The Representation of Hypergeometric Random Variables using Independent Bernoulli Random Variables

In this paper, we show that a hypergeometric random variable can be represented as a sum of independent Bernoulli random variables that are, except in degenerate cases, not identically distributed. In the proof, we use the factorial moment generating function. An asymptotic result on the probabilities of the Bernoulli random variables in the sum is also presented. Numerical examples are used to illustrate the results.     

Impact of misspecified residual correlation structure on the parameter estimates in a shared spatial frailty model

In practice, survival data are often collected over geographical regions. Shared spatial frailty models have been used to model spatial variation in survival times, which are often implemented using the Bayesian Markov chain Monte Carlo method. However, this method comes at the price of slow mixing rates and heavy computational cost, which may render it impractical for data-intensive application. Alternatively, a frailty model assuming an independent and identically distributed (iid) random effect can be easily and efficiently implemented. Therefore, we used simulations to assess the bias and efficiency loss in the estimated parameters, if residual spatial correlation is present but using an iid random effect. Our simulations indicate that a shared frailty model with an iid random effect can estimate the regression coefficients reasonably well, even with residual spatial correlation present, when the percentage of censoring is not too high and the number of clusters and cluster size are not too low. Therefore, if the primary goal is to assess the covariate effects, one may choose the frailty model with an iid random effect; whereas if the goal is to predict the hazard, additional care needs to be given due to the efficiency loss in the parameter(s) for the baseline hazard.     

Ordering Properties of Convolutions of Exponential Random Variables

Convolutions of independent random variables are usually compared. In this paper, after a synthetic comparison with respect to hazard rate ordering between sums of independent exponential random variables, we focus on the special case where one sum is identically distributed. So, for a given sum of n independent exponential random variables, we deduce the "best" Erlang-n bounds, with respect to each of the usual orderings: mean ordering, stochastic ordering, hazard rate ordering and likelihood ratio ordering.     

Probability inequalities for continuous unimodal random variables with finite support

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X₁ ,X₂,…,X_n, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for X_i. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffding's inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).     

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

In this article, we investigate the precise large deviations for a sum of independent but not identical distributed random variables. {X _n , n ≥ 1} are independent non-negative random variables with distribution functions {F _n , n ≥ 1}. We assume that the average of right tails of distribution functions F _n is equivalent to some distribution function F with consistently varying tails. In applications, we apply our main results to a realistic example (Pareto-type distribution) and obtain a specific result.     

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.     

Convergence and Divergence of Random Series

This paper amplifies Daley's (1981) criteria for absolute convergence of certain random series by providing a sufficient condition which also is necessary if the summands are independent. Conditions for unconditional and conditional convergence are also given. These results are used to obtain a substantially complete picture of the behaviour of random Dirichlet series of a fairly general type. Behaviour of the partial sums of divergent series is discussed, with particular attention to Dirichlet series.     

On Large Deviations of Sums of Independent Random Variables

Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér's condition. The large deviation x-region under consideration is broader than in the classical Cramér's theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.     

Estimating the negative binomial dispersion parameter with highly stratified surveys

We investigate several estimators of the negative binomial (NB) dispersion parameter for highly stratified count data for which the statistical model has a separate mean parameter for each stratum. If the number of samples per stratum is small then the model is highly parameterized and the maximum likelihood estimator (MLE) of the NB dispersion parameter can be biased and inefficient. Some of the estimators we investigate include adjustments for the number of mean parameters to reduce bias. We extend other estimators that were developed for the iid case, to reduce bias when there are many mean parameters. We demonstrate using simulations that an adjusted double extended quasi-likelihood estimator we proposed gives much improved estimates compared to the MLE. Adjusted extended quasi-likelihood and adjusted maximum likelihood estimators also give much-improved results. We illustrate the various estimators with stratified random bottom trawl survey data for cod (Gadus morhua) off the south coast of Newfoundland, Canada.     

A Threshold Dose-Response Model with Random Effects in Teratological Experiments

Abstract

Teratological experiments are controlled dose-response studies in which impregnated animals are randomly assigned to various exposure levels of a toxic substance. Subsequently, both continuous and discrete responses are recorded on the litters of fetuses that these animals produce. Discrete responses are usually binary in nature, such as the presence or absence of some fetal anomaly. This clustered binary data usually exhibits over-dispersion (or under-dispersion), which can be interpreted as either variation between litter response probabilities or intralitter correlation. To model the correlation and/or variation, the beta-binomial distribution has been assumed for the number of positive fetal responses within a litter. Although the mean of the beta-binomial model has been linked to dose-response functions, in terms of measuring over-dispersion, it may be a restrictive method in modeling data from teratological studies. Also for certain toxins, a threshold effect has been observed in the dose-response pattern of the data. We propose to incorporate a random effect into a general threshold dose-response model to account for the variation in responses, while at the same time estimating the threshold effect. We fit this model to a well-known data set in the field of teratology. Simulation studies are performed to assess the validity of the random effects threshold model in these types of studies.     

Asymptotics for Tail Probability of Random Sums with a Heavy-Tailed Number and Dependent Increments

This article obtains the asymptotics for the tail probability of random sums, where the random number and the increments are all heavy tailed, and the increments follow a certain wide dependence structure. This dependence structure can contain some commonly used negatively dependent random variables as well as some positively dependent random variables.     

Wild Bootstrap of the Sample Mean in the Infinite Variance Case

It is well known that the standard independent, identically distributed (iid) bootstrap of the mean is inconsistent in a location model with infinite variance (α-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the iid bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986 Wu , C. F. J. ( 1986 ). Jackknife, bootstrap, and other resampling methods . Annals of Statistics 14 : 1261 – 1295 .[Crossref], [Web of Science ®] , [Google Scholar]) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the iid bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower confidence sets than other consistent methods.     

Random Weighting Estimation for Quantile Processes and Negatively Associated Samples

This article presents new theories of random weighting estimation for quantile processes and negatively associated samples. Under the condition that X ₁, X ₂,…, X _n are independent random variables with a common distribution, the consistency for random weighting estimation of quantile processes is rigorously proved. When X ₁, X ₂,…, X _n are not independent of each other, random weighting estimation of sample mean is established for negatively associated samples.     

On some probability distributions associated with random walks

The probability distribution of the total number of games to ruin in a gambler's ruin random walk with initial position n, the probability distribution of the total size of an epidemic starting with n cases and the probability distribution of the number of customers served during a busy period M/M/1 when the service starts with n waiting customers are identical. All these can be easily obtained by using Lagrangian expansions instead of long combinatorial methods. The binomial, trinomial, quadrinomial and polynomial random walks of a particle have been considered with an absorbing barrier at 0 when the particle starts its walks from a point n, and the pgfs. and the probability distributions of the total number of jumps (trials) before absorption at 0 have been obtained. The values for the mean and variance of such walks have also been given.     

The Superiorities of Empirical Bayes Estimation of Variance Components in Random Effects Model

In this article, the Bayes estimators of variance components are derived and the parametric empirical Bayes estimators (PEBE) for the balanced one-way classification random effects model are constructed. The superiorities of the PEBE over the analysis of variance (ANOVA) estimators are investigated under the mean square error (MSE) criterion, some simulation results for the PEBE are obtained. Finally, a remark for the main results is given.     

Regeneration-based bootstrap for Markov chains

We consider a bootstrap method for Markov chains where the original chain is broken into a (random) number of cycles based on an atom (regeneration point) and the bootstrap scheme resamples from these cycles. We investigate the asymptotic accuracy of this method for the case of a sum (or a sample mean) related to the Markov chain. Under some standard moment conditions, the method is shown to be at least as good as the normal approximation, and better (second-order accurate) in the case of nonlattice summands. We give three examples to illustrate the applicability of our results.     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

A method to generate autocorrelated uniform random numbers

We present a simple, fast method to generate autocorrelated uniform random numbers. The “sum of uniforms” method adds a pair of U(0,1) random numbers, transforms the sum to a third U(0,1) random number, and uses this third random number as one member of the next pair. The method produces any desired level of positive or negative correlation between successive random numbers.     

A modified random-effect procedure for combining risk difference in sets of 2×2 tables from clinical trials

Summary Meta-analyses of sets of clinical trials often combine risk differences from several 2×2 tables according to a random-effects model. The DerSimonian-Laird random-effects procedure, widely used for estimating the populaton mean risk difference, weights the risk difference from each primary study inversely proportional to an estimate of its variance (the sum of the between-study variance and the conditional within-study variance). Because those weights are not independent of the risk differences, however, the procedure sometimes exhibits bias and unnatural behavior. The present paper proposes a modified weighting scheme that uses the unconditional within-study variance to avoid this source of bias. The modified procedure has variance closer to that available from weighting by ideal weights when such weights are known. We studied the modified procedure in extensive simulation experiments using situations whose parameters resemble those of actual studies in medical research. For comparison we also included two unbiased procedures, the unweighted mean and a sample-size-weighted mean; their relative variability depends on the extent of heterogeneity among the primary studies. An example illustrates the application of the procedures to actual data and the differences among the results. This research was supported by Grant HS 05936 from the Agency for Health Care Policy and Research to Harvard University.     

Analysis of Linear Models with One Factor Having Both Fixed and Random Levels

ABSTRACT

A general theory for a case where a factor has both fixed and random effect levels is developed under one-way treatment structure model. Estimation procedures for the fixed effects and variance components are consider for the model. The testing of fixed effects is considered when the variance–covariance matrix is known and unknown. Confidence intervals for estimable functions and prediction intervals for predictable functions are constructed. The computational procedures are illustrated using data from an on-farm trial.