Showing papers in "International Statistical Review in 2008"

Robust Likelihood Methods Based on the Skew-t and Related Distributions

Abstract: Summary The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew-t distribution is explored in more detail and reasons are given to adopt this option as a sensible general-purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide-ranging examples with real data are provided in support of the claim. Resume Le probleme de la robustesse est attaque en adoptant une classe parametrique de distributions qui sont suffisamment flexibles pour representer le comportement des observations. Dans une variete de cas pratiques, une option raisonnable est de considerer des distributions qui incluent des parametres pour regler leur asymetrie et leur aplatissement. Comme representant specifique de cette approche, la distribution t asymetrique est exploree plus en detail et des raisons sont apportees pour adopter cette option comme un compromis judicieux et a tous usages entre la robustesse et la simplicite du traitement et de l'interpretation des resultats. Quelques arguments theoriques, les resultats de simulations et divers exemples sur des donnees reelles sont fournis afin de soutenir cette affirmation.

An Overview of Six Sigma

Abstract: Summary WeprovideanintroductoryoverviewoftheSixSigmadevelopmentandimprovementprocesses.A historical perspective is provided. We also discuss the statistical methods frequently used within this framework and briefly comment on the impact of Six Sigma on the practice of statistics in industry.

Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables

Abstract: Summary The aim of this paper is to convey to a wider audience of applied statisticians that nonparametric (matching) estimation methods can be a very convenient tool to overcome problems with endogenous control variables. In empirical research one is often interested in the causal effect of a variable X on some outcome variable Y. With observational data, i.e. in the absence of random assignment, the correlation between X and Y generally does not reflect the treatment effect but is confounded by differences in observed and unobserved characteristics. Econometricians often use two different approaches to overcome this problem of confounding by other characteristics. First, controlling for observed characteristics, often referred to as selection on observables, or instrumental variables regression, usually with additional control variables. Instrumental variables estimation is probably the most important estimator in applied work. In many applications, these control variables are themselves correlated with the error term, making ordinary least squares and two-stage least squares inconsistent. The usual solution is to search for additional instrumental variables for these endogenous control variables, which is often difficult. We argue that nonparametric methods help to reduce the number of instruments needed. In fact, we need only one instrument whereas with conventional approaches one may need two, three or even more instruments for consistency. Nonparametric matching estimators permitconsistent estimation without the need for (additional) instrumental variables and permit arbitrary functional forms and treatment effect heterogeneity. Resume Cet article demontre que l'estimation non parametrique peut etre utile pour resoudre le probleme lie aux variables de controle endogenes. L'objectif de nombreux travaux empiriques est d'identifier l'effet causal d'une variable X sur une variable dependante Y. La correlation entre X et Y qui est observee dans les donnees ne reflete generalement pas l'effet du traitement car celui-ci est masque par les differences dans les caracteristiques (observables ou non) des deux groupes. Les econometres resolvent souvent ce probleme d'une des deux facons suivantes: (1) en controlant pour la selection qui est liee aux caracteristiques observees ou (2) en utilisant des instruments, qui ne sont frequemment valides que conditionnellement a d'autres variables de controle. L'estimation basee sur des instruments (IV) est probablement la methode la plus importante dans la recherche appliquee. Dans beaucoup d'applications ces variables de controle sont elles-memes suspectees d'endogeneite ce qui rendrait OLS et 2SLS inconsistants. La solution habituelle est de chercher des instruments supplementaires pour ces variables de controle endogenes, mais cette approche est tres difficile en pratique. Nous montrons dans cet article qu'utiliser une methode instrumentale non parametrique reduit le nombre des instruments necessaires. En effet, nous n'avons besoin dans ce cas que d'un seul instrument alors que les methodes conventionnelles necessitent deux, trois ou plus encore d'instruments pour garantir leur consistance. Il existe des estimateurs non parametriques bases sur le matching qui convergent a la vitesse racine de n sans exiger des instruments supplementaires et qui ne restreignent ni la forme fonctionnelle ni l'heterogeneite de l'effet du traitement.

Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model

Abstract: A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 ?? level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation.

Stationary-Increment Variance-Gamma and t Models: Simulation and Parameter Estimation

Abstract: Summary We detail a method of simulating data from long range dependent processes with variance-gamma or t distributed increments, test various estimation procedures [method of moments (MOM), product-density maximum likelihood (PMLE), non-standard minimumχ2and empirical characteristic function estimation] on the data, and assess the performance of each. The investigation is motivated by the apparent poor performance of the MOM technique using real data (Tjetjep & Seneta, 2006); and the need to assess the performance of PMLE for our dependent data models. In the simulations considered the product-density method performs favourably. Resume Nous detaillons une methode de simulation de donnees relatives a des processus a accroissements de lois Variance-Gamma ou t. Nous testons sur ces donnees diverses procedures d'estimation (methode des moments, maximum de vraisemblance, χ2 non standard minimum, et fonction caracteristique empirique) et nous evaluons la performance de chacune. Cette etude est motivee par le peu d'efficacite de la technique des moments appliquee a des donnees reelles (Tjetjep et Seneta 2006) et par le besoin d'evaluer la performance de la methode du maximum de vraisemblance relative a une densite produit appliquee a nos modeles de donnees dependantes. Dans les simulations que nous avons faites la methode de la densite produit donne des resultats satisfaisants.

Basic Process Capability Indices: An Expository Review

Abstract: Summary A review of the four basic process capability indices has been made. The interrelationship among these indices has been highlighted. Attention has been drawn to their drawbacks. The relation of these indices to the proportion nonconforming has been dwelt upon and the requirement of the adequate sample size has been emphasized. Cautionary remarks on the use of these indices in the case of nonnormal distributions, skewed distributions, and autocorrelated data are also presented. The effect of measurement error on process capability indices has been dealt with in great detail.

Anomalies in the Foundations of Ridge Regression

Abstract: Summary Errors persist in ridge regression, its foundations, and its usage, as set forth in Hoerl & Kennard (1970) and elsewhere. Ridge estimators need not be minimizing, nor a prospective ridge parameter be admissible. Conventional estimators are not LaGrange's solutions constrained to fixed lengths, as claimed, since such solutions are singular. Of a massive literature on estimation, prediction, cross–validation, choice of ridge parameter, and related issues, little emanates from constrained optimization to include inequality constraints. The problem traces to a misapplication of LaGrange's Principle, unrecognized singularities, and misplaced links between constraints and ridge parameters. Alternative principles, based on condition numbers, are seen to validate both conventional ridge and surrogate ridge regression to be defined. Numerical studies illustrate that ridge regression as practiced often exhibits pathologies it is intended to redress. Resume Les erreurs persistent dans la regression ridge, ses bases, et son utilisation, comme determine en Hoerl et Kennard (1970) et plus tard. Il ne faut ni que les estimateurs ridge se reduisent au minimum ni qu'un parametre ridge soit admissible. Les estimateurs conventionnels ne sont pas les solutions de Lagrange contraintes aux longueurs fixes, comme souvent pretendu, car de telles solutions sont singulieres. D'une litterature vaste—sur l'evaluation, la prevision, la validation croisee, le choix du parametre ridge, et sujets allies, sujets collectivement connus sous le nom de regression ridge—peu est issu de la minimisation contrainte, meme vis a vis les contraintes d'inegalitie. Le probleme remonte a une mauvaise application du principe de Lagrange, au manque d'identifier des singularites, et aux liens mal places entre les contraintes et les parame tres ridge. Des principes alternatifs, bases sur des numeraux de condition, peuvent etre vus comme validant ridge conventionnelle et la regression de ridge succedanee, ce dernier aetre defini. Les etudes numeriques illustrent que la regression ridge, comme practiquee, montrent souvent des pathologies qu'il vise a redresser.

On the Practice of Rescaling Covariates

Abstract: Summary Whether doing parametric or nonparametric regression with shrinkage, thresholding, penalized likelihood, Bayesian posterior estimators (e.g., ridge regression, lasso, principal component regression, waveshrink or Markov random field), it is common practice to rescale covariates by dividing by their respective standard errors ρ. The stated goal of this operation is to provide unitless covariates to compare like with like, especially when penalized likelihood or prior distributions are used. We contend that this vision is too simplistic. Instead, we propose to take into account a more essential component of the structure of the regression matrix by rescaling the covariates based on the diagonal elements of the covariance matrix Σ of the maximum-likelihood estimator. We illustrate the differences between the standard ρ- and proposed Σ-rescalings with various estimators and data sets. Resume Que l'on utilise un modele de regression parametrique ou non-parametrique, par retrecissement, seuillage, vraisemblance penalisee ou Bayesien (ex. regression ridge, lasso, regression en composantes principales, waveshrink, champ Markovien), il est commun de standardiser les variables explicatives en les divisant par leurs ecarts types ρ respectifs. Le but affiche de cette operation est de creer des variables sans unites pour pouvoir les comparer entre elles, en particulier quand l'estimateur est base sur la vraisemblance penalisee ou une distribution a priori. Nous attendons prouver que cette vision est trop simpliste. Nous proposons de plutot considerer un element plus essentiel de la matrice de regression en standardisant les variables explicatives a partir des elements diagonaux de la matrice de covariance Σ de l'estimateur du maximum de vraisemblance. Nous illustrons les differences entre la standardisation ρ et la standarisation Σ avec des estimateurs et des donnees varies. Mots cles: champ markovien, distribution a priori lη, lasso, ondelettes, regression en composantes principales, regression ridge.

Estimating Trends with Percentage of Smoothness Chosen by the User

Abstract: Summary This work presents a method for estimating trends of economic time series that allows the user to fix at the outset the desired percentage of smoothness for the trend. The calculations are based on the Hodrick-Prescott (HP) filter usually employed in business cycle analysis. The situation considered here is not related to that kind of analysis, but with describing the dynamic behaviour of the series by way of a smooth curve. To apply the filter, the user has to specify a smoothing constant that determines the dynamic behaviour of the trend. A new method that formalizes the concept of trend smoothness is proposed here to choose that constant. Smoothness of the trend is measured in percentage terms with the aid of an index related to the underlying statistical model of the HP filter. Empirical illustrations are provided using data on Mexico's GDP.

An Example of Data Technology Product: A Control Chart for Weibull Processes

Abstract: Summary This work proposes a new Shewhart-type control chart of the Weibull percentik (i.e. the reliable life) as a practical example of a product attained following the Data Technology (DT) approach. DT is briefly introduced as a new discipline defined apart from Information Technology (IT). Following this approach, some specific Bayes estimators are selected from literature and then used to build the above new chart. These estimators allow to improve the control making use of any available kind of data (statistical and non-statistical). The operative steps of DT approach are fully explained. The results are illustrated by means of a real applicative example.

Methods for Generating Longitudinally Correlated Binary Data

Abstract: Summary The analysis of longitudinally correlated binary data has attracted considerable attention of late. Since the estimation of parameters in models for such data is based on asymptotic theory, it is necessary to investigate the small-sample properties of estimators by simulation. In this paper, we review the mechanisms that have been proposed for generating longitudinally correlated binary data. We compare and contrast these models with regard to various features, including computational efficiency, flexibility and the range restrictions that they impose on the longitudinal association parameters. Some extensions to the data generation mechanism originally suggested by Kanter (1975) are proposed. Resume L'analyse des donnees longitudinales correlees fait recemment l'objet d'un grand interet. Comme l'estimation des parametres des modeles pour de telles donnees est souvent basee sur des etudes asymptotiques, il est necessaire de proceder a des simulations pour explorer les proprietes des estimateurs en petits echantillonages. Dans ce papier, nous presentons une revue des methodes qui ont ete proposees pour generer des donnees binaires longitudinales correlees. Nous les comparons sous differents aspects, notamment en termes d'efficience, flexibilite, et des restrictions qu'elles peuvent avoir sur les parametres dits d'association longitudinale. Quelques extensions, de la methode suggeree par Kanter (1975) pour generer de telles donnees, sont aussi proposees.

Regression Revisited

Abstract: Summary Sir Francis Galton introduced median regression and the use of the quantile function to describe distributions. Very early on the tradition moved to mean regression and the universal use of the Normal distribution, either as the natural ‘error’ distribution or as one forced by transformation. Though the introduction of ‘quantile regression’ refocused attention on the shape of the variability about the line, it uses nonparametric approaches and so ignores the actual distribution of the ‘error’ term. This paper seeks to show how Galton's approach enables the complete regression model, deterministic and stochastic elements, to be modelled, fitted and investigated. The emphasis is on the range of models that can be used for the stochastic element. It is noted that as the deterministic terms can be built up from components, so to, using quantile functions, can the stochastic element. The model may thus be treated in both modelling and fitting as a unity. Some evidence is presented to justify the use of a much wider range of distributional models than is usually considered and to emphasize their flexibility in extending regression models. Resume Sir Francis Galton (1822–1911) introduisit la regression mediane et l'utilisation de la fonction quartile pour decrire les distributions. Peu apres, on entendait que ces termes signifiaient la regression et l'utilisation universelle de la distribution normale; cette distribution etait consideree soit comme la distribution naturelle des erreurs, soit comme celle forcee par des transformations. Bien que l'introduction de la regression quantile ait attire une nouvelle fois l'attention sur la forme de la dispersion autour de la droite de regression, elle utilise des methodes nonparametriques et ne tient donc pas compte de la distribution reelle du terme d'erreur. Cet article cherche a demontrer comment la methode galtonienne permet la modelisation, le lissage et l'etude du modele de regression dans son entier, c'est -a-dire l'element deterministe ainsi que l'element stochastique. Une importance particuliere est accordee a l'ensemble des modeles dont on peut se servir pour considerer l'element stochastique. De meme qu'il est possible d'etablir les termes deterministes a partir des composantes, nous notons que l'element stochastique peut aussi etre aborde de la meme facon, a l'aide des fonctions quartiles. Le modele peut donc etre considere comme une entite integrale, tant pour la modelisation que pour le lissage. Nous apportons des preuves pour justifier l'utilisation d'un plus vaste ensemble de modeles distributionnels que l'on aborde d'habitude et pour souligner leur flexibilite en ce qui concerne l'extension des modeles de regression.

On Average Predictive Comparisons and Interactions

Abstract: Summary In a regression context, consider the difference in expected outcome associated with a particular difference in one of the input variables. If the true regression relationship involves interactions, then this predictive comparison can depend on the values of the other input variables. Therefore, one may wish to consider an average predictive comparison as a target of inference, where the averaging is with respect to the population distribution of the input variables. We consider inferences about such targets, with emphasis on inferential performance when the regression model is misspecified. Particularly, in light of the difficulties in dealing with interaction terms in regression models, we examine inferences about average predictive comparisons when additive models are fitted to relationships truly involving pairwise interaction terms. We identify some circumstances where such inferences are consistent despite the model misspecification, notably when the input variables are independent, or have a multivariate normal distribution. Resume Dans un contexte de regression, considerons la difference de la valeur esperee de la variable d'interet, associee a une difference donnee d'une des variables d'entree. Si la relation de regression veritable implique des termes d'interaction, alors cette comparaison predictive peut dependre de la valeur des autres variables d'entree. Ainsi, il peut etre souhaitable de considerer la comparaison predictive moyenne comme objet d'inference, ou la moyenne est calculee sur la distribution des variables d'entree dans la population. Nous nous interessons a ce contexte d'inference, et plus particulierement a la performance inferentielle lorsque le modele de regression est mal specifie. A la lumiere des difficultes existantes de traiter les termes d'interaction dans les modeles de regression, nous examinons l'inference sur les comparaisons predictives moyennes lorsque des modeles additifs sont ajustes, alors que les vraies relations impliquent des termes d'interaction deux-a-deux. Nous identifions quelques situations ou l'inference est convergente malgre la mauvaise specification du modele, notamment lorsque les variables d'entree sont independantes ou possedent une loi normale multivariee.

A Bayesian Multiple Comparison Procedure for Order‐Restricted Mixed Models

Abstract: Summary A Bayesian hierarchical mixed model is developed for multiple comparisons under a simple order restriction. The model facilitates inferences on the successive differences of the population means, for which we choose independent prior distributions that are mixtures of an exponential distribution and a discrete distribution with its entire mass at zero. We employ Markov Chain Monte Carlo (MCMC) techniques to obtain parameter estimates and estimates of the posterior probabilities that any two of the means are equal. The latter estimates allow one both to determine if any two means are significantly different and to test the homogeneity of all of the means. We investigate the performance of the model-based inferences with simulated data sets, focusing on parameter estimation and successive-mean comparisons using posterior probabilities. We then illustrate the utility of the model in an application based on data from a study designed to reduce lead blood concentrations in children with elevated levels. Our results show that the proposed hierarchical model can effectively unify parameter estimation, tests of hypotheses and multiple comparisons in one setting. Resume Un modele Bayesien hierarchique mixte est developpe pour des comparaisons multiples avec une simple restriction d'ordre. Le modele facilite les inferences sur les differences successives des moyennes de population, pour lesquelles nous choisissons des distributions independantes prealables qui sont des melanges d'une distribution exponentielle et d'une distribution discrete avec sa masse entiere a zero. Nous employons les techniques de la chaine de Markov Monte Carlo pour obtenir des estimations des parametres et des estimations des probabilites posterieures que deux quelconques des moyennes sont egales. Les seconds estimateurs permettent a chacun de determiner si deux quelconques des moyennes sont significativement differentes et de tester l'homogeneite de toutes les moyennes. Nous etudions la performance des inferences basees sur des modeles avec des jeux de donnees simulees, en se concentrant sur l'estimation du parametre et des comparaisons successives moyennes utilisant des probabilites posterieures. Nous illustrons ensuite l'utilite du modele dans une application basee sur les donnees d'une etude destinee a reduire les concentrations de plomb sanguin chez les enfants avec des niveaux eleves. Nos resultats montrent que le modele hierarchique propose peut efficacement unifier l'estimation des parametres, les hypotheses de tests et les comparaisons multiples dans un seul cadre.

Probability Integrals of the Multivariate t Distribution

Abstract: Summary Results on probability integrals of multivariate t distributions are reviewed. The results discussed include: Dunnett and Sobel's probability integrals, Gupta and Sobel's probability integrals, John's probability integrals, Amos and Bulgren's probability integrals, Steffens' non-central probabilities, Dutt's probability integrals, Amos' probability integral, Fujikoshi's probability integrals, probabilities of cone, probabilities of convex polyhedra, probabilities of linear inequalities, maximum probability content, and Monte Carlo evaluation. Resume On examine les resultats d'integrales de probabilite de distributions multivariees t. Les resultats discutes incluent: integrales de probabilite de Dunnett et Sobel, integrales de probabilite de Gupta et Sobel, integrales de probabilite de John, integrales de probabilite de Amos et Bulgren, probabilites non centrees de Steffen, integrales de probabilite de Dutt, integrale de probabilite de Amos, integrales de probabilite de Fujikoshi, probabilites de cones, probabilites de polyedres convexes, probabilites d'inegalites lineaires, probabilite maximale, evaluation de Monte Carlo.

Tests of Bivariate Exchangeability

Abstract: Summary We discuss the hypothesis of bivariate exchangeability and show that testing bivariate exchangeability is related to the two-sample testing of equality of distribution functions. We consider three test statistics based on the ordering of the Euclidean interpoint distances. The runs test of exchangeability counts the runs among the observations and their mirror images on the minimal spanning tree. The nearest neighbour test of exchangeability is based on the number of nearest neighbour type coincidences among the observations and their folded images on the plane. The rank test of exchangeability compares the within and between ranks of the interpoint distances. We also consider the sign test of exchangeability, which uses the signs of the observations in specific regions, and a bootstrap test of exchangeability based on the maximum distance between the mirror images. We compare the power of these methods in a Monte Carlo study which shows different power orderings of the methods, depending on the alternative hypothesis. Resume Nous discutons le probleme du test de l'hypothese d'echangeabilite bivariee, en liaison avec celui de l'hypothese d'egalite de deux fonctions de repartition. Nous considerons trois types de statistiques de test fondees sur les distances euclidiennes entre les observations et leurs symetriques. Les tests de sequences denombrent les sequences dans les suites formees par les observations et leurs images-miroir dans l'arbre generateur minimal. Les tests de plus proches voisins considerent le cardinal de l'intersection des ensembles de plus proches voisins dans l'ensemble forme par les observations et leurs images-miroir. Les tests de rangs sont fondes sur les rangs des distances euclidiennes entre les observations. Nous prenons aussi en consideration un test de signes dans lequel les signes sont ceux des observations appartenant a certaines regions specifiques, et un test de type reechantillonnage fonde sur les distances maximales entre les observations et leur image-miroir. Une etude par simulation des puissances comparees de ces divers tests est presentee et commentee.

Aristotle and Earthquake Data: A Historical Note

Abstract: Summary The earliest known written work on earthquake theory was written by Aristotle (c. 330 B.C.). Aristotle had collected a lot of earthquake data and therefore we assume (or rather speculate) he had to organize, to classify and to summarize them in order to use and describe them in his work. With respect to the history of Statistics, the Aristotelian text on earthquake theory is probably the earliest written application of statistical tables on collected data. Resume Le tout premier travail ecrit qu'on connait sur la theorie de tremblement de terre, a eteecrit par Aristote (330 avant J.C.). Aristote avait rassemble beaucoup de donnees sur le tremblement de terre, donc nous supposons (ou plutot speculons) qu' il devait les organiser, classifier et resumer afin de les employer et les decrire dans son travail. En ce qui concerne l'histoire des statistiques, le texte aristotelicien sur la theorie de tremblement de terre est probablement la toute premiere application ecrite sur les tableaux statistiques des donnees rassemblees.

Many-valued Logic in Multistate and Vague Stochastic Systems

Abstract: Summary The state of the art in coherent structure theory is driven by two assertions, both of which are limiting: (1) all units of a system can exist in one of two states, failed or functioning; and (2) at any point in time, each unit can exist in only one of the above states. In actuality, units can exist in more than two states, and it is possible that a unit can simultaneously exist in more than one state. This latter feature is a consequence of the view that it may not be possible to precisely define the subsets of a set of states; such subsets are called vague. The first limitation has been addressed via work labeled ‘multistate systems’; however, this work has not capitalized on the mathematics of many-valued propositions in logic. Here, we invoke its truth tables to define the structure function of multistate systems and then harness our results in the context of vagueness. A key contribution of this paper is to argue that many-valued logic is a common platform for studying both multistate and vague systems but, to do so, it is necessary to lean on several principles of statistical inference. Resume L'etat de l'art dans la theorie de structure coherente est guide par deux assertions qui sont tous deux limitants : (1) toutes les unites d'un systeme peuvent exister dans un de deux etats, defaillant ou fonctionnant; et (2) a n'importe quel moment, chaque unite peut seulement exister dans un des susdits etats. En realite, les unites peuvent exister dans plus de deux etats et c'est possible qu'une unite puisse simultanement exister dans plus d'un etat. Cette derniere caracteristique est une consequence de l'opinion qu'il ne soit peut-etre pas possible de definir avec precision les sous-ensembles d'un ensemble d'etats; on appelle de tels sous-ensembles vagues. La premiere restriction a ete adressee par les methodes appelees “systemes multi-etats”; pourtant, ces methodes n'ont pas pris avantage des mathematiques sur les propositions multivalues en logique. Ici, nous invoquons ses tables de verite pour definir la fonction des systemes multi-etats et exploiter ensuite nos resultats dans le contexte d'ambiguite. Une contribution cle de ce papier est d'argumenter que la logique de plusieurs values est une plateforme commune pour etudier tant les systemes multi-etats que les systemes vagues, mais pour faire ceci, il est necessaire de se baser sur plusieurs principes d'inference statistique.