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.     

Limit theorems for dependent Bernoulli variables with statistical inference

We consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.     

Generalized binomial distributions

In many cases where the binomial dismbution fails to apply to real world data it is because of more variability in the data than can be explained by that dismbution. Several authors have proposed models that are useful in explaining extra-binomial variation. In this paper we point out a characterization of sequences of exchangeable Bernoulli random variables which can be used to develop models which show more variability than the binomial. We give sufficient conditions which will yield such models and show how existig models can be combined to generate further models. The usefulness of some of these models is illustrated by fitting them to sets of real data.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Models for papilloma multiplicity and regression: applications to transgenic mouse studies

In cancer studies that use transgenic or knockout mice, skin tumour counts are recorded over time to measure tumorigenicity. In these studies cancer biologists are interested in the effect of endogenous and/or exogenous factors on papilloma onset, multiplicity and regression. In this paper an analysis of data from a study conducted by the National Institute of Environmental Health Sciences on the effect of genetic factors on skin tumorigenesis is presented. Papilloma multiplicity and regression are modelled by using Bernoulli, Poisson and binomial latent variables, each of which can depend on covariates and previous outcomes. An EM algorithm is proposed for parameter estimation, and generalized estimating equations adjust for extra dependence between outcomes within individual animals. A Cox proportional hazards model is used to describe covariate effects on the onset of tumours.     

A note on dependence modeling for Bernoulli variables

Understanding and modeling multivariate dependence structures depending upon the direction are challenging but an interest of theoretical and applied researchers. In this paper, we propose a characterization of tables generated by Bernoulli variables through the uniformization of the marginals and refer to them as Q-type tables. The idea is similar to the copulas. This approach helps to see the dependence structure clearly by eliminating the effect of the marginals that have nothing to do with the dependence structure. We define and study conditional and unconditional Q-type tables and provide various applications for them. The limitations of existing approaches such as Cochran-Mantel-Haenszel pooled odds ratio are discussed, and a new one that stems naturally from our approach is introduced.     

Estimation of Parameters of a Clipped MA(1) Process

Let {X _t , t ∈ ?} be a sequence of iid random variables with an absolutely continuous distribution. Let a > 0 and c ∈ ? be some constants. We consider a sequence of 0-1 valued variables {ξ _t , t ∈ ?} obtained by clipping an MA(1) process X _t  ? aX _t?1 at the level c, i.e., ξ _t  = I[X _t  ? aX _t?1 < c] for all t ∈ ?. We deal with the estimation problem in this model. Properties of the estimators of the parameters a and c, the success probability p, and the 1-lag autocorrelation r ₁ are investigated. A numerical study is provided as an illustration of the theoretical results.     

A generalized binomial distribution determined by a two-state Markov chain and a distribution by the Bayesian approach

For dependent Bernoulli random variables, the distribution of a sum of the random variables is obtained as a generalized binomial distribution determined by a two-state Markov chain. Asymptotic distributions of the sum are derived from the central limit theorem and the Edgeworth expansion. A numerical comparison of the exact and asymptotic distributions of the sum is also given. Further a distribution of the sum by the Bayesian approach is derived and its asymptotic distributions are provided. Numerical results are given.     

A general maximum likelihood analysis of measurement error in generalized linear models

This paper describes an EM algorithm for maximum likelihood estimation in generalized linear models (GLMs) with continuous measurement error in the explanatory variables. The algorithm is an adaptation of that for nonparametric maximum likelihood (NPML) estimation in overdispersed GLMs described in Aitkin (Statistics and Computing 6: 251–262, 1996). The measurement error distribution can be of any specified form, though the implementation described assumes normal measurement error. Neither the reliability nor the distribution of the true score of the variables with measurement error has to be known, nor are instrumental variables or replication required.Standard errors can be obtained by omitting individual variables from the model, as in Aitkin (1996).Several examples are given, of normal and Bernoulli response variables.     

A note on the almost sure central limit theorem for the product of partial sums of m-dependent random variables

We present an almost sure central limit theorem for the product of the partial sums of m-dependent random variables. In order to obtain the main result, we prove a corresponding almost sure central limit theorem for a triangular array.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.     

On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

Forecasting future values of changing sequences

Sequences of independent random variables are observed and on the basis of these observations future values of the process are forecast. The Bayesian predictive density of k future observations for normal, exponential, and binomial sequences which change exactly once are analyzed for several cases. It is seen that the Bayesian predictive densities are mixtures of standard probability distributions. For example, with normal sequences the Bayesian predictive density is a mixture of either normal or t-distributions, depending on whether or not the common variance is known. The mixing probabilities are the same as those occurring in the corresponding posterior distribution of the mean(s) of the sequence. The predictive mass function of the number of future successes that will occur in a changing Bernoulli sequence is computed and point and interval predictors are illustrated.     

BERNOULLI SAMPLING

The concepts of the Bernoulli count process of a point process and Bernoulli sampling of a discrete parameter stochastic process are introduced. The Bernoulli count process determines the stochastic structure of the point process, and a process obtained by thinning a discrete parameter stochastic process by Bernoulli sampling satisfies the same property. Stationarity and the Markov property remain invariant under Bernoulli sampling.     

Counting certain binary strings

Consider a sequence of exchangeable or independent binary (i.e. zero-one) random variables. Numbers of strings with a fixed number of ones between two subsequent zeros are studied under an overlapping enumeration scheme. The respective waiting times are examined as well. Exact probability functions are obtained by means of combinatorial analysis and via recursive schemes in the case of an exchangeable and of an independent sequence, respectively. Explicit formulae for the mean values and variances of the number of strings are provided for both types of the sequences. For a Bernoulli sequence the asymptotic normality of the numbers of strings is established too. Indicative exchangeable and independent sequences, combined with numerical examples, clarify further the theoretical results.     

Estimation for Partially Linear Single-index Instrumental Variables Models

In this article, we generalize the partially linear single-index models to the scenario with some endogenous covariates variables. It is well known that the estimators based on the existing methods are often inconsistent because of the endogeneity of covariates. To deal with the endogenous variables, we introduce some auxiliary instrumental variables. A three-stage estimation procedure is proposed for partially linear single-index instrumental variables models. The first stage is to obtain a linear projection of endogenous variables on a set of instrumental variables, the second stage is to estimate the link function by using local linear smoother for given constant parameters, and the last stage is to obtain the estimators of constant parameters based on the estimating equation. Asymptotic normality is established for the proposed estimators. Some simulation studies are undertaken to assess the finite sample performance of the proposed estimation procedure.     

Additional variables and adjusted estimates with arbitrary known variance-covariance structure

When additional variables are fitted in a linear model under arbitrary known variance-covariance struture, the extra sum of squares due to fitting the new variables and adjusted parameter estimates can be computed in an efficient manner without actually explicity fitting the entire augmented model. When the additional variables are specific dummy variables, downdating formulae are readily obtained, thus generatina methods which are well known for the linear model with variance-covariance structure σ²I. Two different methods to downdate a linear model are presented.     

Use of multi-auxiliary variables as a condensed auxiliary variable in selecting a sample

Information on several auxiliary variables correlated with the variable under study is available in most of the sample survey studies. This paper attempts an optimal use of several auxiliary variables in the form of a single auxiliary variable obtained as a linear function of these variables. The performance of this condensed auxiliary variable has been studied in selecting the sample.     

Sharp bounds on l-estimates and their expectations for dependent samples

Sharp bounds on the expectation of each L-estimate are determined for samples of identically distributed random variables with a given expectation and a central absolute moment of order p > 1, The distributions achieving these bounds are constructed. Analogous results are derived in the cases of symmetric, noraiegative and bounded random variables, Accurate bounds for L-estimates of arbitrary finite samples are concluded.     

The distribution of the first j digits of beta related random variables

This paper gives the discrete distribution of the first j significant digits of two random variables: (1) a beta variable with integer parameter n and the other parameter m > 0, and (2) the reciprocal of (1). As a special case for n=1, we obtain the distribution of the first j significant digits of the pwoers of uniformly distributed random variables. These generalize the results of Kennard and Reith (1981) and Friedberg (1984), who considered only uniformly distributed random variables.