190 research outputs found
A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation
Approximate Bayesian computation (ABC) methods make use of comparisons
between simulated and observed summary statistics to overcome the problem of
computationally intractable likelihood functions. As the practical
implementation of ABC requires computations based on vectors of summary
statistics, rather than full data sets, a central question is how to derive
low-dimensional summary statistics from the observed data with minimal loss of
information. In this article we provide a comprehensive review and comparison
of the performance of the principal methods of dimension reduction proposed in
the ABC literature. The methods are split into three nonmutually exclusive
classes consisting of best subset selection methods, projection techniques and
regularization. In addition, we introduce two new methods of dimension
reduction. The first is a best subset selection method based on Akaike and
Bayesian information criteria, and the second uses ridge regression as a
regularization procedure. We illustrate the performance of these dimension
reduction techniques through the analysis of three challenging models and data
sets.Comment: Published in at http://dx.doi.org/10.1214/12-STS406 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Conditional Transformation Models
The ultimate goal of regression analysis is to obtain information about the
conditional distribution of a response given a set of explanatory variables.
This goal is, however, seldom achieved because most established regression
models only estimate the conditional mean as a function of the explanatory
variables and assume that higher moments are not affected by the regressors.
The underlying reason for such a restriction is the assumption of additivity of
signal and noise. We propose to relax this common assumption in the framework
of transformation models. The novel class of semiparametric regression models
proposed herein allows transformation functions to depend on explanatory
variables. These transformation functions are estimated by regularised
optimisation of scoring rules for probabilistic forecasts, e.g. the continuous
ranked probability score. The corresponding estimated conditional distribution
functions are consistent. Conditional transformation models are potentially
useful for describing possible heteroscedasticity, comparing spatially varying
distributions, identifying extreme events, deriving prediction intervals and
selecting variables beyond mean regression effects. An empirical investigation
based on a heteroscedastic varying coefficient simulation model demonstrates
that semiparametric estimation of conditional distribution functions can be
more beneficial than kernel-based non-parametric approaches or parametric
generalised additive models for location, scale and shape
Endogenous semiparametric binary choice models with heteroscedasticity
In this paper we consider endogenous regressors in the binary choice model under a weak median exclusion restriction, but without further specification of the distribution of the unobserved random components. Our reduced form specification with heteroscedastic residuals covers various heterogeneous structural binary choice models. As a particularly relevant example of a structural model where no semiparametric estimator has of yet been analyzed, we consider the binary random utility model with endogenous regressors and heterogeneous parameters. We employ a control function IV assumption to establish identification of a slope parameter 'â' by the mean ratio of derivatives of two functions of the instruments. We propose an estimator based on direct sample counterparts, and discuss the large sample behavior of this estimator. In particular, we show '√'n consistency and derive the asymptotic distribution. In the same framework, we propose tests for heteroscedasticity, overidentification and endogeneity. We analyze the small sample performance through a simulation study. An application of the model to discrete choice demand data concludes this paper.
Generalized Matrix Decomposition Regression: Estimation and Inference for Two-way Structured Data
This paper studies high-dimensional regression with two-way structured data.
To estimate the high-dimensional coefficient vector, we propose the generalized
matrix decomposition regression (GMDR) to efficiently leverage any auxiliary
information on row and column structures. The GMDR extends the principal
component regression (PCR) to two-way structured data, but unlike PCR, the GMDR
selects the components that are most predictive of the outcome, leading to more
accurate prediction. For inference on regression coefficients of individual
variables, we propose the generalized matrix decomposition inference (GMDI), a
general high-dimensional inferential framework for a large family of estimators
that include the proposed GMDR estimator. GMDI provides more flexibility for
modeling relevant auxiliary row and column structures. As a result, GMDI does
not require the true regression coefficients to be sparse; it also allows
dependent and heteroscedastic observations. We study the theoretical properties
of GMDI in terms of both the type-I error rate and power and demonstrate the
effectiveness of GMDR and GMDI on simulation studies and an application to
human microbiome data.Comment: 25 pages, 6 figures, Accepted by the Annals of Applied Statistic
Pénalités minimales et heuristique de pente
International audienceBirgé and Massart proposed in 2001 the slope heuristics as a way to choose optimally from data an unknown multiplicative constant in front of a penalty. It is built upon the notion of minimal penalty, and it has been generalized since to some "minimal-penalty algorithms". This paper reviews the theoretical results obtained for such algorithms, with a self-contained proof in the simplest framework, precise proof ideas for further generalizations, and a few new results. Explicit connections are made with residual-variance estimators-with an original contribution on this topic, showing that for this task the slope heuristics performs almost as well as a residual-based estimator with the best model choice-and some classical algorithms such as L-curve or elbow heuristics, Mallows' C p , and Akaike's FPE. Practical issues are also addressed, including two new practical definitions of minimal-penalty algorithms that are compared on synthetic data to previously-proposed definitions. Finally, several conjectures and open problems are suggested as future research directions.Birgé et Massart ont proposé en 2001 l'heuristique de pente, pour déterminer à l'aide des données une constante multiplicative optimale devant une pénalité en sélection de modèles. Cette heuristique s'appuie sur la notion de pénalité minimale, et elle a depuis été généralisée en "algorithmes à base de pénalités minimales". Cet article passe en revue les résultats théoriques obtenus sur ces algorithmes, avec une preuve complète dans le cadre le plus simple, des idées de preuves précises pour généraliser ce résultat au-delà des cadres déjà étudiés, et quelques résultats nouveaux. Des liens sont faits avec les méthodes d'estimation de la variance résiduelle (avec une contribution originale sur ce thème, qui démontre que l'heuristique de pente produit un estimateur de la variance quasiment aussi bon qu'un estimateur fondé sur les résidus d'un modèle oracle) ainsi qu'avec plusieurs algorithmes classiques tels que les heuristiques de coude (ou de courbe en L), Cp de Mallows et FPE d'Akaike. Les questions de mise en oeuvre pratique sont également étudiées, avec notamment la proposition de deux nouvelles définitions pratiques pour des algorithmes à base de pénalités minimales et leur comparaison aux définitions précédentes sur des données simulées. Enfin, des conjectures et problèmes ouverts sont proposés comme pistes de recherche pour l'avenir
Minimal penalties and the slope heuristics: a survey
Birg{\'e} and Massart proposed in 2001 the slope heuristics as a way to
choose optimally from data an unknown multiplicative constant in front of a
penalty. It is built upon the notion of minimal penalty, and it has been
generalized since to some "minimal-penalty algorithms". This paper reviews the
theoretical results obtained for such algorithms, with a self-contained proof
in the simplest framework, precise proof ideas for further generalizations, and
a few new results. Explicit connections are made with residual-variance
estimators-with an original contribution on this topic, showing that for this
task the slope heuristics performs almost as well as a residual-based estimator
with the best model choice-and some classical algorithms such as L-curve or
elbow heuristics, Mallows' C p , and Akaike's FPE. Practical issues are also
addressed, including two new practical definitions of minimal-penalty
algorithms that are compared on synthetic data to previously-proposed
definitions. Finally, several conjectures and open problems are suggested as
future research directions
Efficient Nonparametric and Semiparametric Regression Methods with application in Case-Control Studies
Regression Analysis is one of the most important tools of statistics which is widely used in other scientific fields for projection and modeling of association between two variables. Nowadays with modern computing techniques and super high performance devices, regression analysis on multiple dimensions has become an important issue. Our task is to address the issue of modeling with no assumption on the mean and the variance structure and further with no assumption on the error distribution. In other words, we focus on developing robust semiparametric and nonparamteric regression problems. In modern genetic epidemiological association studies, it is often important to investigate the relationships among the potential covariates related to disease in case-control data, a study known as "Secondary Analysis". First we focus to model the association between the potential covariates in univariate dimension nonparametrically. Then we focus to model the association in mulivariate set up by assuming a convenient and popular multivariate semiparametric model, known as Single-Index Model. The secondary analysis of case-control studies is particularly challenging due to multiple reasons (a) the case-control sample is not a random sample, (b) the logistic intercept is practically not identifiable and (c) misspecification of error distribution leads to inconsistent results. For rare disease, controls (individual free of disease) are typically used for valid estimation. However, numerous publication are done to utilize the entire case-control sample (including the diseased individual) to increase the efficiency. Previous work in this context has either specified a fully parametric distribution for regression errors or specified a homoscedastic distribution for the regression errors or have assumed parametric forms on the regression mean.
In the first chapter we focus on to predict an univariate covariate Y by another potential univariate covariate X neither by any parametric form on the mean function nor by any distributional assumption on error, hence addressing potential heteroscedasticity, a problem which has not been studied before. We develop a tilted Kernel based estimator which is a first attempt to model the mean function nonparametrically in secondary analysis. In the following chapters, we focus on i.i.d samples to model both the mean and variance function for predicting Y by multiple covariates X without assuming any form on the regression mean. In particular we
model Y by a single-index model m(X^T ϴ), where ϴ is a single-index vector and m is unspecified. We also model the variance function by another flexible single index model. We develop a practical and readily applicable Bayesian methodology based on penalized spline and Markov Chain Monte Carlo (MCMC) both in i.i.d set up and in case-control set up. For efficient estimation, we model the error distribution by a Dirichlet process mixture models of Normals (DPMM). In numerical examples, we illustrate the finite sample performance of the posterior estimates for both i.i.d and for case-control set up. For single-index set up, in i.i.d case only one existing work based on local linear kernel method addresses modeling of the variance function. We found that our method based on DPMM vastly outperforms the other existing method in terms of mean square efficiency and computation stability. We develop the single-index modeling in secondary analysis to introduce flexible mean and variance function modeling in case-control studies, a problem which has not been studies before. We showed that our method is almost 2 times efficient than using only controls, which is typically used for many cases. We use the real data example from NIH-AARP study on breast cancer, from Colon Cancer Study on red meat consumption and from National Morbidity Air Pollution Study to illustrate the computational efficiency and stability of our methods
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