Description of purchase incidence by multivariate heterogeneous Poisson processes

In this paper the application of bivariate Poisson heterogeneous models to budget data is studied. This study was motivated from inconsistencies that we encountered when univariate Poisson based models were applied to cumulative data sets. Application of a multivariate Poisson based model is a possible solution to this problem. In this paper we will study the feasibility of estimators based on these models.     

A Poisson mixture model of discrete choice

In this paper, we introduce a new Poisson mixture model for count panel data where the underlying Poisson process intensity is determined endogenously by consumer latent utility maximization over a set of choice alternatives. This formulation accommodates the choice and count in a single random utility framework with desirable theoretical properties. Individual heterogeneity is introduced through a random coefficient scheme with a flexible semiparametric distribution. We deal with the analytical intractability of the resulting mixture by recasting the model as an embedding of infinite sequences of scaled moments of the mixing distribution, and newly derive their cumulant representations along with bounds on their rate of numerical convergence. We further develop an efficient recursive algorithm for fast evaluation of the model likelihood within a Bayesian Gibbs sampling scheme. We apply our model to a recent household panel of supermarket visit counts. We estimate the nonparametric density of three key variables of interest-price, driving distance, and their interaction-while controlling for a range of consumer demographic characteristics. We use this econometric framework to assess the opportunity cost of time and analyze the interaction between store choice, trip frequency, search intensity, and household and store characteristics. We also conduct a counterfactual welfare experiment and compute the compensating variation for a 10%-30% increase in Walmart prices.     

Consistent estimation in the bilinear multivariate errors-in-variables model

A bilinear multivariate errors-in-variables model is considered. It corresponds to an overdetermined set of linear equations AXB=C, A∈ℝ^m×n, B∈ℝ^p×q, in which the data A, B, C are perturbed by errors. The total least squares estimator is inconsistent in this case.  An adjusted least squares estimator is constructed, which converges to the true value X, as m →∞, q →∞. A small sample modification of the estimator is presented, which is more stable for small m and q and is asymptotically equivalent to the adjusted least squares estimator. The theoretical results are confirmed by a simulation study. Acknowledgements. We thank two anonymous reviewers for their suggestions and corrections.? A. Kukush is supported by a postdoctoral research fellowship of the Belgian office for Scientific, Technical and Cultural Affairs, promoting Scientific and Technical Collaboration with Central and Eastern Europe.? S. Van Huffel is a full professor with the Katholieke Universiteit Leuven.? I. Markovsky is a research assistant with the Katholieke Universiteit Leuven.? This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP V-22), initiated by the Belgian State, Prime Minister's Office – Federal Office for Scientific, Technical and Cultural Affairs of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U. Leuven) “Predictive computer models for medical classification problems using patient data and expert knowledge”, of the FWO projects G.0078.01, G.0200.00, and G0.0270.02.? The scientific responsibility is assumed by its authors.     

A bivariate Poisson count data model using conditional probabilities

The applied econometrics of bivariate count data predominantly focus on a bivariate Poisson density with a correlation structure that is very restrictive. The main limitation is that this bivariate distribution excludes zero and negative correlation. This paper introduces a new model which allows for a more flexible correlation structure. To this end the joint density is decomposed by means of the multiplication rule in marginal and conditional densities. Simulation experiments and an application of the model to recreational data are presented.     

Introducing a general class of species diversification models for phylogenetic trees

Phylogenetic trees are types of networks that describe the temporal relationship between individuals, species, or other units that are subject to evolutionary diversification. Many phylogenetic trees are constructed from molecular data that is often only available for extant species, and hence they lack all or some of the branches that did not make it into the present. This feature makes inference on the diversification process challenging. For relatively simple diversification models, analytical or numerical methods to compute the likelihood exist, but these do not work for more realistic models in which the likelihood depends on properties of the missing lineages. In this article, we study a general class of species diversification models, and we provide an expectation-maximization framework in combination with a uniform sampling scheme to perform maximum likelihood estimation of the parameters of the diversification process.     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.     

A Selective Overview and Comparison of Robust Mixture Regression Estimators

Mixture regression models have been widely used in business, marketing and social sciences to model mixed regression relationships arising from a clustered and thus heterogeneous population. The unknown mixture regression parameters are usually estimated by maximum likelihood estimators using the expectation–maximisation algorithm based on the normality assumption of component error density. However, it is well known that the normality-based maximum likelihood estimation is very sensitive to outliers or heavy-tailed error distributions. This paper aims to give a selective overview of the recently proposed robust mixture regression methods and compare their performance using simulation studies.     

Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.     

Distribution of correlation coefficient under mixture normal model

The present paper obtains the nonnull distribution of the product moment correlation coefficient r when sample is drawn from a mixture of two bivariate Gaussian distributions. The moments of 1−r ² have been used to derive the nonnull density of r. Received September 2000     

Locally D- and c-optimal designs for Poisson and negative binomial regression models

The research on optimal experimental designs for nonlinear regression models is of great interest because these models are used to characterize chemical, biological or agricultural phenomena. Much of them involve an exponential decay. In this paper, locally D- and c-optimal designs are derived analytically for Poisson and negative binomial regression models.     

Graphical interaction models for multivariate time series1

In this paper we extend the concept of graphical models for multivariate data to multivariate time series. We define a partial correlation graph for time series and use the partial spectral coherence between two components given the remaining components to identify the edges of the graph. As an example we consider multivariate autoregressive processes. The method is applied to air pollution data. Received: June 1999     

Influence analysis for Poisson inverse Gaussian regression models based on the EM algorithm

For Poisson inverse Gaussian regression models, it is very complicated to obtain the influence measures based on the traditional method, because the associated likelihood function involves intractable expressions, such as the modified Bessel function. In this paper, the EM algorithm is employed as a basis to derive diagnostic measures for the models by treating them as a mixed Poisson regression with the weights from the inverse Gaussian distributions. Several diagnostic measures are obtained in both case-deletion model and local influence analysis, based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Two numerical examples are given to illustrate the results.     

Tests for zero inflation in a bivariate zero‐inflated Poisson model

The score test statistics for testing zero inflation and covariance parameter are proposed in the bivariate zero‐inflated Poisson (BZIP) regression model. The Monte Carlo studies show that the score test and likelihood ratio test for testing zero inflation underestimate the nominal significance level, while the score test for covariance parameter keeps the significance level close to the nominal one. To overcome this nominal level underestimation, we propose a bootstrap method of the score test for the testing problem of zero inflation. An empirical example with covariates is provided to illustrate the results. In addition, score test for zero inflation is also proposed in the BZIP model, which allows a flexible dependence structure using copula.     

A model for identifying and ranking dangerous accident locations: a case study in Flanders

These days, road safety has become a major concern in most modern societies. In this respect, the determination of road locations that are more dangerous than others (black spots or also called sites with promise) can help in better scheduling road safety policies. The present paper proposes a multivariate model to identify and rank sites according to their total expected cost to the society. Bayesian estimation of the model via a Markov Chain Monte Carlo approach is discussed in this paper. To illustrate the proposed model, accident data from 23,184 accident locations in Flanders (Belgium) are used and a cost function proposed by the European Transport Safety Council is adopted to illustrate the model. It is shown in the paper that the model produces insightful results that can help policy makers in prioritizing road infrastructure investments.     

A TVM-Copula-MIDAS-GARCH model with applications to VaR-based portfolio selection

This paper develops a novel time-varying multivariate Copula-MIDAS-GARCH (TVM-Copula-MIDAS-GARCH) model with exogenous explanatory variables to model the joint distribution of returns. The model accounts for mixed frequency factors that affect the time-varying dependence structure of financial assets. Furthermore, we examine the effectiveness of the proposed model in VaR-based portfolio selection. We conduct an empirical analysis on estimating the 90%, 95%, 99% VaRs of the portfolio constituted of the Shanghai Composite Index, Shanghai SE Fund Index, and Shanghai SE Treasury Bond Index. The empirical results show that the proposed TVM-Copula-MIDAS-GARCH model is effective to investigate the nonlinear time-varying dependence among those three indices and performs better in portfolio selection.     

Statistical inference for continuous-time Markov processes with block structure based on discrete-time network data

A widely used approach to modeling discrete-time network data assumes that discrete-time network data are generated by an unobserved continuous-time Markov process. While such models can capture a wide range of network phenomena and are popular in social network analysis, the models are based on the homogeneity assumption that all nodes share the same parameters. We remove the homogeneity assumption by allowing nodes to belong to unobserved subsets of nodes, called blocks, and assuming that nodes in the same block have the same parameters, whereas nodes in distinct blocks have distinct parameters. The resulting models capture unobserved heterogeneity across nodes and admit model-based clustering of nodes based on network properties chosen by researchers. We develop Bayesian data-augmentation methods and apply them to discrete-time observations of an ownership network of non-financial companies in Slovenia in its critical transition from a socialist economy to a market economy. We detect a small subset of shadow-financial companies that outpaces others in terms of the rate of change and the desire to accumulate stocks of other companies.     

Numerical Maximisation of Likelihood: A Neglected Alternative to EM?

There is by now a long tradition of using the EM algorithm to find maximum‐likelihood estimates (MLEs) when the data are incomplete in any of a wide range of ways, even when the observed‐data likelihood can easily be evaluated and numerical maximisation of that likelihood is available as a conceptually simple route to the MLEs. It is rare in the literature to see numerical maximisation employed if EM is possible. But with excellent general‐purpose numerical optimisers now available free, there is no longer any reason, as a matter of course, to avoid direct numerical maximisation of likelihood. In this tutorial, I present seven examples of models in which numerical maximisation of likelihood appears to have some advantages over the use of EM as a route to MLEs. The mathematical and coding effort is minimal, as there is no need to derive and code the E and M steps, only a likelihood evaluator. In all the examples, the unconstrained optimiser nlm available in R is used, and transformations are used to impose constraints on parameters. I suggest therefore that the following question be asked of proposed new applications of EM: Can the MLEs be found more simply and directly by using a general‐purpose numerical optimiser?     

A spatial model for multivariate lattice data

In this article, we develop Markov random field models for multivariate lattice data. Specific attention is given to building models that incorporate general forms of the spatial correlations and cross-correlations between variables at different sites. The methodology is applied to a problem in environmental equity. Using a Bayesian hierarchical model that is multivariate in form, we examine the racial distribution of residents of southern Louisiana in relation to the location of sites listed with the U.S. Environmental Protection Agency's Toxic Release Inventory.     

A Comparative Simulation Study of Wavelet Shrinkage Estimators for Poisson Counts

Using computer simulations, the finite sample performance of a number of classical and Bayesian wavelet shrinkage estimators for Poisson counts is examined. For the purpose of comparison, a variety of intensity functions, background intensity levels, sample sizes, primary resolution levels, wavelet filters and performance criteria are employed. A demonstration is given of the use of some of the estimators to analyse a data set arising in high-energy astrophysics. Following the philosophy of reproducible research, the M atlab programs and real-life data example used in this study are made freely available.