Inference for comparing a multinomial distribution with a known standard

We propose a new type of stochastic ordering which imposes a monotone tendency in differences between one multinomial probability and a known standard one. An estimation procedure is proposed for the constrained maximum likelihood estimate, and then the asymptotic null distribution is derived for the likelihood ratio test statistic for testing equality of two multinomial distributions against the new stochastic ordering. An alternative test is also discussed based on Neyman modified minimum chi-square estimator. These tests are illustrated with a set of heart disease data.     

Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

A Note on Signed Rank Tests for the Changepoint Problem

We consider two classes of signed rank statistics to test for smooth or abrupt changepoints in sequences of independent random variables. We derive asymptotic null distributions and finite sample approximations for the two classes. We infer from a Monte Carlo power study that the signed rank statistics may compare favorably with parametric analogues in detecting abrupt changes in a sequence of independent normal random variables.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.     

Joint distribution of simultaneous exposures to several carcinogens in a case–control study: sample size determination

Consider a population of individuals who are free of a disease under study, and who are exposed simultaneously at random exposure levels, say X,Y,Z,… to several risk factors which are suspected to cause the disease in the populationm. At any specified levels X=x, Y=y, Z=z, …, the incidence rate of the disease in the population ot risk is given by the exposure–response relationship r(x,y,z,…) = P(disease|x,y,z,…). The present paper examines the relationship between the joint distribution of the exposure variables X,Y,Z, … in the population at risk and the joint distribution of the exposure variables U,V,W,… among cases under the linear and the exponential risk models. It is proven that under the exponential risk model, these two joint distributions belong to the same family of multivariate probability distributions, possibly with different parameters values. For example, if the exposure variables in the population at risk have jointly a multivariate normal distribution, so do the exposure variables among cases; if the former variables have jointly a multinomial distribution, so do the latter. More generally, it is demonstrated that if the joint distribution of the exposure variables in the population at risk belongs to the exponential family of multivariate probability distributions, so does the joint distribution of exposure variables among cases. If the epidemiologist can specify the differnce among the mean exposure levels in the case and control groups which are considered to be clinically or etiologically important in the study, the results of the present paper may be used to make sample size determinations for the case–control study, corresponding to specified protection levels, i.e., size α and 1–β of a statistical test. The multivariate normal, the multinomial, the negative multinomial and Fisher's multivariate logarithmic series exposure distributions are used to illustrate our results.     

S–S method for stochastically ordered multinomial populations with missing data

Stochastic ordering is a useful concept in order restricted inferences. In this paper, we propose a new estimation technique for the parameters in two multinomial populations under stochastic orderings when missing data are present. In comparison with traditional maximum likelihood estimation method, our new method can guarantee the uniqueness of the maximum of the likelihood function. Furthermore, it does not depend on the choice of initial values for the parameters in contrast to the EM algorithm. Finally, we give the asymptotic distributions of the likelihood ratio statistics based on the new estimation method.     

A modified chi-square test for testing equality of two multinomial populations against an order restricted alternative

A modified chi-square test for testing the equality of two multinomial populations against an order restricted alternative in one sample and two sample cases is constructed. The relation between the concepts of dependence by cM-square and stochastic ordering is established, The asymptotic distribution of the test statistic is the chi-bar-square type discussed by Robertson, Wright and Dykstra (1988). Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP₂ ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.     

On a Generalization of the Classical Random Sampling Scheme and Unbiased Estimation

A generalization of the classical random sampling scheme is suggested. Based on the proposed generalization one can derive many new minimum variance unbiased estimators for probabilities, as well as for other functions of unknown parameters, for the multivariate Pólya, the multivariate negative Pólya, the multinomial, the multivariate hypergeometric, the multivariate Poisson, and the Wishart probability distributions.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

Using Multinomial Mixture Models to Cluster Internet Traffic

The paper considers the clustering of two large sets of Internet traffic data consisting of information measured from headers of transmission control protocol packets collected on a busy arc of a university network connecting with the Internet. Packets are grouped into 'flows' thought to correspond to particular movements of information between one computer and another. The clustering is based on representing the flows as each sampled from one of a finite number of multinomial distributions and seeks to identify clusters of flows containing similar packet‐length distributions. The clustering uses the EM algorithm, and the data‐analytic and computational details are given.     

Conditioning out rare events for exponential families has negligible effect on inference

ABSTRACT

Suppose that we observe X ～Binomial(n, p). Inference on p is difficult if X = 0 or n. One way around this is to condition on these events not happening. We show that this has only an exponentially small effect on its cumulants. This is also true if we condition away other rare events. Our results are presented for exponential families, with applications to the binomial, multinomial and negative multinomial distributions.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

A changepoint statistic with uniform type I error probabilities

ABSTRACT

Likelihood ratio tests for a change in mean in a sequence of independent, normal random variables are based on the maximum two-sample t-statistic, where the maximum is taken over all possible changepoints. The maximum t-statistic has the undesirable characteristic that Type I errors are not uniformly distributed across possible changepoints. False positives occur more frequently near the ends of the sequence and occur less frequently near the middle of the sequence. In this paper we describe an alternative statistic that is based upon a minimum p-value, where the minimum is taken over all possible changepoints. The p-value at any particular changepoint is based upon both the two-sample t-statistic at that changepoint and the probability that the maximum two-sample t-statistic is achieved at that changepoint. The new statistic has a more uniform distribution of Type I errors across potential changepoints and it compares favorably with respect to statistical power, false discovery rates, and the mean square error of changepoint estimates.     

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.     

A flexible procedure for formulating probability distributions on the unit interval with applications

Abstract

In this paper, we present a flexible mechanism for constructing probability distributions on a bounded intervals which is based on the composition of the baseline cumulative probability function and the quantile transformation from another cumulative probability distribution. In particular, we are interested in the (0, 1) intervals. The composite quantile family of probability distributions contains many models that have been proposed in the recent literature and new probability distributions are introduced on the unit interval. The proposed methodology is illustrated with two examples to analyze a poverty dataset in Peru from the Bayesian paradigm and Likelihood points of view.     

Corrected asymptotic distribution of statistics based on the multinomial law

Investigators and epidemiologists often use statistics based on the parameters of a multinomial distribution. Two main approaches have been developed to assess the inferences of these statistics. The first one uses asymptotic formulae which are valid for large sample sizes. The second one computes the exact distribution, which performs quite well for small samples. They present some limitations for sample sizes N neither large enough to satisfy the assumption of asymptotic normality nor small enough to allow us to generate the exact distribution. We analytically computed the 1/N corrections of the asymptotic distribution for any statistics based on a multinomial law. We applied these results to the kappa statistic in 2×2 and 3×3 tables. We also compared the coverage probability obtained with the asymptotic and the corrected distributions under various hypothetical configurations of sample size and theoretical proportions. With this method, the estimate of the mean and the variance were highly improved as well as the 2.5 and the 97.5 percentiles of the distribution, allowing us to go down to sample sizes around 20, for data sets not too asymmetrical. The order of the difference between the exact and the corrected values was 1/N² for the mean and 1/N³ for the variance.     

The ordering of negative quadrant dependence

A partial ordering is developed among negative quadrant dependent distributions with fixed marginals. Basic properties and closure under certain statistical operations are derived. Applications of the results in statistics and probability are given.     

Non-homogeneous Markov models in the analysis of survival after breast cancer

A cohort of 300 women with breast cancer who were submitted for surgery is analysed by using a non-homogeneous Markov process. Three states are onsidered: no relapse, relapse and death. As relapse times change over time, we have extended previous approaches for a time homogeneous model to a non omogeneous multistate process. The trends of the hazard rate functions of transitions between states increase and then decrease, showing that a changepoint can be considered. Piecewise Weibull distributions are introduced as transition intensity functions. Covariates corresponding to treatments are incorporated in the model multiplicatively via these functions. The likelihood function is built for a general model with k changepoints and applied to the data set, the parameters are estimated and life-table and transition probabilities for treatments in different periods of time are given. The survival probability functions for different treatments are plotted and compared with the corresponding function for the homogeneous model. The survival functions for the various cohorts submitted for treatment are fitted to the mpirical survival functions.     

Mixtures of regressions with changepoints

We introduce an extension to the mixture of linear regressions model where changepoints are present. Such a model provides greater flexibility over a standard changepoint regression model if the data are believed to not only have changepoints present, but are also believed to belong to two or more unobservable categories. This model can provide additional insight into data that are already modeled using mixtures of regressions, but where the presence of changepoints has not yet been investigated. After discussing the mixture of regressions with changepoints model, we then develop an Expectation/Conditional Maximization (ECM) algorithm for maximum likelihood estimation. Two simulation studies illustrate the performance of our ECM algorithm and we analyze a real dataset.