75 research outputs found
De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers
We provide adaptive inference methods, based on regularization, for
regular (semiparametric) and non-regular (nonparametric) linear functionals of
the conditional expectation function. Examples of regular functionals include
average treatment effects, policy effects, and derivatives. Examples of
non-regular functionals include average treatment effects, policy effects, and
derivatives conditional on a covariate subvector fixed at a point. We construct
a Neyman orthogonal equation for the target parameter that is approximately
invariant to small perturbations of the nuisance parameters. To achieve this
property, we include the Riesz representer for the functional as an additional
nuisance parameter. Our analysis yields weak "double sparsity robustness":
either the approximation to the regression or the approximation to the
representer can be "completely dense" as long as the other is sufficiently
"sparse". Our main results are non-asymptotic and imply asymptotic uniform
validity over large classes of models, translating into honest confidence bands
for both global and local parameters
Bootstrap confidence sets under model misspecification
A multiplier bootstrap procedure for construction of likelihood-based
confidence sets is considered for finite samples and a possible model
misspecification. Theoretical results justify the bootstrap validity for a
small or moderate sample size and allow to control the impact of the parameter
dimension : the bootstrap approximation works if is small. The main
result about bootstrap validity continues to apply even if the underlying
parametric model is misspecified under the so-called small modelling bias
condition. In the case when the true model deviates significantly from the
considered parametric family, the bootstrap procedure is still applicable but
it becomes a bit conservative: the size of the constructed confidence sets is
increased by the modelling bias. We illustrate the results with numerical
examples for misspecified linear and logistic regressions.Comment: Published at http://dx.doi.org/10.1214/15-AOS1355 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Robust Estimation and Inference for Expected Shortfall Regression with Many Regressors
Expected Shortfall (ES), also known as superquantile or Conditional
Value-at-Risk, has been recognized as an important measure in risk analysis and
stochastic optimization, and is also finding applications beyond these areas.
In finance, it refers to the conditional expected return of an asset given that
the return is below some quantile of its distribution. In this paper, we
consider a recently proposed joint regression framework that simultaneously
models the quantile and the ES of a response variable given a set of
covariates, for which the state-of-the-art approach is based on minimizing a
joint loss function that is non-differentiable and non-convex. This inevitably
raises numerical challenges and limits its applicability for analyzing
large-scale data. Motivated by the idea of using Neyman-orthogonal scores to
reduce sensitivity with respect to nuisance parameters, we propose a
statistically robust (to highly skewed and heavy-tailed data) and
computationally efficient two-step procedure for fitting joint quantile and ES
regression models. With increasing covariate dimensions, we establish explicit
non-asymptotic bounds on estimation and Gaussian approximation errors, which
lay the foundation for statistical inference. Finally, we demonstrate through
numerical experiments and two data applications that our approach well balances
robustness, statistical, and numerical efficiencies for expected shortfall
regression
Partially linear models
In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models.
We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics.
This monograph can be divided into three parts: part one–Chapter 1 through Chapter 4; part two–Chapter 5; and part three–Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theo- retical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples.
The second part is of more theoretical interest. In this part, we construct several adaptive and efficient estimates for the parametric component. We show that the LS estimator of the parametric component can be modified to have both Bahadur asymptotic efficiency and second order asymptotic efficiency. In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to fit a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a Cross-Validation (CV) based criterion to select the optimum linear subset from a partially linear regression and estab- lish a CV selection criterion for the bandwidth involved in the nonparametric kernel estimation. The CV selection criterion can be applied to the case where the observations fitted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a sepa- rate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression.
This work of the authors was supported partially by the Sonderforschungs- bereich373“QuantifikationundSimulationO ̈konomischerProzesse”.Thesecond author was also supported by the National Natural Science Foundation of China and an Alexander von Humboldt Fellowship at the Humboldt University, while the third author was also supported by the Australian Research Council. The second and third authors would like to thank their teachers: Professors Raymond Car- roll, Guijing Chen, Xiru Chen, Ping Cheng and Lincheng Zhao for their valuable inspiration on the two authors’ research efforts. We would like to express our sin- cere thanks to our colleagues and collaborators for many helpful discussions and stimulating collaborations, in particular, Vo Anh, Shengyan Hong, Enno Mam- men, Howell Tong, Axel Werwatz and Rodney Wolff. For various ways in which they helped us, we would like to thank Adrian Baddeley, Rong Chen, Anthony Pettitt, Maxwell King, Michael Schimek, George Seber, Alastair Scott, Naisyin Wang, Qiwei Yao, Lijian Yang and Lixing Zhu.
The authors are grateful to everyone who has encouraged and supported us to finish this undertaking. Any remaining errors are ours
Honest confidence sets in nonparametric IV regression and other ill-posed models
This paper provides novel methods for inference in a very general class of ill-posed models in econometrics, encompassing the nonparametric instrumental regression, different functional regressions, and the deconvolution. I focus on uniform confidence sets for the parameter of interest estimated with Tikhonov regularization, as in Darolles, Fan, Florens, and Renault (2011). I first show that it is not possible to develop inferential methods directly based on the uniform central limit theorem. To circumvent this difficulty I develop two approaches that lead to valid confidence sets. I characterize expected diameters and coverage properties uniformly over a large class of models (i.e. constructed confidence sets are honest). Finally, I illustrate that introduced confidence sets have reasonable width and coverage properties in samples commonly used in applications with Monte Carlo simulations and considering application to Engel curves
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