Mixture Of Binomial Distributions Research Articles

A Bayesian nonparametric approach for handling item and examinee heterogeneity in assessment data.

We propose a novel nonparametric Bayesian item response theory model that estimates clusters at the question level, while simultaneously allowing for heterogeneity at the examinee level under each question cluster, characterized by a mixture of binomial distributions. The main contribution of this work is threefold. First, we present our new model and demonstrate that it is identifiable under a set of conditions. Second, we show that our model can correctly identify question-level clusters asymptotically, and the parameters of interest that measure the proficiency of examinees in solving certain questions can be estimated at a rate (up to a log term). Third, we present a tractable sampling algorithm to obtain valid posterior samples from our proposed model. Compared to the existing methods, our model manages to reveal the multi-dimensionality of the examinees' proficiency level in handling different types of questions parsimoniously by imposing a nested clustering structure. The proposed model is evaluated via a series of simulations as well as apply it to an English proficiency assessment data set. This data analysis example nicely illustrates how our model can be used by test makers to distinguish different types of students and aid in the design of future tests.

Approximations of the ruin probability in a discrete time risk model

Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.

Alters as species: Predicting personal network size from contact diaries

Like wildlife species in an ecological system, members within a personal network (or alters) constantly shift and often remain hard to count. Previous studies often estimated the size of such personal networks using information given by a focal person (or ego), who names a list of friends and acquaintances, or someone known or related, that meet certain specified criteria. In a search for alternative methods, we estimate the number of alters using contact diaries that help reveal active and comprehensive interactions, which enable us to predict personal network size from a longitudinal perspective. By exploring contact frequencies between ego and alters, we propose a modeling approach based on species accumulation curves from ecology. Under this approach, the contact frequency between ego and alter often turns out to be a mixture of binomial distributions, and the number of alters with whom ego may make contact in the future is assumed to follow a specified discrete distribution. We estimate the model with the Bayesian nonparametric method, in which the distribution of contact probabilities is assumed to be a mixture of Dirichlet processes. We then demonstrate this approach with a data set containing 48 contact diaries collected over three months and discuss how such an ecological analogy may enrich social network studies.

On monotonicity of expected values of some run-related distributions

We prove that the expectation of the binomial distribution of order k with success probability p is monotonically increasing with respect to p for all n and k. The result is extended to the problems on exchangeable random sequences and expectations of distributions of mixtures of binomial distributions of order k are studied. If the mixing measure is stochastically increasing with respect to its parameter, the expectation of the mixture of binomial distributions of order k becomes nondecreasing. As examples of mixing measures two submodels of beta distributions are examined and the resulting expectation of the mixture distribution is monotonically strictly increasing. Further, we prove some properties on the expectation of the \(\ell \)-overlapping 1-runs in a sequence of independent and identically distributed n trials.

$W_{1,+}$-interpolation of probability measures on graphs

We generalize an equation introduced by Benamou and Brenier and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a nal distributions (f_0(x)), (f_1(x)), we prove the existence of a curve (f_t(x)) satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem.

Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters

There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.

Edgeworth Expansion of the Moment-Based Test for Homogeneity in an NEF-QVF Mixture Model

In this article, we study the moment-based test procedure for a mixture distribution for the Natural exponential family with quadratic variance functions (NEF-QVF) family proposed by Ning et al. (2009b) in the small sample size scenario. We derive the approximation for the null distribution of the test statistic by the Edgeworth expansion. The simulations are conducted for a binomial mixture distribution, which includes the situation corresponding to the detection of the linkage in the genetic analysis, with different sample sizes and family sizes at various significance levels. The simulation results show that our test performs reasonably well. We also apply the proposed method to the real clinical data to verify the significant difference between two drug treatments. The critical values associated with a binomial mixture distribution are also provided.

The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion

The binomial outcome data are widely encountered in many real world applications. The Binomial distribution often fails to model the binomial outcomes since the variance of the observed binomial outcome data exceeds the nominal Binomial distribution variance, a phenomenon known as overdispersion. One way of handling overdispersion is modeling the success probability of the Binomial distribution using a continuous distribution defined on the standard unit interval. The resultant general class of univariate discrete distributions is known as the class of Binomial mixture distributions. The Beta-Binomial (BB) distribution is a prominent member of this class of distributions. The Kumaraswamy-Binomial (KB) distribution is another recent member of this class. In this paper we focus the emphasis on the McDonald's Generalized Beta distribution of the first kind as the mixing distribution and introduce a new Binomial mixture distribution called the McDonald Generalized Beta-Binomial distribution(McGBB). Some theoretical properties of McGBB are discussed. The parameters of the McGBB distribution are estimated via maximum likelihood estimation technique. A real world dataset is modeled by using the new McGBB mixture distribution, and it is shown that this model gives better fit than its nested models. Finally, an extended simulation study is presented to compare the McGBB distribution with its nested distributions in handling overdispersed binomial outcome data.

ESTIMATION OF A PROPORTION USING SEVERAL INDEPENDENT SAMPLES OF BINOMIAL MIXTURES

SummaryThis paper considers a sequence of independent counts, with each count arising from a mixture of binomial distributions; the mixing distribution is fixed but the number of trials varies from count to count. In this common situation, an estimate of the underlying mean binomial proportion is needed. Two estimators are in general use: the arithmetic average and a weighted average of the observed proportions. Variances of the two estimators are compared and used to decide which estimator is preferred in a given context. The relative merits depend on the distribution of the proportions and the numbers of trials used.

Binomial mixtures: geometric estimation of the mixing distribution

Given a mixture of binomial distributions, how do we estimate the unknown mixing distribution? We build on earlier work of Lindsay and further elucidate the geometry underlying this question, exploring the approximating role played by cyclic polytopes. Convergence of a resulting maximum likelihood fitting algorithm is proved and numerical examples given; problems over the lack of identifiability of the mixing distribution in part disappear.

Stability and change in longitudinal water-level task performance.

Three longitudinal samples of children (N = 481), 8 to 16 years old, were assessed 3 times at yearly intervals on 8 water-level items. The within-child change in task performance over age is viewed as a stochastic process of the child changing or remaining in 1 of 3 latent (strategy) states: (a) bottom-parallel responders, (b) random responders, or (c) accurate responders. A random-effects binomial mixture distribution is used to model performance at each age. Change over age is gauged by a stochastic transition model. Although there was improvement in task performance over age, the more general finding is that strategy stability, not change, is most typical.

The Subclass Approach for Mutational Spectrum Analysis: Application of the SEM Algorithm

Analysis and comparison of mutational spectra represents an important problem in molecular biology. To analyse a mutational spectra we apply an algorithm based on the SEM subclass approach (Simulation, Expectation, Maximization). The algorithm tries to classify the mutational sites according to different mutation probabilities, and each site should belong to one class. Each class is approximated to binomial distribution and thus any real mutational spectrum is regarded as a mixture of binomial distributions. The separation process runs iteratively. Each iteration includes the simulation, maximization and estimation procedures. To evaluate the quality of the classification results, theX2test is used. The algorithm has been checked on random spectra with preset parameters and on real mutational spectra. As has been shown, 17 out of 19 analysed real mutational spectra can be divided into two or more classes of sites, of which one contains hotspots of mutation. For the G:C→A:T mutational spectra induced by Sn1 alkylating mutagens (11 spectra) the classification accuracy was 0.95. To test different site volumes, each Sn1-induced spectrum was divided into the G→A and C→T spectra. The classification accuracy for these spectra was 0.96. From the analysis of classification errors it is possible to suggest that at least part of them cannot be ascribed to the faults of the algorithm but are caused by some special features of the mutagenesis itself. The results of the real data are in good relation with existing knowledge. The approach we present is an attempt to formalize the concept of a “mutational hotspot”. The program implementing the SEM algorithm is available on the Web server (http://www.itba.mi.cnr.it/webmutation).

The latent class model for multiple binary screening tests.

Given multiple binary tests, such as repeated application of a blind screening test to each individual in a sample, we attempt to estimate the prevalence, sensitivity and specificity of the test without knowing the true disease status of those tested (gold standard). This problem is equivalent to finding the mixing distribution of a mixture of binomial distributions. We suggest a new method to determine the number of latent classes. Our simulations show that the coverage probabilities of the bootstrap confidence intervals of our estimates are correct. Our methods are illustrated by examples from published medical research.

Likelihood ratio tests for genetic linkage

Mixtures of binomial distributions are used to model linkage between genes and markers. In several classical models, we study the likelihood ratio test for linkage, which are homogeneity tests in mixture models. Using ad hoc reparametrizations, we prove that the corresponding test statistics converge to the supremum of squared truncated Gaussian processes under the null hypothesis.

Realistic interpretations of de Finetti's Theorem

Mixtures of binomial distributions have been recently introduced in Statistical Physics both in theoretical and in experimental frames in order to explain physical correlations. The physical interpretation of the mixing distribution is realistic, in contrast with the subjective mood originally attached by De Finetti to his celebrated Representation Theorem. Inductive and physical correlations, despite different in meaning, obey the same mathematical rules. Hence the logical status of the Theorem is worth of reconsideration by non-bayesian statisticians.

Tests for Genetic Linkage and Homogeneity

This report concerns likelihood ratio tests in a heterogeneous model for linkage, where the recombination fraction has a binomial mixture distribution with an unknown proportion of unlinked families. We consider families of unequal sizes with known or unknown phase data. In both cases, the limit distributions of the linkage test statistics are a mixture of a mass at ) and of a X2/1 distribution in equal proportions and homogeneity test statistics tend to the supremum of Gaussian processes. The critical values of the homogeneity tests are simulated, and the power functions of the linkage and homogeneity tests are compared in a simulation study.

Tests for Genetic Linkage and Homogeneity

This report concerns likelihood ratio tests in a heterogeneous model for linkage, where the recombination fraction has a binomial mixture distribution with an unknown proportion of unlinked families. We consider families of unequal sizes with known or unknown phase data. In both cases, the limit distributions of the linkage test statistics are a mixture of a mass at ) and of a X2/1 distribution in equal proportions and homogeneity test statistics tend to the supremum of Gaussian processes. The critical values of the homogeneity tests are simulated, and the power functions of the linkage and homogeneity tests are compared in a simulation study.

Quantifying Counterforce and Active Defense in Countering Theater Ballistic Missiles

This paper contains the formulation and analysis of a model to measure, compare, and contrast the effects of counterforce (pre-launch attack) and active defense (post-launch attack) against tactical ballistic missiles (TBM's). It is shown that without counterforce an active defense system could require an impractical number of weapons to counter incoming missiles and/or their warheads. This number is shown to decrease geometrically as effective counterforce is used, so that the expected number of warheads killed increases dramatically with counterforce that is only modestly effective. Actual distributions of warheads reaching the target area are shown to be complex mixtures of binomial distributions. It is shown that normal approximations to these distributions based on the easily-calculated means and variances often agree poorly with the actual distributions. This is especially true when using effective counterforce.

BINOMIX: A BASIC program for maximum likelihood analyses of finite and beta-binomial mixture distributions

If a discrete random variableY is binomially distributed for each observation unit in a sample, and if the binomial parameterp varies between different units, then the distribution ofY underlying a sample of observations based on different units is called a binomial mixture distribution. BINOMIX is a BASIC program to estimate parameters and to evaluate the goodness of fit for the two most important types of binomial mixtures, finite and beta-binomial.