Showing papers in "Canadian Journal of Statistics-revue Canadienne De Statistique in 1992"

Deconvolution with supersmooth distributions

Abstract: Nonparametric deconvolution problems require one to recover an unknown density when the data are contaminated with errors Optimal global rates of convergence are found under the weighted Lp-loss (1 ≤ p ≤ ∞) It appears that the optimal rates of convergence are extremely low for supersmooth error distributions To resolve this difficulty, we examine how high the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate It is shown that if the noise level is not too high, nonparametric Gaussian deconvolution can still be practical Several simulation studies are also presented Lorsque l'on desire estimer une densite a partir d'observations sujettes a des erreurs, on fait face a un probleme de deconvolution nonparametrique On s'interesse ici au taux global de convergence lorsqu'une fonction de perte de type Lp ponderee est utilisee Le taux de convergence optimal est tres lent pour des distributions d'erreurs tres lisses Pour bien cemer cette difficulte, on examine quel niveau de bruit peut tout de měme permettre une estimation par deconvolution qui soit aussi bonne que celle donnee par l'estimation usuelle d'une densite On montre que si le niveau de bruit n'est pas trop important la deconvolution nonparametrique gaussienne peut ětre utilisee Plusieurs simulations sont presentees

A Bayesian method for classification and discrimination

Abstract: We discuss Bayesian analyses of traditional normal-mixture models for classification and discrimination. The development involves application of an iterative resampling approach to Monte Carlo inference, commonly called Gibbs sampling, and demonstrates routine application. We stress the benefits of exact analyses over traditional classification and discrimination techniques, including the ease with which such analyses may be performed in a quite general setting, with possibly several normal-mixture components having different covariance matrices, the computation of exact posterior classification probabilities for observed data and for future cases to be classified, and posterior distributions for these probabilities that allow for assessment of second-level uncertainties in classification.

Comparing three bootstrap methods for survey data

Abstract: Various bootstrap methods for variance estimation and confidence intervals in complex survey data, where sampling is done without replacement, have been proposed in the literature. The oldest, and perhaps the most intuitively appealing, is the without-replacement bootstrap (BWO) method proposed by Gross (1980). Unfortunately, the BWO method is only applicable to very simple sampling situations. We first introduce extensions of the BWO method to more complex sampling designs. The performance of the BWO and two other bootstrap methods, the rescaling bootstrap (Rao and Wu 1988) and the mirror-match bootstrap (Sitter 1992), are then compared through a simulation study. Together these three methods encompass the various bootstrap proposals. Differentes variantes de la methode du bootstrap ont ete proposees afin d'estimer la variance et construire des intervalles de confiance dans le contexte de sondages complexes ou l'echantillonnage se fait sans remise. La plus ancienne et probablement la plus naturelle est le bootstrap sans remise (BWO) propose par Gross (1980). Malheureusement cette methode n'est applicable qu'a des plans d'echantillonnage tres simples. Nous proposons une generalisation de la methode BWO a des plans d'echantillonnage plus complexes. Cette nouvelle methode et deux autres variantes du bootstrap, proposees respectivement par Rao et Wu (1988) et Sitter (1992), sont comparees a l'aide de simulations. Ces trois methodes englobent plusieurs des differentes variantes proposees.

Fourier smoother and additive models

Abstract: Suppose the observations (ti,yi), i = 1,… n, follow the model where gj are unknown functions. The estimation of the additive components can be done by approximating gj, with a function made up of the sum of a linear fit and a truncated Fourier series of cosines and minimizing a penalized least-squares loss function over the coefficients. This finite-dimensional basis approximation, when fitting an additive model with r predictors, has the advantage of reducing the computations drastically, since it does not require the use of the backfitting algorithm. The cross-validation (CV) [or generalized cross-validation (GCV)] for the additive fit is calculated in a further 0(n) operations. A search path in the r-dimensional space of degrees of freedom is proposed along which the CV (GCV) continuously decreases. The path ends when an increase in the degrees of freedom of any of the predictors yields an increase in CV (GCV). This procedure is illustrated on a meteorological data set.

Applications of empirical characteristic functions in some multivariate problems

Abstract: In this note we extend univariate tests for normality and symmetry based on empirical characteristic functions to the multivariate case. RESUME Cette note vise a etendre au cas multidimensionnel certains tests de normalite et de symetrie univaries construits a partir de fonctions caracteristiques experimentales.

Linear representation of M-estimates in linear models

Abstract: Consider the linear regression model, yi = xiβ0 + ei, i = l,…,n, and an M-estimate β of βo obtained by minimizing Σρ(yi — xiβ), where ρ is a convex function. Let Sn = ΣXiXiXi and rn = Sn½ (β — β0) — Sn2 Σxih(ei), where, with a suitable choice of h(.), the expression Σ xix(e,) provides a linear representation of β. Bahadur (1966) obtained the order of rn as n ∞ when βo is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of rn as n ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them. Soient p une fonction convexe et β un M-estimateur de βo obtenu en minimisant Σρ(yi-xiβ)xiβ) dans le cadre du modele de regression lineaire yi=xiβ+ ei,i=[,…,n. En introduisant une fonction h(.) convenable, il est possible de reprdsenter β sous la forme ΣXih(ei(ei). Soient alors Sn =ΣXiXiet rn=Sn½(β — β0) — Sn2Σxih(ei)Σ xih(ei)Dans le cas particulier ou β0est un parametre de localisation unidimensionnel representant une mediane, l'ordre asymptotique de rna ete etabli par Bahadur (1966) et un resultat semblable a ete demontre par Babu (1989) pour un parametre de regression plus general estime par la methode des moindres ecarts absolus. Cet article a pour objectif d'etendre ces resultats en etablissant l'ordre stochastique de rnlorsque n ∞ dans le cadre abstrait decrit ci-haut.

Resolving the conflict over fisher's exact test†

Abstract: Fisher's exact test for two-by-two contingency tables has repeatedly been criticized as being too conservative. These criticisms arise most frequently in the context of a planned experiment for which the numbers of successes in each of two experimental groups are assumed to be binomially distributed. It is argued here that the binomial model is often unrealistic, and that the departures from the binomial assumptions reduce the conservatism in Fisher's exact test. Further discussion supports a recent claim of Barnard (1989) that the residual conservatism is attributable, not to any additional information used by the competing method, but to the discrete nature of the test, and can be drastically reduced through the use of Lancaster's mid-p-value. The binomial model is not recommended in that it depends on extra, questionable assumptions. On a souvent pretendu que le test exact de Fisher pour les tableaux de contingence 2 × 2 est trop conservateur. Ces critiques surviennent le plus souvent dans le contexte d'une experience planifiee ou on suppose que le nombre de succes dans chacun des deux groupes experimentaux suit une loi binomiale. On soutient ici que le modele binomial est souvent irrealiste et que les ecarts par rapport aux lois binomiales ont pour effet de reduire le conservatisme du test exact de Fisher. De plus, l'argumentation appuie une affirmation recente de Barnard (1989) a l'effet que ce qui reste de conservatisme est attribuable a la nature discrete du test et non pas a une information additionnelle quelconque utilisee par la methode alternative. Ainsi, ce conservatisme residuel peut ětre considerablement reduit en utilisant la probabilite milieu de depassement au sens de Lancaster. Le modele binomial n'est pas recommande car il depend d'hypotheses supplementaires dont la validite est douteuse.

Nonparametric density estimation for censored survival data: Regression‐spline approach

Abstract: A method for nonparametric estimation of density based on a randomly censored sample is presented. The density is expressed as a linear combination of cubic M -splines, and the coefficients are determined by pseudo-maximum-likelihood estimation (likelihood is maximized conditionally on data-dependent knots). By using regression splines (small number of knots) it is possible to reduce the estimation problem to a space of low dimension while preserving flexibility, thus striking a compromise between parametric approaches and ordinary nonparametric approaches based on spline smoothing. The number of knots is determined by the minimum AIC. Examples of simulated and real data are presented. Asymptotic theory and the bootstrap indicate that the precision and the accuracy of the estimates are satisfactory. Nous proposons une methode pour l'estimation non-parametrique d'une densite quand l'echan-tillon est sujet a une censure aleatoire. La densite est ecrite comme une combinaison lineaire de M-splines cubiques et les coefficients sont determines en optimisant une pseudo-vraisemblance (la vraisemblance est maximisee conditionnellement a des noeuds determines par les donnees). L'utilisation de splines de regression (caracterisees par un petit nombre de noeuds) rend possible la reduction de la dimension de l'espace d'estimation, tout en preservant la flexibilite de la modelisation. Cette approche peut ětre vue comme un compromis entre l'approche parametrique et les approches non-parametriques ordinaires, utilisant les splines dites “de lissage” (smoothing splines) qui font intervenir un nombre important de noeuds. Dans notre methode le nombre de noeuds est determine par le biais du critere d'information d'Akaike (AIC). Nous presentons des exemples d'estimation a partir de donnees soit reelles soit simulees. La theorie asymptotique et le bootstrap indiquent que les estimations obtenues sont satisfaisantes.

Multivariate distributions with generalized inverse gaussian marginals, and associated poisson mixtures

Abstract: Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.

On prediction and mean squared error

Abstract: Practical questions motivate the search for predictors either of an as yet unobserved random vector, or of a random function of a parameter. An extension of the classical UMVUE theory is presented to cover such situations. In includes a Rao-Blackwell-type theorem, a Cramer-Rao-type inequality, and necessary and sufficient conditions for a predictor to minimize the mean squared error uniformly in the parameter. Applications are considered to the problem of selected means, the species problem, and the examination of some u-v estimates of Robbins (1988). Des problemes pratiques ont suggere la construction des estimateurs d'un vecteur aleatoire non-observe ou d'une fonction aleatoire d'un parametre. On presente une extension de la theorie classique des estimateurs sans biais minimisant l'erreur quadratique moyenne. Alors on obtient des theoremes du type de ceux de Rao-Blackwell et de Cramer-Rao ainsi que des conditions necessaires et suffisantes pour qu'un estimateur minimise l'erreur quadratique moyenne uniformement pour le parametre. On considere comme applications le probleme des moyennes selectionnees et le probleme des especes et on examine certains des estimateurs u-v de Robbins (1988).

Gaussian deconvolution via differentiation

Abstract: Suppose that ϕ is a Gaussian density and that g = f * ϕ, where * denotes convolution. From observations with density g, one wishes to estimate f. We analyze an estimate which is a linear combination of estimates of derivatives of g and show that this estimate converges in an L2 norm at a rate which is compatible with the pointwise optimal rate established by Fan (1991). Soit ϕ la densite d'une loi gaussienne et g = f * ϕ ou * denote la convolution. A partir d'observations suivant la densite g on desire faire l'estimation def. Un estimateur ayant la forme d'une combinaison lineaire d'estimateurs des derivees de g est etudie. On montre que cet estimateur converge, suivant la norme 2, a un taux compatible avec la taux pontuel optimal etabli par Fan (1991).

Goodness of fit for the inverse Gaussian distribution

Abstract: For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A2 is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A2 These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A2. This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.

On the identification of ARMA echelon-form models*

Abstract: An identification procedure for multivariate autoregressive moving average (ARMA) echelonform models is proposed. It is based on the study of the linear dependence between rows of the Hankel matrix of serial correlations. To that end, we define a statistical test for checking the linear dependence between vectors of serial correlations. It is shown that the test statistic TN considered is distributed asymptotically as a finite linear combination of independent chi-square random variables with one degree of freedom under the null hypothesis, whereas under the alternative hypothesis, TN/N converges in probability to a positive constant. These results allow us, in particular, to compute the asymptotic probability of making a specification error with the proposed procedure. Links to other methods based on the application of canonical analysis are discussed. A simulation experiment was done in order to study the performance of the procedure. It is seen that the graphical representation of TN, as a function of N, can be very useful in identifying the dynamic structure of ARMA models. Furthermore, for the model considered, the proposed identification procedure performs very well for series of 100 observations or more and reasonably well with short series of 50 observations.

On the Edgeworth expansion and bootstrap approximation for the cox regression model under random censorship

Abstract: We establish the one-term Edgeworth expansion for various statistics related to Cox semipara-metric regression model when the covariate is one-dimensional and the observations are i.i.d. We show that the bootstrap approximation method is second-order correct. The second-order-correct estimates of the sampling distribution can be obtained without Monte Carlo simulation. We pay special attention to the Studentized version of the statistics and show that their distributions are different from those of the original statistics to order n-½ RESUME Cet article developpe le premier terme de l'expansion de Edgeworth de certaines statistiques associees au modele de reegression semi-parametrique de Cox lorsque Ton dispose d'observations independantes et identiquement distribuees et d'une seule variable explicative. II y est montre que la methode d'approximation de Cyrano est correcte au deuxieme ordre et qu'a cet ordre, les estimations correctes de la loi d'echantillonnage peuvent ěbtenues sans avoir recours a des simulations de type Monte-Carlo. On etablit en outre que la loi de chaque statistique consideree et celle de sa version studentisee different a l'ordre n-½.

Asymptotic properties of a conditional maximum‐likelihood estimator

Abstract: Inference for a scalar interest parameter in the presence of nuisance parameters is considered in terms of the conditional maximum-likelihood estimator developed by Cox and Reid (1987). Parameter orthogonality is assumed throughout. The estimator is analyzed by means of stochastic asymptotic expansions in three cases: a scalar nuisance parameter, m nuisance parameters from m independent samples, and a vector nuisance parameter. In each case, the expansion for the conditional maximum-likelihood estimator is compared with that for the usual maximum-likelihood estimator. The means and variances are also compared. In each of the cases, the bias of the conditional maximum-likelihood estimator is unaffected by the nuisance parameter to first order. This is not so for the maximum-likelihood estimator. The assumption of parameter orthogonality is crucial in attaining this result. Regardless of parametrization, the difference in the two estimators is first-order and is deterministic to this order. On s'interesse a l'inference a propos d'un parametre particulier en presence d'un parametre de nuisance, cela en utilisant I'estimateur a vraisemblance maximale conditionnelle propose par Cox et Reid (1987). On suppose l'orthogonalite des parametres. Au moyen de developpements asymptotiques stochastiques I'estimateur est etudie dans trois situations: un parametre de nuisance scalaire, m parametres de nuisance provenant de m echantillons independants et un parametre de nuisance vectoriel. Dans chaque cas on compare le developpement associe a l'estimateur a vraisemblance maximale conditionnelle a celui associe a I'estimateur a vraisemblance maximale habituel. Les moyennes et les variances sont aussi comparees. Dans les trois situations le biais de I'estimateur a vraisemblance maximale conditionelle n'est pas affecte, au premier ordre, par le parametre de nuisance; ce n'est pas le cas pour l'estimateur a vraisemblance maximale. L'hypothese d'orthogonalite des parametres est cruciate pour la validite de ce resultat. Independamment de la parametrisation, la difference entre les deux estimateurs est du premier ordre et de type deterministe.

On the curtailment of sampling

Abstract: We address the problem of the curtailment or continuation of an experiment or trial at some interim point where say N observations are in hand and at least S > N observations had originally been scheduled for a decision. A Bayesian predictive approach is used to determine the probability that if one continued the trial with a further sample of size M where N +M ≥S, one would come to a particular decision regarding a parameter or a future observable. This point of view can also be applied to significance tests if one is willing to admit the calculation as a subjective assessment. Supposons qu'il soit prevu qu'une experience (ou essai) implique le prelevement de S observations avant la prise d'une decision. On considers la situation ou N > S observations sont deja prises et on se demande si, a la lumiere de celles-ci, il vaut mieux ecourter l'experience ou la poursuivre. A l'aide d'une approche bayesienne predictive on determine la probabilite que dans le cas ou un echantillon supplemental de taille M(N +M > S) est preleve, on arriverait a une decision particuliere a propos d'un parametre ou d'une valeur future. Ce point de vue peut ětre applique aux tests de signification si lon est prět a considerer les calculs comme une appreciation subjective.

Hierarchical bayesian curve fitting and smoothing

Abstract: Estimation of a smooth function is considered when observations on this function added with Gaussian errors are observed. The problem is formulated as a general linear model, and a hierarchical Bayesian approach is then used to study it. Credible bands are also developed for the function. Sensitivity analysis is conducted to determine the influence of the choice of priors on hyperparameters. Finally, the methodology is illustrated using real and simulated examples where it is compared with classical cubic splines. It is also shown that our approach provides a Bayesian solution to some problems in discrete time series. Nous etudierons le lissage d'une fonction lorsque les observations de cette fonction sont sujettes a des erreurs gaussiennes. Le probleme sera formule a l'aide d'un modele lineaire et nous utiliserons l'approche bayesienne hierarchique pour l'etudier. De plus nous developperons des bandes de credibilite pour le lissage. Une analyse de sensibilite sera faite pour determiner l'influence sur le lissage de la densite a priori sur les hyperparametres. Pour conclure, nous illustrerons cette nouvelle methodologie a l'aide de donnees reelles et d'une simulation; nous comparerons les resultats obtenus avec ceux fournis par les splines cubiques. Il sera aussi montre que cette approche foumit une solution bayesienne a quelques problemes en series chronologiques.

Orthogonalizing parametric link transformation families in binary regression analysis

Abstract: This paper studies the application of the orthogonalization technique of Cox and Reid (1987) to parametric families of link functions used in binary regression analysis. The explicit form of Cox and Reid's condition (4), for orthogonality at a point, is derived for arbitrary link families. This condition is used to determine a transform of a family introduced by Burr (1942) and Prentice (1975, 1976) which is locally orthogonal when the regression parameter is zero. Thus the benefits of having orthogonal parameters are limited to “small” regression effects. The extent to which approximate orthogonality holds for nonzero regression coefficients is investigated for two data sets from the literature. Two specific issues considered are: (1) the ability of orthogonal reparametrization to reduce the variability of the regression parameters caused by estimation of the link parameter and (2) the improved numerical stability (and hence interpretability) of regression estimates corresponding to different link parameters. On traite de l'application de la technique d'orthogonalisation de Cox et Reid (1987) a des families de fonctions de lien utilisees en regression binaire. Dans le cas de families arbitraires, on determine la forme explicite de la condition d'orthogonalite en un point (4) de Cox et Reid. Cette condition est utilisee afin de trouver une transformation pour une famille introduite par Burr (1942) et Prentice (1975, 1976), celle-ci etant localement orthogonale lorsque le parametre de regression est nul. Ainsi, les avantages decoulant de parametres orthogonaux sont limites aux “petites” composantes de regression. A partir de deux ensembles de donnees publiees on examine jusqu'a quel point on a approximativement orthogonalite lorsque les coefficients de regression ne sont pas nuls. Deux elements particuliers sont consideres: (1) la capacite de la reparametrisation orthogonale de reduire la variabilite des parametres de regression due a l'estimation d'un parametre de lien, et (2) l'amelioration de la stabilite numerique (et alors de la fiabilite de l'interpretation) des estimes correspondant a differents parametres de lien.

A robust biplot

Abstract: This paper introduces a robust biplot which is related to multivariate M-estimates. The n × p data matrix is first considered as a sample of size n from some p-variate population, and robust M-estimates of the population location vector and scatter matrix are calculated. In the construction of the biplot, each row of the data matrix is assigned a weight determined in the preliminary robust estimation. In a robust biplot, one can plot the variables in order to represent characteristics of the robust variance-covariance matrix: the length of the vector representing a variable is proportional to its robust standard deviation, while the cosine of the angle between two variables is approximately equal to their robust correlation. The proposed biplot also permits a meaningful representation of the variables in a robust principal-component analysis. The discrepancies between least-squares and robust biplots are illustrated in an example. Cet article propose un biplot robuste construit a l'aide de M-estimateurs multivaries. On considere d'abord la matrice nxp des donnees comme un echantillon de taille n d'une population ap variables et on calcule des M-estimations robustes du vecteur des parametres de position et de la matrice de dispersion theorique. Dans la construction du biplot, chaque ligne de la matrice des donnees recoit un poids determine dans l'estimation robuste preliminaire. Le biplot robuste peut ětre construit de telle sorte que le graphique des variables represente les composantes de la matrice de variances-covariances robuste: la longueur du vecteur representant une variable est proportionnelle a sa deviation standard robuste tandis que le cosinus de l'angle entre deux variables est egal au coefficient de correlation robuste entre ces deux variables. Le biplot propose permet en outre des representations des variables dans une analyse en composantes principales robuste. Un exemple permet de faire une comparaison du biplot des moindres carres avec le biplot robuste.

Limit theorems for a general weighted process under random censoring

Abstract: Necessary and sufficient conditions for weak and strong convergence are derived for the weighted version of a general process under random censoring. To be more explicit, this means that for this process complete analogues are obtained of the Chibisov-O'Reilly theorem, the Lai-Wellner Glivenko-Cantelli theorem, and the James law of the iterated logarithm for the empirical process. The process contains as special cases the so-called basic martingale, the empirical cumulative hazard process, and the product-limit process. As a tool we derive a Kiefer-process-type approximation of our process, which may be of independent interest.

Bayesian estimates of predictive value and related parameters of a diagnostic test

Abstract: Bayesian analysis of predictive values and related parameters of a diagnostic test are derived. In one case, the estimates are conditional on values of the prevalence of the disease; in the second case, the corresponding unconditional estimates are presented. Small-sample point estimates, posterior moments, and credibility intervals for all related parameters are obtained. Numerical methods of solution are also discussed. On s'interesse a une analyse bayesienne de la capacite de predire d'un test diagnostique, ainsi qu'a des parametres qui lui sont associes. On considere a la fois des estimateurs qui sont conditionnels a la frequence de la maladie et des estimateurs qui ne le sont pas. On obtient des estimateurs ponctuels pour de petits echantillons, des moments a posteriori et des intervalles de credibilite pour tous les parametres pertinents. Des methodes numeriques conduisant aux solutions sont aussi presentees.

Consistency of least‐squares estimator and its jackknife variance estimator in nonlinear models

Abstract: The purpose of this paper is twofold: (1) We establish the consistency of the least-squares estimator in a nonlinear modelyi = f(xi,θ) +σiei where the range of the parameter θ is noncompact, the regression function is unbounded, and the σi,'s are not necessarily equal. This extends the results in Jennrich (1969) and Wu (1981). (2) Under the same model, the jackknife estimator of the asymptotic covariance matrix of the least-squares estimator is shown to be consistent, which provides a theoretical justification of the empirical results in Duncan (1978) and the use of the jackknife method in large-sample inferences. RESUME Cet article a un double objectif: (1) de generaliser les resultats de Jennrich (1969) et de Wu (1981) en etablissant la convergence de l'estimateur des moindres carres dans le cadre du modele non lineaire y, = f(Xi,θ) + xie, lorsque le domaine de θ n'est pas compact, que la fonctionf de regression n'est pas bornee et que les σi ne sont pas forcement egaux; (2) sous les měmes conditions, de demontrer la convergence de l'estimateur jackknife de la matrice de variances-covariances asymptotique de l'estimateur des moindres carres, de facon a confirmer les resultats empiriques de Duncan (1978) et a justifier, au plan theorique, l'utilisation de la methode du jackknife dans les grands echantillons.

Simultaneous estimation of the hardy‐weinberg proportions

Abstract: We consider the problem of simultaneously estimating k + 1 related proportions, with a special emphasis on the estimation of Hardy-Weinberg (HW) proportions. We prove that the uniformly minimum-variance unbiased estimator (UMVUE) of two proportions which are individually admissible under squared-error loss are inadmissible in estimating the proportions jointly. Furthermore, rules that dominate the UMVUE are given. A Bayesian analysis is then presented to provide insight into this inadmissibility issue: The UMVUE is undesirable because the two estimators are Bayes rules corresponding to different priors. It is also shown that there does not exist a prior which yields the maximum-likelihood estimators simultaneously. When the risks of several estimators for the HW proportions are compared, it is seen that some Bayesian estimates yield significantly smaller risks over a large portion of the parameter space for small samples. However, the differences in risks become less significant as the sample size gets larger. On s'interesse a l'estimation simultanee de k + 1 proportions reliees. Une attention particuliere est apportee au cas des proportions de Hardy-Weinberg (HW). Dans le cas de deux proportions, on deemontre que l'estimateur sans biais a variance uniformement minimale (UMVUE) donne des estimateurs individuels admissibles sous la perte quadratique, mais un estimateur inadmissible pour le vecteur des proportions. Des regies de decision dominant le UMVUE sont donnees. Une analyse bayesienne vise a eclairer la question de l'inadmissibilite: le UMVUE engendre des estimateurs individuels qui sont bayesiens par rapport a des lois a priori differentes, c'est ce qui le rend indesirable en tant qu'estimateur du vecteur. On montre aussi qu'il n'existe pas de loi a priori engendrant simultanement les estimateurs de vraisemblance maximale. En comparant plusieurs estimateurs pour les proportions HW, on constate que pour de petits echantillons les estimateurs bayesiens engendrent des risques passablement inferieurs sur une portion appreciable de l'espace des parametres. Les differences au niveau du risque s'attenuent lorsque la taille echantillonnale augmente.

Small‐sample intervals for regression

Abstract: For the general linear regression model Y = Xη + e, we construct small-sample exponentially tilted empirical confidence intervals for a linear parameter 6 = aTη and for nonlinear functions of η. The coverage error for the intervals is Op(1/n), as shown in Tingley and Field (1990). The technique, though sample-based, does not require bootstrap resampling. The first step is calculation of an estimate for η. We have used a Mallows estimate. The algorithm applies whenever η is estimated as the solution of a system of equations having expected value 0. We include calculations of the relative efficiency of the estimator (compared with the classical least-squares estimate). The intervals are compared with asymptotic intervals as found, for example, in Hampel et at. (1986). We demonstrate that the procedure gives sensible intervals for small samples. Pour le modele de regression lineaire Y = Xeta; + e, on construit, a partir de petits echantillons, des intervalles de confiance pour un parametre lineaire θ+Tη ainsi que pour des fonctions non lineaires de η. Ces intervalles font intervenir une transformation de type exponentiel appliquee a une distribution empirique. L'erreur relativement a la couverture est Op(1/n), voir Tingley et Field (1990). Bien que la technique proposee soit basee sur l'echantillon, elle ne fait pas intervenir le reechantillonnage “bootstrap”. La premiere etape consiste a obtenir une estimation de η; l'estimateur de Mallows est utilise. L'algorithme propose s'applique dans tous les cas ou η est estime par la solution d'un systeme d'equations ayant une esperance nulle. Des calculs illustrent l'efficacite relative de l'estimateur (par rapport a l'estimateur de moindres carres usuel). Les intervalles sont compares aux intervalles bases sur les lois asymptotiques tels que presentes, par exemple, par Hampel et al. (1986). On deemontre que la procedure suggeree donne des intervalles raisonnables lorsque Ton travaille a partir de petits echantillons.

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

Abstract: We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order Op(T-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed. Nous considerons l'estimateur des moindres carres du parametre autoregressif dans un modele saisonnier quasi-integre. En se basant sur l'etude de Chan (1989) qui a obtenu la distribution limite, nous donnons une expression explicite pour la fonction generatrice de moments limite appropriee. Nous utilisons cette fonction pour calculer, par integration numerique, les valeurs critiques de la distribution asymptotique pour diverses periodes saisonnieres. Ces resultats sont generalises en considerant le developpement asymptotique stochastique d'ordre Op(T-1). Les valeurs critiques de cette distribution sont aussi calculees a partir de la fonction generatrice de moments correspondante. Nous discutons aussi la qualite de ces approximations pour des echantillons finis.

Graphical methods for ranking data

Abstract: Graphical methods are presented for the analysis of ranking data collected from g groups of rankers. The data provided by a single individual consist of the ranks of r objects. The sample space is the space of all permutations and has cardinality r! In order to reduce the dimensionality of the data and to study the interrelationships among rankers and items, a two-stage approach is proposed. First, transformations motivated by various metrics on permutations are defined. In particular, the Kendall metric gives rise to pairwise comparisons. Then, the transformed data are analyzed using results in connection with the generalized singular-value decomposition of a matrix. The methods are illustrated on two examples. RESUME Des methodes graphiques sont presentees pour l'analyse de donnees sous forme de rang provenant de g groupes de juges. Chaque juge fournit individuellement un classement de r objets distincts. L'espace echantillonnal, forme de toutes les permutations des r premiers entiers, est alors de dimension r! Afin de reduire la dimension de l'espace des observations et d'etudier Ie rapport entre les individus et les rangs assignes, on introduit d'abord certaines transformations motivees par des mesures de distance particulieres sur l'espace des permutations. La metrique de Kendall conduit notamment a des comparaisons par paires. Une fois transformees, les donnees sont ensuite analysees a l'aide de resultats portant sur la decomposition en valeurs signulieres generalisee d'une matrice. Les methodes qui en decoulent sont illustrees par deux exemples.

A general method of finding a minimax estimator of a distribution function when no equalizer rule is available

Abstract: We propose a feasible general method for finding a minimax estimator of an unknown distribution function F in the nonparametric problem. As an application, some minimax estimators are proposed. Furthermore, some minimax binomial parametric problems are studied. Nous proposons une methode generate afin de trouver, dans un contexte non-parametrique, un estimateur minimax pour une fonction de distribution inconnue. Plusieurs exemples illustrent comment cette methode peut ětre mise en application. Un lien est fait avec certains problemes de type minimax dans le contexte binomial.

13 Zero-one laws for large order statistics

Abstract: Fix r ≥ 1, and let {Mnr} be the rth largest of {X1,X2,…Xn}, where X1,X2,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[Mnr ≤ un i.o.] = 0 or 1 according as the series Σ∞n=3Fn(un)(log log n)r/n converges or diverges, for any real sequence {un} such that n{1 -F(un)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1. Soit X1,X2,… une suite de variables aleatoires i.i.d. de distribution F; etant donne un r ≥ 1 fixe, denotons par {Mnr} les rieme plus grands elements de {X1,…Xn}. On demontre que P[Mnr ≤ un inf.s.] = 0 ou 1 suivant que la serie Σ∞n=3Fn(un)(log log n)r/n converge ou diverge, cela pour toute suite reelle {un} telle que n{1 — F(un)} est non decroissante et divergente. Cela generalise un resultat de Bamdorff-Nielsen (1961) pour le cas r = 1.

Testing retrospective hypotheses

Abstract: This paper develops a conditional approach to testing hypotheses set up after viewing the data. For example, suppose Xi are estimates of location parameters θi, i = 1,…n. We show how to compute p-values for testing whether θ1 is one of the three largest θi after observing that X1 is one of the three largest Xi, or for testing whether θ1 > θ2 > … > θn after observing X1 >X2> … >Xn. On s'interesse au probleme du test d'hypotheses formulees apres l'examen des donnees. Une approche conditionnelle est proposee. Supposons que Xi- est une estimation d'un parametre de position θ1, i = 1…n. On montre comment tester si θ1 est parmi les trois plus grands θ1, apres avoir observe que X1 est une des trois plus grandes Xi, ou encore comment tester θ1 > θ2 > … > θn apres avoir observe que X1 > X2 > … > Xn.

Some topics of current interest in clinical trials

Abstract: After a short review of the development of the modern approach to the design of clinical trials, attention is focused on the wide variety of trials, their ethical constraints, the criteria for stopping trials, and the possible use of Bayesian methods. Brief discussions are then presented of two specific topics: the analysis of categorical data, and the replication of trials with the consequent need for overviews. Apres un survol des progres recents a propos de la planification des essais cliniques, on s'interesse a la grande variete des essais, aux considerations d'ordre ethique, aux criteres d'arrět pour ces essais et a l'utilisation possible de methodes bayesiennes. Deux sujets specifiques sont alors abordes: I'analyse des donnees categorielles et la repetition des essais avec les analyses qui en decoulent.