution is normal. When the normality assumption is not true, using the usual normal case method can be quite misleading. In this paper we deal with the Laplace distribution for both the standard deviation known and then unknown. We establish a decision rule for accepting a lot of product containing a defective proportion p. We determine the density function of the decision rule statistic, for small and large sample sizes. We give some practical ways to choose the sample size and the acceptance constant to obtain a desired operating characteristic curve     

A modified ziggurat algorithm for generating exponentially and normally distributed pseudorandom numbers

The ziggurat algorithm is a very fast rejection sampling method for generating pseudorandom numbers (PRNs) from statistical distributions. In the algorithm, rectangular sampling domains are layered on top of each other (resembling a ziggurat) to encapsulate the desired probability density function. Random values within these layers are sampled and then returned if they lie beneath the graph of the probability density function. Here, we present an implementation where ziggurat layers reside completely beneath the probability density function, thereby eliminating the need for any rejection test within the ziggurat layers. In the new algorithm, small overhanging segments of probability density remain to the right of each ziggurat layer, which can be efficiently sampled with triangularly shaped sampling domains. Median runtimes of the new algorithm for exponential and normal variates is reduced to 58% and 53%, respectively (collective range: 41–93%). An accessible C library, along with extensions into Python and MATLAB/Octave are provided.     

On the distribution of products of independent beta variates

This paper concerns an inquiry into the problem of generating closed form expressions for the cumulative distribution and probability density functions of products of independent beta variates. Recursive analytical procedures for constructing the equational forms of these functions-from their Mellin inversion integral representations, via the Cauchy residue theorem-are described. A numerical example illustrating details of the construction of a computable form of the distribution function of the product of three independent beta variates is also included.     

Universal methods for generating random variables with a given characteristic function

Universal generators for absolutely-continuous and integer-valued random variables are introduced. The proposal is based on a generalization of the rejection technique proposed by Devroye [The computer generation of random variables with a given characteristic function. Computers and Mathematics with Applications. 1981;7:547–552]. The method involves a dominating function solely requiring the evaluation of integrals which depend on the characteristic function of the underlying random variable. The proposal gives rise to simple algorithms which may be implemented in a few code lines and which may show noticeable performance even if some classical families of distributions are considered.     

On nonparametric Bayesian inference for the distribution of a random sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet-process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution, which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet-process mixtures of normals to recover this distribution and its features.