Semiparametric model

In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components. A statistical model is a parameterized family of distributions: {P_\theta: \theta \in \Theta} indexed by a parameter \theta.

A parametric model is a model in which the indexing parameter \theta is a vector in k-dimensional Euclidean space, for some nonnegative integer k. Thus, \theta is finite-dimensional, and \Theta \subseteq \mathbb{R}^k.
With a nonparametric model, the set of possible values of the parameter \theta is a subset of some space V, which is not necessarily finite-dimensional. For example, we might consider the set of all distributions with mean 0. Such spaces are vector spaces with topological structure, but may not be finite-dimensional as vector spaces. Thus, \Theta \subseteq V for some possibly infinite-dimensional spac

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MATH-449: Biostatistics

This course covers statistical methods that are widely used in medicine and biology. A key topic is the analysis of longitudinal data: that is, methods to evaluate exposures, effects and outcomes that are functions of time. While motivated by real-life problems, some of the material will be abstract

MICRO-513: Signal processing for functional brain imaging

Computational methods for the analysis of human brain imaging data

Estimateur (statistique)

En statistique, un estimateur est une fonction permettant d'estimer un moment d'une loi de probabilité (comme son espérance ou sa variance). Il peut par exemple servir à estimer certaines caractérist

Modèle statistique

Un modèle statistique est une description mathématique approximative du mécanisme qui a généré les observations, que l'on suppose être un processus stochastique et non un processus déterministe. Il s

Statistique

La statistique est la discipline qui étudie des phénomènes à travers la collecte de données, leur traitement, leur analyse, l'interprétation des résultats et leur présentation afin de rendre ces don

Safe density ratio modeling

Kjell Konis

An important problem in logistic regression modeling is the existence of the maximum likelihood estimators. In particular, when the sample size is small, the maximum likelihood estimator of the regression parameters does not exist if the data are completely, or quasicompletely separated. Recognizing that this phenomenon has a serious impact on the fitting of the density ratio model - which is a semiparametric model whose profile empirical log-likelihood has the logistic form because of the equivalence between prospective and retrospective sampling - we suggest a linear programming methodology for examining whether the maximum likelihood estimators of the finite dimensional parameter vector of the model exist. It is shown that the methodology can be effectively utilized in the analysis of case-control gene expression data by identifying cases where the density ratio model cannot be applied. It is demonstrated that naive application of the density ratio model yields erroneous conclusions. (C) 2009 Elsevier B.V. All rights reserved.

2009

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Mixture models for multivariate extremes

Marc-Olivier Boldi

This thesis is a contribution to multivariate extreme value statistics. The tail of a multivariate distribution function is characterized by its spectral distribution, for which we propose a new semi-parametric model based on mixtures of Dirichlet distributions. To estimate the components of this model, reversible jump Monte Carlo Markov chain and EM algorithms are developed. Their performances are illustrated on real and simulated data, obtained using new representations of the extremal logistic and Dirichlet models. In parallel with the estimation of the spectral distribution, extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not. This selection is achieved by a new method based on heuristic arguments. It allows a selection independent of the dimension of the data. Its performance is illustrated on real and simulated data. Primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure, for which the estimation of the spectral measure is a means rather than an end. These two issues are addressed. For the first, a Monte Carlo method is developed based on simulation of extremes. It is compared with classical and new methods of the literature. For the second one, an original conditional dependence analysis is proposed, which enlightens various aspects of the dependence structure of the data. Examples using real data sets are given. In the last part, the semi-parametric model and the presented methods are extended to spatial extremes. It is made possible by considering the spectral distribution as the distribution of a random probability, an original viewpoint adopted throughout this thesis. Classical multivariate extremes are extended to extremes of random measures. The application is illustrated on rainfall data in China.

EPFL2004

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A mixture model for multivariate extremes

Marc-Olivier Boldi, Anthony Christopher Davison

The spectral density function plays a key role in ﬁtting the tail of multivariate extremal data and so in estimating probabilities of rare events. This function satisﬁes moment constraints but unlike the univariate extreme value distributions has no simple parametric form. Parameterized subfamilies of spectral densities have been suggested for use in applications, and nonparametric estimation procedures have been proposed, but semiparametric models for multivariate extremes have hitherto received little attention. We show that mixtures of Dirichlet distributions satisfying the moment constraints are weakly dense in the class of all nonparametric spectral densities, and discuss frequentist and Bayesian inference in this class based on the EM algorithm and reversible jump Markov chain Monte Carlo simulation. We illustrate the ideas using simulated and real data.

2007

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