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Large Sample Sieve Estimation of SemiNonparametric Models
 Handbook of Econometrics
, 2007
"... Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; seminonparametric models are more flexible and robust, but lead to other complications such as introducing infinite dimensional parameter spaces that may not be compact. The method o ..."
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Cited by 181 (19 self)
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Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; seminonparametric models are more flexible and robust, but lead to other complications such as introducing infinite dimensional parameter spaces that may not be compact. The method of sieves provides one way to tackle such complexities by optimizing an empirical criterion function over a sequence of approximating parameter spaces, called sieves, which are significantly less complex than the original parameter space. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated econometric models. For example, it can simultaneously estimate the parametric and nonparametric components in seminonparametric models with or without constraints. It can easily incorporate prior information, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and nonnegativity. This chapter describes estimation of seminonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve Mestimates, pointwise normality of series estimates of regression functions, rootn asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite dimensional parameters. Examples are used to illustrate the general results.
Variable Selection for Cox's Proportional Hazards Model and Frailty Model
 ANNALS OF STATISTICS
, 2002
"... A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an o ..."
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Cited by 83 (14 self)
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A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an oracle property. The proposed procedures were illustrated in the context of linear regression, robust linear regression and generalized linear models. In this paper, the nonconcave penalized likelihood approach is extended further to the Cox proportional hazards model and the Cox proportional hazards frailty model, two commonly used semiparametric models in survival analysis. As a result, new variable selection procedures for these two commonlyused models are proposed. It is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function. Further, with a proper choice of the regularization parameter and the penalty function, the proposed estimators possess an oracle property. Standard error formulae are derived and their accuracies are empirically tested. Simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection, and they reduce model complexity as effectively as the best subset variable selection. Compared with the LASSO, which is the penalized likelihood method with the L1penalty, proposed by Tibshirani, the newly proposed approaches have better theoretic properties and finite sample performance.
Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals
 FORTHCOMING IN JOURNAL OF ECONOMETRICS
, 2008
"... For semi/nonparametric conditional moment models containing unknown parametric components (θ) and unknown functions of endogenous variables (h), Newey and Powell (2003) and Ai and Chen (2003) propose sieve minimum distance (SMD) estimation of (θ, h) and derive the large sample properties. This paper ..."
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Cited by 43 (7 self)
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For semi/nonparametric conditional moment models containing unknown parametric components (θ) and unknown functions of endogenous variables (h), Newey and Powell (2003) and Ai and Chen (2003) propose sieve minimum distance (SMD) estimation of (θ, h) and derive the large sample properties. This paper greatly extends their results by establishing the followings: (1) The penalized SMD (PSMD) estimator ( ˆ θ, ˆ h) can simultaneously achieve rootn asymptotic normality of ˆ θ and nonparametric optimal convergence rate of ˆ h, allowing for models with possibly nonsmooth residuals and/or noncompact infinite dimensional parameter spaces. (2) A simple weighted bootstrap procedure can consistently estimate the limiting distribution of the PSMD ˆ θ. (3) The semiparametric efficiency bound results of Ai and Chen (2003) remain valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bounds. (4) The profiled optimally weighted PSMD criterion is asymptotically Chisquare distributed, which implies an alternative consistent estimation of confidence region of the efficient PSMD estimator of θ. All the theoretical results are stated in terms of any consistent nonparametric estimator of conditional mean functions. We illustrate our general theories using a partially linear quantile instrumental variables regression, a Monte Carlo study, and an
Large Sample Theory for Semiparametric Regression Models with TwoPhase, Outcome Dependent Sampling
, 2000
"... Outcomedependent, twophase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and inuenc ..."
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Cited by 30 (9 self)
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Outcomedependent, twophase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and inuence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under twophase sampling designs. We relate the efficient score to the leastfavorable parametric submodel by use of formal calculations suggested by Newey (1994). We then proceed to show that the maximum likelihood estimators proposed by Lawless, Kalbfleisch, and Wild (1999) for both the parametric and nonparametric parts of the model are asymptotically normal and efficient, and that the efficient influence function for the parametric part agrees with the more general calculations of Robins, Hsieh, and Newey (1995).
Efficient independent component analysis (I
, 2003
"... Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on Mestimates have been proposed for estimating the mixing matrix. Recently, several nonpar ..."
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Cited by 23 (4 self)
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Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on Mestimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but indepth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using Bspline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations.
Maximum likelihood estimation in semiparametric regression models with censored data
, 2007
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Empirical Evidence Concerning the Finite Sample Performance of ELType Structural Equation Estimation and Inference Methods,” in Identification and Inference for Econometric Models, Essays
 in Honor of Thomas Rothenberg, Andrews and Stock (eds). Lecture Notes 15, Summer ’07 23
, 2005
"... This paper presents empirical evidence concerning the finite sample performance of empirical likelihoodtype estimators when the estimating functions are well determined and the parameters are over identified. There are suggestions in the literature that traditional and nontraditional asymptoticall ..."
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Cited by 19 (1 self)
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This paper presents empirical evidence concerning the finite sample performance of empirical likelihoodtype estimators when the estimating functions are well determined and the parameters are over identified. There are suggestions in the literature that traditional and nontraditional asymptotically efficient estimators based on moment equations may, for the relatively small sample sizes usually encountered in econometric practice, have relatively large biases and/or variances and provide an inadequate basis for estimation and inference. Given this uncertainty we use a range of data sampling processes and Monte Carlo sampling procedures to accumulate finite sample empirical evidence concerning these questions for a family of empirical likelihoodtype estimators. Solutions to ELtype empirical momentconstrained optimization problems present formidable numerical challenges. We identify effective optimization algorithms for meeting these challenges.
Huang (2010): “Bootstrap Consistency for General Semiparametric MEstimation,”Annals of Statistics
"... Consider Mestimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinitedimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found wide applications in semiparametric Mestimation and, because of ..."
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Cited by 16 (5 self)
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Consider Mestimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinitedimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found wide applications in semiparametric Mestimation and, because of its simplicity, provides an attractive alternative to the inference approach based on the asymptotic distribution theory. The purpose of this paper is to provide theoretical justifications for the use of bootstrap as a semiparametric inferential tool. We show that, under general conditions, the bootstrap is asymptotically consistent in estimating the distribution of the Mestimate of Euclidean parameter; that is, the bootstrap distribution asymptotically imitates the distribution of the Mestimate. We also show that the bootstrap confidence set has the asymptotically correct coverage probability. These general conclusions hold, in particular, when the nuisance parameter is not estimable at rootn rate, and apply to a broad class of bootstrap methods with exchangeable bootstrap weights. This paper provides a first general theoretical study of the bootstrap in semiparametric models.
Likelihood based inference for monotone response models
 Annals of Statistics
, 2007
"... The behavior of maximum likelihood estimates (MLE’s) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the ..."
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Cited by 16 (8 self)
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The behavior of maximum likelihood estimates (MLE’s) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate n 1/3 (slower than the usual √ n rate) with a nonGaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLE’s and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer χ 2 but can be explicitly characterized in terms of a functional of Brownian motion. 1