Compound Poisson Models Research Articles

Host manipulation by Ligulaintestinalis: a cause or consequence of parasite aggregation?

Previous investigations suggest that the infection of the cyprinid roach, Rutilusrutilus, with the larval plerocercoid forms of the cestode, Ligulaintestinalis, creates behavioural and morphological changes in the fish host, potentially of adaptive significance to the parasite in promoting transmission to definitive avian hosts. Here we consider whether these behavioural changes are important in shaping the distribution of parasite individuals across the fish population. An examination of field data illustrates that fish infected with a single parasite were more scarce than expected under the negative binomial distribution, and in many months were more scarce than burdens of two, three or more, leading to a bimodal distribution of worm counts (peaks at 0 and >1). This scarcity of single-larval worm infections could be accounted for a priori by a predominance of multiple infection. However, experimental infections of roach gave no evidence for the establishment of multiple worms, even when the host was challenged with multiple intermediate crustacean hosts, each multiply infected. A second hypothesis assumes that host manipulation following an initial single infection leads to an increased probability of subsequent infection (thus creating a contagious distribution). If manipulated fish are more likely to encounter infected first-intermediate hosts (through microhabitat change, increased ingestion, or both), then host manipulation could act as a powerful cause of aggregation. A number of scenarios based on contagious distribution models of aggregation are explored, contrasted with alternative compound Poisson models, and compared with the empirical data on L.intestinalis aggregation in their roach intermediate hosts. Our results indicate that parasite-induced host manipulation in this system can function simultaneously as both a consequence and a cause of parasite aggregation. This mutual interaction between host manipulation and parasite aggregation points to a set of ecological interactions that are easily missed in most experimental studies of either phenomenon.

Actuarial statistics I: likelihood characterizations.

The present work establishes some likelihood properties of Bernoulli mixed discretecontinuous models with special emphasis on actuarial models. Potential statistical applications include one-parametric deformations from discrete to continuous families of distributions, perturbed models for Robust Statistics, models with unobservable events of use in the (re)insurance of large claims, insurance claims models with outliers, distribution-free reinsurance models via stop-loss extremal distributions (Section 1). Based on a mean equivalence relation for parametric random functions, a mean equivalence class of log-likelihoods is considered. The corresponding systems of sample mean equations lead to a class of maximum likelihood mean equivalent estimators, or shortly likelihood estimators (Section 2). For Bernoulli mixed discrete-continuous models a likelihood estimator, which is asymptotically equal to the maximum likelihood estimator, is constructed. Sometimes it is equal to the maximum likelihood estimator. This is the case for a specific insurance claims model with outliers. Furthermore this estimator appears to be useful as estimation method for the unknown parameters of a stop-loss ordered extremal distribution associated to the class of positive risks with fixed mean and variance (Section 3). The mainly theoretical Section 4 studies the relationship between pseudo-estimators and orthogonal parameters. After some motivation for the use of pseudo-estimators has been given, the well-known information inequality for pseudo-estimators is recalled. It extends the classical Cramer-Rao lower bound for unbiased estimators. Several examples, which illustrate the usefulness of the generalized inequality, are presented. As a main result, a substantial generalization of a characterization obtained previously by the author is derived. It characterizes unbiased pseudo-estimators related to the class of maximum likelihood mean equivalent estimators (introduced in Section 2) under an orthogonal condition of the parameter of interest. A thorough discussion and several examples follow. In particular a complete likelihood characterization of one-parametric maximum likelihood estimators is formulated (Corollary 4.2). The latter result is applied in Section 5 to provide rational justifications for the use of some categories of statistical models with location or scale parameters encountered in the fields of Economy and Social Sciences. First two simpler variants of classical results are formulated. Then practically quite useful characterizations of the compound Poisson, binomial and negative binomial gamma models in collective risk theory are derived. In Section 6 sufficient conditions for the sample mean to be a somewhat robust likelihood estimator in a Bernoulli mixed discrete-continuous model are given. These conditions characterize in Section 7 a mean scaled gamma outlier model encountered in the recent actuarial literature.

Flexible Purchase Frequency Modeling

The authors present a general framework for purchase frequency modeling that enables flexible fitting and convenient computation. Their easily described purchase frequency distributions subsume many previous models and provide a connection between mixed Poisson marketing models and the conceptually distinct compound Poisson models. These distributions provide simple parametric equations for individual-level prediction of second-period purchase frequency based on observed first-period purchase frequencies. The results are applied to four marketing panel data sets.

Econometric models based on count data. Comparisons and applications of some estimators and tests

AbstractThis paper deals with specification, estimation and tests of single equation reduced form type equations in which the dependent variable takes only non‐negative integer values. Beginning with Poisson and compound Poisson models, which involve strong assumptions, a variety of possible stochastic models and their implications are discussed. A number of estimators and their properties are considered in the light of uncertainty about the data generation process. The paper also considers the role of tests in sequential revision of the model specification beginr ing with the Poisson case and provides a detailed application of the estimators and tests to a model of the number of doctor consultations.

Compound poisson models in actuarial risk theory

This paper deals with the relationship between the Compound Poisson distribution, Bayesian uncertainty (or mixing) models and infinite divisibility. It is shown that the Compound Poisson distribution arises in many situations in the theory of risk. A large number of recent results are reviewed and some new results are given.