Semiparametric Binary Regression Models Under Shape Constraints
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Instrumental Regression in Partially Linear Models
10-167 Research Group: Econometrics and Statistics September, 2009 Instrumental Regression in Partially Linear Models JEAN-PIERRE FLORENS, JAN JOHANNES AND SEBASTIEN VAN BELLEGEM INSTRUMENTAL REGRESSION IN PARTIALLY LINEAR MODELS∗ Jean-Pierre Florens1 Jan Johannes2 Sebastien´ Van Bellegem3 First version: January 24, 2008. This version: September 10, 2009 Abstract We consider the semiparametric regression Xtβ+φ(Z) where β and φ( ) are unknown · slope coefficient vector and function, and where the variables (X,Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parameterization of φ may generally lead to an inconsistent estimator of β, whereas even consistent nonparametric estimators for φ imply a slow rate of convergence of the estimator of β. An additional complication is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated nonparametri- cally. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interprated in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroscedastic. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples. Keywords: Partially linear model, semiparametric regression, instrumental variables, endo- geneity, ill-posed inverse problem, Tikhonov regularization, root-N consistent estimation, semiparametric efficiency bound JEL classifications: Primary C14; secondary C30 ∗We are grateful to R. -
On Assessing Binary Regression Models Based on Ungrouped Data
Biometrics 000, 1{18 DOI: 000 000 0000 On Assessing Binary Regression Models Based on Ungrouped Data Chunling Lu Division of Global Health, Brigham and Women's Hospital & Department of Global Health and Social Medicine Harvard University, Boston, U.S. email: chunling [email protected] and Yuhong Yang School of Statistics, University of Minnesota, Minnesota, U.S. email: [email protected] Summary: Assessing a binary regression model based on ungrouped data is a commonly encountered but very challenging problem. Although tests, such as Hosmer-Lemeshow test and le Cessie-van Houwelingen test, have been devised and widely used in applications, they often have low power in detecting lack of fit and not much theoretical justification has been made on when they can work well. In this paper, we propose a new approach based on a cross validation voting system to address the problem. In addition to a theoretical guarantee that the probabilities of type I and II errors both converge to zero as the sample size increases for the new method under proper conditions, our simulation results demonstrate that it performs very well. Key words: Goodness of fit; Hosmer-Lemeshow test; Model assessment; Model selection diagnostics. This paper has been submitted for consideration for publication in Biometrics Goodness of Fit for Ungrouped Data 1 1. Introduction 1.1 Motivation Parametric binary regression is one of the most widely used statistical tools in real appli- cations. A central component in parametric regression is assessment of a candidate model before accepting it as a satisfactory description of the data. In that regard, goodness of fit tests are needed to reveal significant lack-of-fit of the model to assess (MTA), if any. -
Arxiv:1609.06421V4 [Math.ST] 26 Sep 2019 Semiparametric Identification and Fisher Information∗
Semiparametric Identification and Fisher Information∗ Juan Carlos Escanciano† Universidad Carlos III de Madrid September 25th, 2019 Abstract This paper provides a systematic approach to semiparametric identification that is based on statistical information as a measure of its “quality”. Identification can be regular or irregular, depending on whether the Fisher information for the parameter is positive or zero, respectively. I first characterize these cases in models with densities linear in a nonparametric parameter. I then introduce a novel “generalized Fisher information”. If positive, it implies (possibly irregular) identification when other conditions hold. If zero, it implies impossibility results on rates of estimation. Three examples illustrate the applicability of the general results. First, I find necessary conditions for semiparametric regular identification in a structural model for unemployment duration with two spells and nonparametric heterogeneity. Second, I show irregular identification of the median willingness to pay in contingent valuation studies. Finally, I study identification of the discount factor and average measures of risk aversion in a nonparametric Euler Equation with nonparametric measurement error in consumption. Keywords: Identification; Semiparametric Models; Fisher Information. arXiv:1609.06421v4 [math.ST] 26 Sep 2019 JEL classification: C14; C31; C33; C35 ∗First version: September 20th, 2016. †Department of Economics, Universidad Carlos III de Madrid, email: [email protected]. Research funded by the Spanish Programa de Generaci´on de Conocimiento, reference number PGC2018-096732-B-I00. I thank Michael Jansson, Ulrich M¨uller, Whitney Newey, Jack Porter, Pedro Sant’Anna, Ruli Xiao, and seminar par- ticipants at BC, Indiana, MIT, Texas A&M, UBC, Vanderbilt and participants of the 2018 Conference on Identification in Econometrics for useful comments. -
Semiparametric Efficiency
Faculteit Wetenschappen Vakgroep Toegepaste Wiskunde en Informatica Semiparametric Efficiency Karel Vermeulen Prof. Dr. S. Vansteelandt Proefschrift ingediend tot het behalen van de graad van Master in de Wiskunde, afstudeerrichting Toegepaste Wiskunde Academiejaar 2010-2011 To my parents, my brother Lukas To my best friend Sara A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street::: David Hilbert Preface My interest in semiparametric theory awoke several years ago, two and a half to be precise. That time, I was in my third year of mathematics. I had to choose a subject for my Bachelor thesis. My decision was: A geometrical approach to the asymptotic efficiency of estimators, based on the monograph by Anastasios Tsiatis, [35], under the supervision of Prof. Dr. Stijn Vansteelandt. In this manner, I entered the world of semiparametric efficiency. However, at that point I did not know it was just the beginning. Shortly after I wrote this Bachelor thesis, Prof. Dr. Stijn Vansteelandt asked me to be involved in research on semiparametric inference for so called probabilistic index models, in the context of a one month student job. Under his guidance, I applied semiparametric estimation theory to this setting and obtained the semiparametric efficiency bound to which efficiency of estimators in this model can be measured. Some results of this research are contained within this thesis. While short, this experience really convinced me I wanted to write a thesis in semiparametric efficiency. That feeling was only more encouraged after following the course Causality and Missing Data, taught by Prof. -
A Weakly Informative Default Prior Distribution for Logistic and Other
The Annals of Applied Statistics 2008, Vol. 2, No. 4, 1360–1383 DOI: 10.1214/08-AOAS191 c Institute of Mathematical Statistics, 2008 A WEAKLY INFORMATIVE DEFAULT PRIOR DISTRIBUTION FOR LOGISTIC AND OTHER REGRESSION MODELS By Andrew Gelman, Aleks Jakulin, Maria Grazia Pittau and Yu-Sung Su Columbia University, Columbia University, University of Rome, and City University of New York We propose a new prior distribution for classical (nonhierarchi- cal) logistic regression models, constructed by first scaling all nonbi- nary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-t prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with cen- ter 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. Cross- validation on a corpus of datasets shows the Cauchy class of prior dis- tributions to outperform existing implementations of Gaussian and Laplace priors. We recommend this prior distribution as a default choice for rou- tine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of pre- dictors is small), and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analy- sis as well as in automated procedures such as chained equations for missing-data imputation. We implement a procedure to fit generalized linear models in R with the Student-t prior distribution by incorporating an approxi- mate EM algorithm into the usual iteratively weighted least squares. -
Efficient Estimation in Single Index Models Through Smoothing Splines
arXiv: 1612.00068 Efficient Estimation in Single Index Models through Smoothing splines Arun K. Kuchibhotla and Rohit K. Patra University of Pennsylvania and University of Florida e-mail: [email protected]; [email protected] Abstract: We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the para- metric and the nonparametric components given independent and identically distributed (i.i.d.) data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under under mild assumption and prove asymptotic effi- ciency of the parametric component under homoscedastic errors. A finite sample simulation corroborates our asymptotic theory. We also analyze a car mileage data set and a Ozone concentration data set. The identifiability and existence of the PLSEs are also investigated. Keywords and phrases: least favorable submodel, penalized least squares, semiparamet- ric model. 1. Introduction Consider a regression model where one observes i.i.d. copies of the predictor X ∈ Rd and the response Y ∈ R and is interested in estimating the regression function E(Y |X = ·). In nonpara- metric regression E(Y |X = ·) is generally assumed to satisfy some smoothness assumptions (e.g., twice continuously differentiable), but no assumptions are made on the form of dependence on X. While nonparametric models offer flexibility in modeling, the price for this flexibility can be high for two main reasons: the estimation precision decreases rapidly as d increases (“curse of dimensionality”) and the estimator can be hard to interpret when d > 1. -
Measures of Fit for Logistic Regression Paul D
Paper 1485-2014 SAS Global Forum Measures of Fit for Logistic Regression Paul D. Allison, Statistical Horizons LLC and the University of Pennsylvania ABSTRACT One of the most common questions about logistic regression is “How do I know if my model fits the data?” There are many approaches to answering this question, but they generally fall into two categories: measures of predictive power (like R-square) and goodness of fit tests (like the Pearson chi-square). This presentation looks first at R-square measures, arguing that the optional R-squares reported by PROC LOGISTIC might not be optimal. Measures proposed by McFadden and Tjur appear to be more attractive. As for goodness of fit, the popular Hosmer and Lemeshow test is shown to have some serious problems. Several alternatives are considered. INTRODUCTION One of the most frequent questions I get about logistic regression is “How can I tell if my model fits the data?” Often the questioner is expressing a genuine interest in knowing whether a model is a good model or a not-so-good model. But a more common motivation is to convince someone else--a boss, an editor, or a regulator--that the model is OK. There are two very different approaches to answering this question. One is to get a statistic that measures how well you can predict the dependent variable based on the independent variables. I’ll refer to these kinds of statistics as measures of predictive power. Typically, they vary between 0 and 1, with 0 meaning no predictive power whatsoever and 1 meaning perfect predictions. -
BUGS Example 1: Linear Regression Length 1.8 2.0 2.2 2.4 2.6
BUGS Example 1: Linear Regression length 1.8 2.0 2.2 2.4 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 log(age) For n = 27 captured samples of the sirenian species dugong (sea cow), relate an animal’s length in meters, Yi, to its age in years, xi. Intermediate WinBUGS and BRugs Examples – p. 1/35 BUGS Example 1: Linear Regression length 1.8 2.0 2.2 2.4 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 log(age) For n = 27 captured samples of the sirenian species dugong (sea cow), relate an animal’s length in meters, Yi, to its age in years, xi. To avoid a nonlinear model for now, transform xi to the log scale; plot of Y versus log(x) looks fairly linear! Intermediate WinBUGS and BRugs Examples – p. 1/35 Simple linear regression in WinBUGS Yi = β0 + β1 log(xi) + ǫi, i = 1,...,n iid where ǫ N(0,τ) and τ = 1/σ2, the precision in the data. i ∼ Prior distributions: flat for β0, β1 vague gamma on τ (say, Gamma(0.1, 0.1), which has mean 1 and variance 10) is traditional Intermediate WinBUGS and BRugs Examples – p. 2/35 Simple linear regression in WinBUGS Yi = β0 + β1 log(xi) + ǫi, i = 1,...,n iid where ǫ N(0,τ) and τ = 1/σ2, the precision in the data. i ∼ Prior distributions: flat for β0, β1 vague gamma on τ (say, Gamma(0.1, 0.1), which has mean 1 and variance 10) is traditional posterior correlation is reduced by centering the log(xi) around their own mean Intermediate WinBUGS and BRugs Examples – p. -
Introduction to Empirical Processes and Semiparametric Inference1
This is page i Printer: Opaque this Introduction to Empirical Processes and Semiparametric Inference1 Michael R. Kosorok August 2006 1c 2006 SPRINGER SCIENCE+BUSINESS MEDIA, INC. All rights re- served. Permission is granted to print a copy of this preliminary version for non- commercial purposes but not to distribute it in printed or electronic form. The author would appreciate notification of typos and other suggestions for improve- ment. ii This is page iii Printer: Opaque this Preface The goal of this book is to introduce statisticians, and other researchers with a background in mathematical statistics, to empirical processes and semiparametric inference. These powerful research techniques are surpris- ingly useful for studying large sample properties of statistical estimates from realistically complex models as well as for developing new and im- proved approaches to statistical inference. This book is more a textbook than a research monograph, although some new results are presented in later chapters. The level of the book is more introductory than the seminal work of van der Vaart and Wellner (1996). In fact, another purpose of this work is to help readers prepare for the mathematically advanced van der Vaart and Wellner text, as well as for the semiparametric inference work of Bickel, Klaassen, Ritov and Wellner (1997). These two books, along with Pollard (1990) and chapters 19 and 25 of van der Vaart (1998), formulate a very complete and successful elucida- tion of modern empirical process methods. The present book owes much by the way of inspiration, concept, and notation to these previous works. What is perhaps new is the introductory, gradual and unified way this book introduces the reader to the field. -
Variable Selection in Semiparametric Regression Modeling
The Annals of Statistics 2008, Vol. 36, No. 1, 261–286 DOI: 10.1214/009053607000000604 c Institute of Mathematical Statistics, 2008 VARIABLE SELECTION IN SEMIPARAMETRIC REGRESSION MODELING By Runze Li1 and Hua Liang2 Pennsylvania State University and University of Rochester In this paper, we are concerned with how to select significant vari- ables in semiparametric modeling. Variable selection for semipara- metric regression models consists of two components: model selection for nonparametric components and selection of significant variables for the parametric portion. Thus, semiparametric variable selection is much more challenging than parametric variable selection (e.g., linear and generalized linear models) because traditional variable selection procedures including stepwise regression and the best subset selection now require separate model selection for the nonparametric compo- nents for each submodel. This leads to a very heavy computational burden. In this paper, we propose a class of variable selection proce- dures for semiparametric regression models using nonconcave penal- ized likelihood. We establish the rate of convergence of the resulting estimate. With proper choices of penalty functions and regulariza- tion parameters, we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures per- form as well as an oracle procedure. A semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distri- bution follows a chi-square distribution which is independent of the nuisance parameters. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures. -
1D Regression Models Such As Glms
Chapter 4 1D Regression Models Such as GLMs ... estimates of the linear regression coefficients are relevant to the linear parameters of a broader class of models than might have been suspected. Brillinger (1977, p. 509) After computing β,ˆ one may go on to prepare a scatter plot of the points (βxˆ j, yj), j =1,...,n and look for a functional form for g( ). Brillinger (1983, p. 98) · This chapter considers 1D regression models including additive error re- gression (AER), generalized linear models (GLMs), and generalized additive models (GAMs). Multiple linear regression is a special case of these four models. See Definition 1.2 for the 1D regression model, sufficient predictor (SP = h(x)), estimated sufficient predictor (ESP = hˆ(x)), generalized linear model (GLM), and the generalized additive model (GAM). When using a GAM to check a GLM, the notation ESP may be used for the GLM, and EAP (esti- mated additive predictor) may be used for the ESP of the GAM. Definition 1.3 defines the response plot of ESP versus Y . Suppose the sufficient predictor SP = h(x). Often SP = xT β. If u only T T contains the nontrivial predictors, then SP = β1 + u β2 = α + u η is often T T T T T T used where β = (β1, β2 ) = (α, η ) and x = (1, u ) . 4.1 Introduction First we describe some regression models in the following three definitions. The most general model uses SP = h(x) as defined in Definition 1.2. The GAM with SP = AP will be useful for checking the model (often a GLM) with SP = xT β. -
An Averaging Estimator for Two Step M Estimation in Semiparametric Models
An Averaging Estimator for Two Step M Estimation in Semiparametric Models Ruoyao Shi∗ February, 2021 Abstract In a two step extremum estimation (M estimation) framework with a finite dimen- sional parameter of interest and a potentially infinite dimensional first step nuisance parameter, I propose an averaging estimator that combines a semiparametric esti- mator based on nonparametric first step and a parametric estimator which imposes parametric restrictions on the first step. The averaging weight is the sample analog of an infeasible optimal weight that minimizes the asymptotic quadratic risk. I show that under mild conditions, the asymptotic lower bound of the truncated quadratic risk dif- ference between the averaging estimator and the semiparametric estimator is strictly less than zero for a class of data generating processes (DGPs) that includes both cor- rect specification and varied degrees of misspecification of the parametric restrictions, and the asymptotic upper bound is weakly less than zero. Keywords: two step M estimation, semiparametric model, averaging estimator, uniform dominance, asymptotic quadratic risk JEL Codes: C13, C14, C51, C52 ∗Department of Economics, University of California Riverside, [email protected]. The author thanks Colin Cameron, Xu Cheng, Denis Chetverikov, Yanqin Fan, Jinyong Hahn, Bo Honoré, Toru Kitagawa, Zhipeng Liao, Hyungsik Roger Moon, Whitney Newey, Geert Ridder, Aman Ullah, Haiqing Xu and the participants at various seminars and conferences for helpful comments. This project is generously supported by UC Riverside Regents’ Faculty Fellowship 2019-2020. Zhuozhen Zhao provides great research assistance. All remaining errors are the author’s. 1 1 Introduction Semiparametric models, consisting of a parametric component and a nonparametric com- ponent, have gained popularity in economics.