Showing papers in "Canadian Journal of Statistics-revue Canadienne De Statistique in 1999"

Beyond kappa: A review of interrater agreement measures

Abstract: In 1960, Cohen introduced the kappa coefficient to measure chance-corrected nominal scale agreement between two raters. Since then, numerous extensions and generalizations of this interrater agreement measure have been proposed in the literature. This paper reviews and critiques various approaches to the study of interrater agreement, for which the relevant data comprise either nominal or ordinal categorical ratings from multiple raters. It presents a comprehensive compilation of the main statistical approaches to this problem, descriptions and characterizations of the underlying models, and discussions of related statistical methodologies for estimation and confidence-interval construction. The emphasis is on various practical scenarios and designs that underlie the development of these measures, and the interrelationships between them.

Sequential importance sampling for nonparametric Bayes models: The next generation

Abstract: There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators. Deux generations d'echantillonneurs de Gibbs ont ete popularisees pour les modeles semi-parametriques ou intervient le processus de Dirichlet. La premiere generation etait lourdement handicapee du fait que la localisation des grappes, ou groupes de parametres, pouvait devenir quasi-stationnaire. La deuxieme generation d'echantillonneurs a comble cette lacune en incorporant deux strategies: I'integration et I'ajout d'une eeape de relocalisation des grappes. Les auteurs montrent ici que ces strategies sont egalement applicables a la methode d'echantillonnage sequentiel pondere et que I'integration est particulierement efficace dans ce contexte. Comme dans le cas de l'echantillonnage de Gibbs, les deux strategies s'appliquent en fait a une classe beaucoup plus grande de modeles. Comme les auteurs le font valoir, elles produisent des poids d'echantillonnage plus uniformes et permettent une Rao-Blackwellisation additionnelle des estimateurs.

Bayesian density estimation using bernstein polynomials

Abstract: We propose a Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval, say [0,1]. To this aim, we use a prior based on Bemstein polynomials. This corresponds to expressing the density of the data as a mixture of given beta densities, with random weights and a random number of components. The density estimate is then obtained as the corresponding predictive density function. Comparison with classical and Bayesian kernel estimates is provided. The proposed procedure is illustrated in an example; an MCMC algorithm for approximating the estimate is also discussed.

A prior for the variance in hierarchical models

Abstract: The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix. Le choix d'une loi a priori pour les variances peut s'averer a la fois difficile et important dans le cadre d'une analyse bayesienne hierarchique ou d'un modele des composantes de la variance. En l'absence totale ou quasi-totale d'information a priori, l'emploi d'une loi ‘non informative’ est de mise. Plusieurs lois de ce type ont ete proposees dans differents contextes, mais leur utilisation est delicate, puisque certaines d'entre elles sont impropres et peuvent conduire a des lois a posteriori non integrables. Dans de petits echantillons, ces lois peuvent aussi se reveler ‘informatives’. Cet article est consacre a l'etude d'une loi a priori a la fois vague et integrable, la loi a priori a retrecissement uniforme (Strawderman 1971; Christiansen & Morris, 1997). Certaines de ses proprietes sont evoquees, notamment le fait que les lois a posteriori auxquelles elle conduit dans certains modeles hierarchiques classiques sont bel et bien integrables. Ses proprietes frequentistes sont egalement mises en valeur dans des situations d'estimation et de test au sein du modele hierarchique gaussien. On montre en outre comment elle peut ětre generalisee au cas multivarie d'une matrice de covariance.

Bayesian variable selection for proportional hazards models

Abstract: The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semi-parametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coefficients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coefficients and the model space, they propose a semi-automatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology. Les auteurs abordent d'un point de vue bayesien le probleme de la selection de variables dans les modeles de regression des risques proportionnels en presence de censure a droite. Ils proposent une approche semi-parametrique dans laquelle la loi a priori du taux de base est non parametrique, mais celle des coefficients de regression est completement parametrique. L'information concernant le taux de base est representee par la loi a priori issue d'un processus gamma discret; quant a la loi a priori des parametres du modele de regression, elle est choisie dans une classe de lois parameriques au moyen d'une procedure semi-automatique centree sur les donnees plutǒt que sur les parametres. La mise a jour de rinformation se fait au moyen d'un algorithme de Monte-Carlo a chaǐne de Markov. Des donnees reelles et simulees permettent d'illustrer la methode.

A full Bayesian analysis of circular data using the von Mises distribution

Abstract: This paper presents a full Bayesian analysis of circular data, paying special attention to the von Mises distribution. We obtain samples from the posterior distribution using the Gibbs sampler which, after the introduction of strategic latent variables, has all full conditional distributions of known type. Les auteurs montrent comment analyser des donnees circulates de facon bayesienne, notamment au moyen de la loi de von Mises. Us montrent comment exploiter l'echantillonneur de Gibbs pour obtenir des observations de la loi a posteriori, laquelle peut avoir des lois conditionnelles de tous les types connus, une fois prises en compte certaines variables latentes.

Convergence properties of the EM algorithm in constrained parameter spaces

Abstract: The established general results on convergence properties of the EM algorithm require the sequence of EM parameter estimates to fall in the interior of the parameter space over which the likelihood is being maximized. This paper presents convergence properties of the EM sequence of likelihood values and parameter estimates in constrained parameter spaces for which the sequence of EM parameter estimates may converge to the boundary of the constrained parameter space contained in the interior of the unconstrained parameter space. Examples of the behavior of the EM algorithm applied to such parameter spaces are presented. Les resultats de convergence existants concernant l'algorithme EM presupposent que les estimations successives du parametre appartiennent a l'interieur de l'espace parametrique sur lequel la vraisemblance est maximisee. Cet article examine la convergence de cette suite d'estimations et celle des valeurs de la vraisemblance qui leur sont associees dans le cas ou la suite des estimations du parametre peut converger vers un point qui se situe a la frontiere de l'espace parametrique contraint tout en appartenant a l'interieur de l'espace parametrique complet. Le comportement de l'algorithme EM est illustre dans de tels cas.

Asymptotic behavior of the Grenander estimator at density flat regions

Abstract: Over forty years ago, Grenander derived the MLE of a monotone decreasing density f with known mode. Prakasa Rao obtained the asymptotic distribution of this estimator at a fixed point x where f' (x) < 0. Here, we obtain the asymptotic distribution of this estimator at a fixed point x when f is constant and nonzero in some open neighborhood of x. This limiting distribution is expressible as the convolution of a closed-form density and a rescaled standard normal density. Groeneboom (1983) derived the aforementioned closed-form density and we provide an alternative, more direct derivation. II y a plus de quarante ans que Grenander a donne la forme de l'estimateur du maximum de vraisemblance d'une densite f monotone decroissante de mode connu. Prakasa Rao a determine la loi asymptotique de cet estimateur en tout point x tel que f'(x) ≤ 0. Dans cet article, les auteurs font de měme pour les points x dans un voisinage ouvert desquels f est constante et non nulle. La loi limite est ta convoluee d'une loi normale centree et d'une densite que Groeneboom (1983) avait deja explicitee et qui est ici obtenue de maniere plus directe.

Confidence estimation of a normal mean vector with incomplete data

Abstract: The problem of confidence estimation of a normal mean vector when data on different subsets of response variables are missing is considered. A simple approximate confidence region is proposed when the data matrix is of monotone pattern. Simultaneous inferential procedures based on Scheffe's method and Bonferroni's method are outlined. Further, applications of the results to a repeated measurements model are given. The results are illustrated using a practical example. Les auteurs s'interessent a la construction de regions de confiance pour le vecteur moyenne d'une population normale dans la situation ou certaines donnees sont manquantes pour des sous-ensembles particuliers de variables-reponses. Us proposent une solution simple, quoiqu'approximative, a ce probleme dans le cas ou la matrice des observations possede une structure monotone. Us presentent en outre des procedures d'inference simultanee s'appuyant sur les methodes de Scheffe et de Bonferroni. Ces resultats sont appliques a un modele de mesures repetees et illustres au moyen d'un exemple concret.

Empirical bayes estimation of reliability characteristics for an exponential family

Abstract: This paper considers empirical Bayes (EB) squared-error-loss estimations of mean lifetime, variance and reliability function for failure-time distributions belonging to an exponential family, which includes gamma and Weibull distributions as special cases. EB estimators are proposed when the prior distribution of the lifetime parameter is completely unknown but has a compact (known or unknown) support. Asymptotic optimality and rates of convergence of these estimators are investigated. The rates established here under the compact support restriction are better than the polynomial rates of convergence obtained previously. Les auteurs abordent l'estimation bayesienne empirique, sous fonction de perte quadratique, de la duree de vie moyenne, de la variance et de la fiabilite de distributions de temps de panne appartenant a une famille exponentielle, comme e'est le cas pour les lois gamma ou de Weibull par exemple. lls proposent des estimateurs de Bayes empiriques pour les situations ou la loi a priori du parametre de duree de vie est completement inconnue, sauf pour le fait qu'elle possede un support compact (precise ou non). lls etudient I'optimalite asymptotique et les taux de convergence de tels estimateurs. Les taux qu'ils obtiennent en postulant la compacite du support sont meilleurs que les taux de convergence polynomiaux dej a connus.

Asymptotics for L1‐estimators of regression parameters under heteroscedasticityY

Abstract: We consider the asymptotic behaviour of L1 -estimators in a linear regression under a very general form of heteroscedasticity. The limiting distributions of the estimators are derived under standard conditions on the design. We also consider the asymptotic behaviour of the bootstrap in the heteroscedastic model and show that it is consistent to first order only if the limiting distribution is normal. L'auteur etudie le comportement asymptotique d'estimateurs L1 dans le cadre de la regression lineaire, sous des hypotheses d'heteroscedasticite tres generates. Il determine leur loi limite dans des conditions classiques portant sur la matrice d'incidence. Il examine en outre le comportement asymptotique du bootstrap dans le modele heteroscedastique et montre que la convergence du premier ordre ne se produit que si la loi limite est gaussienne.

Quasi‐universal bandwidth selection for kernel density estimators

Abstract: Let fn, h denote the kernel density estimate based on a sample of size n drawn from an unknown density f. Using techniques from L2 projection density estimators, the author shows how to construct a data-driven estimator fn, h which satisfies This paper is inspired by work of Stone (1984), Devroye and Lugosi (1996) and Birge and Massart (1997). Soil fn, h l'estimateur a noyau de la densite construit a partir d'un echantillon aleatoire de taille d'une densite f inconnue. L'auteur montre comment l'emploi de techniques d'estimation par projection dans L2 permet de bǎtir un estimateur guide par les donnees tel que Ces travaux font suite a ceux deja publies par Stone (1984), Devroye et Lugosi (1996), ainsi que Birge et Massart (1997).

Minimax designs for approximately linear models with AR(1) errors

Abstract: We obtain designs for linear regression models under two main departures from the classical assumptions: (1) the response is taken to be only approximately linear, and (2) the errors are not assumed to be independent, but to instead follow a first-order autoregressive process. These designs have the property that they minimize (a modification of) the maximum integrated mean squared error of the estimated response, with the maximum taken over a class of departures from strict linearity and over all autoregression parameters p, Ipl < 1, of fixed sign. Specific methods of implementation are discussed. We find that an asymptotically optimal procedure for AR(1) models consists of choosing points from that design measure which is optimal for uncorrelated errors, and then implementing them in an appropriate order.

Robust inference for bivariate point processes

Abstract: In the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures. La fonction de moyenne cumulative (FMC) d'un processus (Lawless & Nadeau 1995) est un bon outil de modelisation dans les situations ou Ton s'interesse a l'effet des covariables sur le nombre espere d'evenements de type recurrent realises au cours d'une periode de temps donnee. On developpe ici une procedure d'inference robuste pour des modeles de regression conjoints pour des FMC emanant d'un processus ponctuel bivarie, afin de pouvoir traiter les cas, comme celui de maladies chroniques, ou plusieurs types d'evenements differents peuvent survenir. On deduit de fonctions d'estimation sans biais pour les FMC des estimateurs convergents des parametres et des estimations robustes de leur variance. Dans la plupart des cas, il est difficile de specifier correctement la structure de covariance d'un processus ponctuel bidimensionnel, mais lorsque celle-ci est connue, il est possible de trouver une fonction d'estimation optimale pour les FMC. On suggere l'emploi de fonctions d'estimation pertinentes pour les processus de melanges de Poisson bivaries comme modele commode dans des situations plus generates. Des simulations montrent que les estimateurs derives de ce modele sont pratiquement sans biais et que l'estimation de leur variance est tres robuste. On montre de plus comment il est possible, dans ce contexte, de confronter des hypotheses au moyen des tests score ou de Wald generalises. La methodologie est illustree au moyen de donnees recueillies sur des patients souffrant d'asthme bronchique.

Some linear model techniques for analyzing small‐circle spherical data

Abstract: The author investigates least squares as a method for fitting small-circle models to a sample of unit vectors in R3. He highlights a local linear model underlying the estimation of the parameters of a circle. This model is used to construct an estimation algorithm and regression-type inference procedures for the parameters of a circle. It makes it possible to compare the fit of a small circle with that of a spherical ellipse. The limitations of the least-squares approach are emphasized: when the errors are bounded away from 0, the least-squares estimators are not consistent as the sample size goes to infinity. Two examples, concerned with the migration of elephant seals and with the classification of geological folds, are analyzed using the linear model techniques proposed in this work. L'auteur etudie l'ajustement d'un petit cercle a la surface de la sphere unite a un echantillon de vecteurs unitaires de R3 a l'aide de la methode des moindres carres. Il met en lumiere un modele lineaire local sous-jacent a l'estimation des parametres du cercle. Ce modele est utilise pour construire un algorithme pour le calcul des estimations de měme que des tests et des intervalles de confiance concernant les parametres du cercle. Il permet egalement de determiner si une ellipse spherique donne un meilleur ajustement qu'un petit cercle. L'auteur montre egalement que la methode des moindres carres ne s'applique qu'a des echantillons ou les vecteurs sont tres proches du cercle echantillonnal. Si les erreurs experimentales sont grandes, les estimateurs des moindres carres ne sont pas convergents lorsque la taille d'echantillon tend vers l'infini. Deux exemples, portant sur la migration des phoques et sur la classification des plis geologiques, sont traites a l'aide des methodes des moindres carres mises de l'avant dans cet article.

Bayesian computation for the superposition of nonhomogeneous poisson processes

Abstract: Bayesian inference for the superposition of nonhomogeneous Poisson processes is studied. A Markov-chain Monte Carlo method with data augmentation is developed to compute the features of the posterior distribution. For each observed failure epoch, a latent variable is introduced that indicates which component of the superposition model gives rise to the failure. This data-augmentation approach facilitates specification of the transitional kernel in the Markov chain. Moreover, new Bayesian tests are developed for the full superposition model against simpler submodels. Model determination by a predictive likelihood approach is studied. A numerical example based on a real data set is given. Cet article concerne l'inference bayesienne dans le cadre des modeles obtenus par superposition de proces-sus de Poisson non homogenes. Les auteurs y montrent comment les principales caracteristiques de la loi a posteriori peuvent ětre determinees au moyen d'une methode de Monte-Carlo a chaǐne de Markov avec accroissement de donnees. Une variable latente permet, a chaque episode de panne observe, d'identifier laquelle des composantes du systeme est en cause. Le recours a une approche d'accroissement de donnees facilite la specification du noyau de transition de la chaǐne de Markov. En plus de proposer de nouveaux tests bayesiens permettant de comparer le modele de superposition complet a certains sous-modeles, les auteurs montrent comment l'evaluation de la vraisemblance previsionnelle peut servir a choisir un modele. La demarche est illustree dans son ensemble a l'aide d'un jeu de donnees reelles.

Fisher information in weighted distributions

Abstract: Suppose that a density fθ (x) belongs to an exponential family, but that inference about θ must be based on data that are obtained from a density that is proportional to W(x)fθ(x). The authors study the Fisher information about θ in observations obtained from such weighted distributions and give conditions under which this information is greater than under the original density. These conditions involve the hazard- and reversed-hazard-rate functions. Supposons qu'une densite fθ (x) appartienne a une famille exponentielle mais que l'inference concernant θ ne puisse s'appuyer que sur des donnees issues d'une densite proportionnelle a W(x)fθ(x). Les auteurs etudient l'information de Fisher pour θ lorsque les observations proviennent de telles lois ponderees et enoncent des conditions sous lesquelles cette information est plus grande que sous la loi de depart. Ces conditions font intervenir le taux de panne et le taux de panne a contre-sens.

Robust extrapolation designs and weights for biased regression models with heteroscedastic errors

Abstract: We consider the construction of designs for the extrapolation of regression responses, allowing both for possible heteroscedasticity in the errors and for imprecision in the specification of the response function. We find minimax designs and correspondingly optimal estimation weights in the context of the following problems: (1) for ordinary least squares estimation, determine a design to minimize the maximum value of the integrated mean squared prediction error (IMSPE), with the maximum being evaluated over both types of departure; (2) for weighted least squares estimation, determine both weights and a design to minimize the maximum IMSPE; (3) choose weights and design points to minimize the maximum IMSPE, subject to a side condition of unbiasedness. Solutions to (1) and (2) are given for multiple linear regression with no interactions, a spherical design space and an annular extrapolation space. For (3) the solution is given in complete generality; as one example we consider polynomial regression. Applications to a dose-response problem for bioassays are discussed. Numerical comparisons, including a simulation study, indicate that, as well as being easily implemented, the designs and weights for (3) perform as well as those for (1) and (2) and outperform some common competitors for moderate but undetectable amounts of model bias. Les auteurs expliquent comment construire des plans d'experience pour l'extrapolation de variables modelisees par regression en presence (i) d'heteroscedasticite' de l'erreur et (ii) d'imprecision dans la specification du modele. Les plans proposes sont minimax et les poids d'estimation correspondants sont optimaux dans les situations ou: (1) on cherche un plan minimisant la valeur maximale sur (i) et (ii) de l'erreur quadratique moyenne de prevision integree (IMSPE) dans un contexte d'estimation par les moindres Carres ordinaires (2) on cherche a la fois un plan et des poids qui minimisent l'IMSPE maximal dans un contexte d'estimation par la methode des moindres carres ponderes; (3) on veut selectionner les points a echantillonner et les poids de facon a minimiser l'IMSPE maximal sous une condition d'absence de biais. Les solutions aux problames (1) et (2) sont donnees dans le cadre de la regression lineaire multiple sans interactions, pour un espace d'echantillonnage spherique et pour un espace d'extrapolation annulaire. La solution au probleme (3) est donnee en toute generalite et illustree dans le cas de la regression polynomiale. Les auteurs presentent en outre des applications ayant trait a un probleme de dose de reponse dans des bio-essais. Des comparaisons numeriques prenant notamment la forme de simulations indiquent qu'en plus de leur facilite d'implantation, les plans et les poids optimaux du cas (3) se comportent aussi bien que ceux correspondant aux cas (1) et (2) en plus de surclasser certains competiteurs d'usage courant dans des situations ou le biais inherent au modele est relativement important sans toutefois ětre detectable.

Generalized orthogonal contrast tests for homogeneity of ordered means

Abstract: Several procedures have been proposed for testing equality of ordered means. The best-known of these is the likelihood-ratio test introduced by Bartholomew, which possesses generally superior power characteristics to those of its competitors. Difficulties in implementing this test have led to the development of alternative approaches, such as tests based on single and multiple contrasts. Some recent approaches have utilized approximations to the polyhedral cone defining the restricted parameter space, including those of Akkerboom (circular cone) and Mudholkar & McDermott (orthant). This article proposes a class of tests based on an improved orthant approximation to the polyhedral cone. These tests may be viewed as generalizations of the orthogonal contrast test proposed by Mukerjee, Robertson & Wright. Studies of the power functions of several competing tests indicate that the generalized orthogonal contrast tests are effective alternatives to the likelihood-ratio test, especially when the latter is difficult to implement. Plusieurs procedures ont ete proposees pour tester l'egalite de moyennes ordonnees. La plus connue a ete popularisee par Bartholomew; elle s'appuie sur le rapport des vraisemblances et est generalement plus puissante que les autres. Les difficultes d'implantation qu'elle pose ont cependant stimule le developpement d'autres tests, dont ceux fondes sur des contrastes. Recemment, on a aussi cherche a approximer le cěne polyedrique definissant l'espace parametrique contraint, soit par un cene circulaire (Akkerboom), soit par un orthant (Mudholkar & McDermott). L'auteur propose ici une meilleure approximation de type orthant du cǒne polyedrique. Les tests qui en decoulent peuvent ětre vus comme des generalisations du test de Mukerjee, Robertson & Wright base sur des contrastes orthogonaux. Une etude de Monte-Carlo comparant la puissance de differents tests montre que la solution proposee offre des avantages par rapport au test du rapport des vraisemblances, particulierement lorsque ce dernier est difficile a mettre en œuvre.

A minimally informative likelihood for decision analysis: illustration and robustness

Abstract: The authors discuss a class of likelihood functions involving weak assumptions on data generating mechanisms. These likelihoods may be appropriate when it is difficult to propose models for the data. The properties of these likelihoods are given and it is shown how they can be computed numerically by use of the Blahut-Arimoto algorithm. The authors then show how these likelihoods can give useful inferences using a data set for which no plausible physical model is apparent. The plausibility of the inferences is enhanced by the extensive robustness analysis these likelihoods permit. Les auteurs montrent comment il est possible, en l'absence de modele naturel pour des observations, de construire une classe de fonctions de vraisemblance a partir d'hypotheses tres faibles concernant l'origine des donnees. Ils presentent les proprietes de ces vraisemblances a information minimale et expliquent comment les calculer a l'aide de l'algorithme de Blahut-Arimoto. Ils illustrent la faisabilite et l'utilite de cette approche au moyen d'un exemple concret. Comme cette methode se prěte bien a une etude de robustesse, les conclusions auxquelles elle conduit sont d'autant plus plausibles.

Extended voting measures

Abstract: This paper extends a number of voting measures used to quantify voting power. The extension is based on the recognition that individuals sometimes vote in coalitions. This observation gives rise to a statistical model which considers past voting patterns of subsets of eligible voters. The model is then used to obtain estimates of the probabilities of all voting combinations from which empirical measures are calculated. The calculation of the estimated probabilities may involve high-dimensional integrations. An example is given based on past decisions arising from the Supreme Court of Canada. Les auteurs generalised un certain nombre d'indices permettant de quantifier le pouvoir electoral. En partant de la constatation que des coalitions se forment parfois, ils formulent un modele statistique qui tient compte des habitudes de vote de divers groupes d'electeurs. Ceci leur permet d'estimer la probabilite de toutes les combinaisons possibles de voix et d'en deduire des mesures empiriques du pouvoir electoral. L'estimation de ces probabilites peut necessiter l'evaluation d'integrales multiples. Certains jugements de la Cour Suprěme du Canada sont reexamines a la lumiere de cette approche.

The robustness of the quasilikelihood estimator

Abstract: The quasilikelihood estimator is widely used in data analysis where a likelihood is not available. We illustrate that with a given variance function it is not only conservative, in minimizing a maximum risk, but also robust against a possible misspecification of either the likelihood or cumulants of the model. In examples it is compared with estimators based on maximum likelihood and quadratic estimating functions.

L'analyse de la variation saisonniére quand l'amplitude et la taille sont faibles

Abstract: Studies of seasonal variation are valuable in biomedical research because they can help to discover the etiology of diseases that are not well understood. Generally in these studies the data have certain characteristics that require specialized tests and methods for the statistical analysis. But the effectiveness of these specialized tests is variable, especially according to the seasonal variation, the dimension of the amplitude in the seasonal variation, and the sample size. The purpose of this paper is to present a test and methods appropriate for the analysis and modeling of data whose seasonal variation has small amplitude and whose sample size is small. This test can detect different kinds of seasonal variation. The results from a simulation study show that the test performs very well. The application of these methods is illustrated by two examples.

Bootstrapping with auxiliary information

Abstract: In the nonparametric setting, the standard bootstrap method is based on the empirical distribution function of a random sample. The author proposes, by means of the empirical likelihood technique, an alternative bootstrap procedure under a nonparametric model in which one has some auxiliary information about the population distribution. By proving the almost sure weak convergence of the modified bootstrapped empirical process, the validity of the proposed bootstrap procedure is established. This new result is used to obtain bootstrap confidence bands for the population distribution function and to perform the bootstrap Kolmogorov test in the presence of auxiliary information. Other applications include bootstrapping means and variances with auxiliary information. Three simulation studies are presented to demonstrate the performance of the proposed bootstrap procedure for small samples.

Convergence rates of change-point estimators and tail probabilities of the first-passage-time process

Abstract: In the classical setting of the change-point problem, the maximum-likelihood estimator and the traditional confidence region for the change-point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times. L'auteur s'interesse a l'estimation a vraisemblance maximale et a l'estimation par intervalie traditionnelle dans le cadre classique du probieme de la recherche du point d'inflexion. II demontre que la probabilite de prendre la bonne decision, que la probabilite de couverture et que la taille esperee de la region de confiance convergent a une vitesse exponentielle a mesure que croǐt la taille de I'echantillon. L'auteur etudie pour ce faire le comportement des ailes de la loi des premiers temps de passage. II obtient des inegalites generates et des resultats asymptotiques exacts dans le cas des lois de Bernoulli. Dans ce cas, il deduit en outre de la loi conditionnelle des premiers temps de passage une forme asymptotique explicite pour la taille espere de la region de confiance.

Multiple roots of estimating functions

Abstract: Estimating functions can have multiple roots. In such cases, the statistician must choose among the roots to estimate the parameter. Standard asymptotic theory shows that in a wide variety of cases, there exists a unique consistent root, and that this root will lie asymptotically close to other consistent (possibly inefficient) estimators for the parameter. For this reason, attention has largely focused on the problem of selecting this root and determining its approximate asymptotic distribution. In this paper, however, we concentrate on the exact distribution of the roots as a random set. In particular, we propose the use of higher-order root intensity functions as a tool for examining the properties of the roots and determining their most problematic features. The use of root intensity functions of first and second order is illustrated by application to the score function for the Cauchy location model.

A generalized‐moments specification test for the logistic link

Abstract: The authors consider the problem of testing the validity of the logistic regression model using a random sample. Given the values of the response variable, they observe that the sample actually consists of two independent subsets of observations whose density ratio has a known parametric form when the model is true. They are thus led to propose a generalized‐moments specification test in detail. In addition, they show that this test can be derived using Neyman's smooth tests for goodness of fit. They present simulation results and apply the methodology to the analysis of two real data sets.

Probability and statistics: A tale of two worlds?

Abstract: This comparative study of research productivity and publication habits in probability and statistics completes the paper that was published in this Journal at the end of 1997. It is based on a ten-year survey of eighteen international journals, half of which are specialized in probability theory and the other half in statistics. Paper, author and adjusted page counts yield cursory measures of productivity for countries and institutions that contributed to fundamental research in these two related fields during the period 1986-1995. These data also reveal significant cultural differences between probabilists and statisticians in the volume of research, the length of papers, coauthorship practices, etc. Canada is seen to be one of the strongest contributors to the development of these two disciplines. Cette etude comparative de la productivite et des habitudes de publication des chercheurs en probabilites et en statistique complete le rapport deja paru dans La revue a la fin de 1997. Pour realiser ce travail, dix-huit revues internationales, neuf de chaque domaine, ont ete recensees sur une periode de dix ans. Un decompte des articles publies dans ces revues, de leur longueur et du nombre de leurs auteurs, permet d'evaluer sommairement la productivite des etablissements et des pays ayant le plus contribee a la recherche fondamentale dans ces deux disciplines entre 1986 et 1995. Ces donnees font aussi ressortir d'importantes differences culturelles dans les pratiques de publication des probabilistes et des statisticiens, notamment en ce qui touche le volume des ecrits, leur longueur et le nombre de leurs signataires. Enfin, Particle met en lumiere l'importance de la contribution canadienne a l'essor de ces deux disciplines.

Order‐restricted hypothesis testing in a variation of the normal mixture model

Abstract: This paper examines likelihood-ratio tests concerning the relationships among a fixed number of univariate normal means given a sample of normal observations whose population membership is uncertain. The asymptotic null distributions of likelihood-ratio test statistics are derived for a class of tests including hypotheses which place linear inequality constraints on the normal means. The use of such tests in the interval mapping of quantitative trait loci is addressed. L'auteur examine le comportement stochastique du rapport des vraisemblances utilise pour tester l'existence de certaines relations entre les moyennes de populations normales univariees dans la situation ou l'echantillon provient d'une loi gaussienne d'origine incertaine. II determine la loi asymptotique de ce rapport, notamment dans les cas ou l'hypothese a tester s'exprime en terme d'inegalites sur des combinaisons lineaires de moyennes. II discute en outre de la pertinence de telles procedures de test pour la representation par intervalle de la localisation de traits quantitatifs.

Dirichlet forms: Some infinite‐dimensional examples

Abstract: The author presents three examples of a Markov process taking values in an infinite-dimensional state space and analyzes the sample path behaviour using the theory of Dirichlet forms. L'auteur presente trois exemples de processus de Markov a valeurs dans un espace d'etats de dimension infinie et il analyse le comportement de leurs trajectoires au moyen de la theorie des formes de Dirichlet.