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This paper presents a model selection technique of estimation in semiparametric regression models of the type$Y_i = \\beta ^\\prime \\b X_i + f(T_i) + W_i$, i = 1,..., n. The parametric and nonparametric components are estimated simultaneously by this procedure. Estimation is based on a collection of finite-dimensional models, using a penalized least squares criterion for selection. We show that by tailoring the penalty terms developed for nonparametric regression to semiparametric models, we can consistently estimate the subset of nonzero coefficients of the linear part. Moreover, the selected estimator of the linear component is asymptotically normal.

The authors propose a two-stage estimation procedure for the partially linear model Y=f0(T)+X′β0+ε . They show how to estimate consistently the location of the nonzero components of β0. Their approach turns out to be compatible with minimax adaptive estimation of f0over Besov balls in the case of penalized least squares. Their proofs are based on a new type of oracle inequality. /// Les auteurs proposent une procédure d'estimation en deux temps pour le modèle partiellement linéaire Y=f0(T)+X′β0+ε . Ils montrent comment estimer de façon convergente la position des composantes non nulles de β0. Leur approche s'avère compatible avec l'estimation minimax adaptative de f0sur les boules de Besov dans le cas des moindres carrés pénalisés. Leurs démonstrations s'appuient sur une inégalité d'oracle d'un nouveau genre.

Many problems in genomics are related to variable selection where high-dimensional genomic data are treated as covariates. Such genomic covariates often have certain structures and can be represented as vertices of an undirected graph. Biological processes also vary as functions depending upon some biological state, such as time. High-dimensional variable selection where covariates are graph-structured and underlying model is nonparametric presents an important but largely unaddressed statistical challenge. Motivated by the problem of regression-based motif discovery, we consider the problem of variable selection for high-dimensional nonparametric varying-coefficient models and introduce a sparse structured shrinkage (SSS) estimator based on basis function expansions and a novel smoothed penalty function. We present an efficient algorithm for computing the SSS estimator. Results on model selection consistency and estimation bounds are derived. Moreover, finite-sample performances are studied via simulations, and the effects of high-dimensionality and structural information of the covariates are especially highlighted. We apply our method to motif finding problem using a yeast cell-cycle gene expression dataset and word counts in genes' promoter sequences. Our results demonstrate that the proposed method can result in better variable selection and prediction for high-dimensional regression when the underlying model is nonparametric and covariates are structured. Supplemental materials for the article are available online.

This article presents an exposition and synthesis of the theory and some applications of the so-called indirect method of inference. These ideas have been exploited in the field of econometrics, but less so in other fields such as biostatistics and epidemiology. In the indirect method, statistical inference is based on an intermediate statistic, which typically follows an asymptotic normal distribution, but is not necessarily a consistent estimator of the parameter of interest. This intermediate statistic can be a naive estimator based on a convenient but misspecified model, a sample moment or a solution to an estimating equation. We review a procedure of indirect inference based on the generalized method of moments, which involves adjusting the naive estimator to be consistent and asymptotically normal. The objective function of this procedure is shown to be interpretable as an \"indirect likelihood\" based on the intermediate statistic. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. This method is often more convenient computationally than maximum likelihood estimation when handling such model complexities as random effects and measurement error, for example, and it can also serve as a basis for robust inference and model selection, with less stringent assumptions on the data generating mechanism. Many familiar estimation techniques can be viewed as examples of this approach. We describe applications to measurement error, omitted covariates and recurrent events. A dataset concerning prevention of mammary tumors in rats is analyzed using a Poisson regression model with overdispersion. A second dataset from an epidemiological study is analyzed using a logistic regression model with mismeasured covariates. A third dataset of exam scores is used to illustrate robust covariance selection in graphical models.

Consider a study whose design calls for the study subjects to be followed from enrollment (time t = 0) to time t = T, at which point a primary endpoint of interest Y is to be measured. The design of the study also calls for measurements on a vector V t) of covariates to be made at one or more times t during the interval [0, T). We are interested in making inferences about the marginal mean μ 0 of Y when some subjects drop out of the study at random times Q prior to the common fixed end of follow-up time T. The purpose of this article is to show how to make inferences about μ 0 when the continuous drop-out time Q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables. In particular, we consider two models for the conditional hazard of drop-out given (V(T), Y), where V(t) denotes the history of the process V t) through time t, t ∈ [0, T). In the first model, we assume that λ Q (t|V(T), Y) exp(α 0 Y), where α 0 is a scalar parameter and λ 0 (t|V(t)) is an unrestricted positive function of t and the process V(t). When the process Vt) is high dimensional, estimation in this model is not feasible with moderate sample sizes, due to the curse of dimensionality. For such situations, we consider a second model that imposes the additional restriction that λ 0 (t|V(t)) = λ 0 (t) exp(γ′ 0 (t)), where λ 0 t) is an unspecified baseline hazard function, W(t) = w(t, V(t)), w(·,·) is a known function that maps (t, V(t)) to R q , and γ 0 is a q × 1 unknown parameter vector. When α 0 ≠ 0, then drop-out is nonignorable. On account of identifiability problems, joint estimation of the mean μ 0 of Y and the selection bias parameter α 0 may be difficult or impossible. Therefore, we propose regarding the selection bias parameter α 0 as known, rather than estimating it from the data. We then perform a sensitivity analysis to see how inference about α 0 changes as we vary α 0 over a plausible range of values. We apply our approach to the analysis of ACTG 175, an AIDS clinical trial.