Error distribution function for parametrically truncated and censored data
|
|
- Jacob Morris
- 5 years ago
- Views:
Transcription
1 Error distribution function for parametrically truncated and censored data Géraldine LAURENT Jointly with Cédric HEUCHENNE QuantOM, HEC-ULg Management School - University of Liège Friday, 14 September 2012
2 The doctor Anders Green et al (1981) studied the mortality of diabetics in the county of Fyn (Denmark) between the 1 July 1973 and 1 January For these data, we have the survival time of diabetic patient but this time can be partially observed, the age at the diabetic diagnosis, the sex for each patient.
3 Duration between the diagnosis and the study end (in years) Female diabetics Censored Observed Age at the diagnosis (in years) Duration between the diagnosis and the study end (in years) Male diabetics Censored Observed Age at the diagnosis (in years)
4 Estimation Asymptotic properties Data Analysis
5 Estimation
6 We consider the nonparametric regression model where Y is the response variable X is the covariate Y = m(x ) + σ(x )ε m( ) = E[Y ] and σ 2 ( ) = Var[Y ] are unknown smooth functions, ε is independent of X, with E[ε] = 0 and Var[ε] = 1.
7 The particularity of (X, Y ) is that (X, Y ) is obtained from cross-sectional sampling Y is subject to right censoring. We study the variable Y delimited by T Y C where T is the truncation variable C is the censoring variable.
8 Real World Time We use as notation F for cdf
9 Real World Truncation Time Time We use as notation F for cdf
10 Real World Truncation Time Time We use as notation F for cdf
11 Intermediate Observed World Y1 C2 Y3 Y4 C5 C6 Truncation Time Time We use as notation H for cdf, n the sample size
12 Observed World Y1 Y3 Y4 Truncation Time Time We use as notation H for cdf
13 We want to estimate the error distribution F ε (e) = IP(ε e) with (X, Y ) satisfying T Y C where the distribution F T X is a parametric distribution the distribution F C T X is completely unknown
14 We suppose that the variables Y and T are independent, conditionally on X for each value x, the support of F Y X ( x) is included into the support of F T X ( x) the lower bound of the T support is zero the variables (T, Y ) and C T are independent, conditionally on T Y, X
15 We have H X,Y (x, y) = IP(X x, Y y T Y C) Z Z = (E[w(X, Y )]) 1 w(r, s)df X,Y (r, s), r x the weight function w(x, y) is defined by w(x, y) = Z t y s y 1 GC T X (y t x) df T X (t x) where G C T X (z x) = IP(C T z X = x, T Y ).
16 We easily obtain F X,Y (x, y) = E[w(X, Y )] Therefore, we have Y m(x ) F ε (e) = IP σ(x ) = = ZZ (x,y): ZZ (x,y): y m(x) e σ(x) Z y m(x) e σ(x) u x e Z v y df X,Y (x, y) dh X,Y (u, v). w(u, v) E[w(X, Y )] dh X,Y (x, y) w(x, y)
17 Thus, the estimator is with ˆF ε (e) = Ê[w(X, Y )] M nx i=1 ˆε i = Y i ˆm(X i ), M = ˆσ(X i ) Ê[w(X, Y )] = 1 M nx i=1 I { ˆε i e, i = 1} ŵ(x i, Y i ) nx i ŵ(x i, Y i ) i, i=1! 1 where the functions ˆm( ), ˆσ( ) and ŵ(, ) are nonparametric estimators.
18 For G C T X (t x), we use the Beran (1981) estimator defined by Y W i (x, h n ) Ĝ C T X (t x) = 1 1 P nj=1 W j (x, h n )I {Z j Z i } where Z i t, i =0 Z i = min(c i T i, Y i T i ) and i = I {Y i C i } W i (x, h n ) = K x Xi hn Š P n j=1 K x Xj K is a kernel function hn Š are the Nadaraya-Watson weights h n is a bandwidth sequence tending to 0 when n => ŵ(x, y) = Z t y 1 ĜC T X (y t x) df T X (t x) Œ
19 The estimators of m( ) and σ( ) are given by ˆm(x) = P ni=1 W i (x,h n)y i i ŵ(x,y i ) P ni=1 W i (x,h n) i ŵ(x,y i ), ˆσ 2 (x) = P ni=1 W i (x,h n) i (Y i ˆm(x)) 2 ŵ(x,y i ) P ni=1 W i (x,h n) i ŵ(x,y i ), extension of the estimators in de Uña-Alvarez and Iglesias-Pérez (2010), described in Heuchenne-Laurent (2012).
20 Asymptotic properties
21 Under some assumptions not mentioned here, we obtain an asymptotic representation ˆF ε (e) F ε (e) = nx i=1 uniformly in < e < where Γ(Z i, Y i, X i, i, e) + o p (n 1 2 ) Γ(Z i, Y i, X i, i, e) = E[w(X, Y )] w(x i, Y i ) (I {ε i e, i = 1} F ε (e)) + f ε(e)f X (X i ) Ψ(Z i, Y i, X i, i, e) σ(x i ) for a given function Ψ.
22 Intermediate results used for the proof Technical lemma sup <e< n 1 nx i=1 E[w(X, Y )] w(x i, Y i ) I {ˆε i e, i = 1} E[w(X, Y )] w(x i, Y i ) I {ε i e, i = 1} IP (ˆε e χ n ) + IP (ε e) = o p n 1 2 Asymptotic representation of ˆm(x) m(x) Asymptotic representation of ˆσ(x) σ(x)
23 Under the assumptions of the previous theorem, the process n 1 2 ( ˆFε (e) F ε (e)) where < e < converges weakly to a zero-mean Gaussian process Ω(e) with a complex covariance function.
24 Data Analysis
25 Between 1987 and 1997 the Spanish Institute for Statistics studied the unemployment of active people, and more especially the one of married women. For these data, we note that the time of unemployment will not be completely observed, the age of the woman acts on the future job.
26 Censored Observed 160 Unemployment duration (in months) Woman age (in months)
27 For the real data, we suppose that the number of time periods is equal to 1009 but only 446 aren t censored. the distribution of T is a uniform one (Wang, 1991); the variable C is defined by C = T + τ where τ is a constant equal to 18 months; Because of the definition of the censoring variable, the weight function is simply defined by w(x, y) = Z y 0 y τ df T X (t x) The Bootstrap approximation gives the value of 70 months as the optimal bandwidth.
28 Representation of ˆF Y X for various ages Cumulative distribution function cdf of Y X on 20 years cdf of Y X on 35 years cdf of Y X on 50 years Unemployment time (in months)
29 For the real data, we suppose that the number diabetics is equal to 716 for female patients and 783 for male patients but only 241 and 256 are not censored for the female diabetics and the male diabetics. for each sex, the distribution of T X = x is supposed to be a gamma one (Wang, 1991) with a decreasing linear function for the mean and a constant function for the variance; the variable C T is supposed to be a variable dependant of the covariate for each sex. The optimal bandwidth obtained by bootstrap approximation is equal to 26 years for the female population and 28 years for the male population.
30 Representation of Ŝ Y X for various ages Est of F Y X ( X=15) female patient Est of F Y X ( X=30) female patient Est of F Y X ( X=50) female patient Est of F Y X ( X=15) male patient Est of F Y X ( X=30) male patient Est of F Y X ( X=50) male patient Survival function Time interval between the age at diagnosis and the death (in years)
31 Thank you for your attention
32 BERAN, R. (1981): Nonparametric regression with randomly censored survival data. Technical Report, University of California, Berkeley. de UNA-ALVAREZ, J., IGLESIAS-PEREZ, M.C. (2010): Nonparametric estimation of a conditional distribution from length-biased data. Annals of the Institute of Statistical Mathematics 62, HEUCHENNE, C., LAURENT, G. (2012): Nonparametric estimation of conditional moments with right-censored election biased data. Submission Journal of Statistical Planning and Inference. WANG, M.-C. (1991): Nonparametric estimation from cross-sectional survival data. Journal of the American Statistical Association 86,
Computational treatment of the error distribution in nonparametric regression with right-censored and selection-biased data
Computational treatment of the error distribution in nonparametric regression with right-censored and selection-biased data Géraldine Laurent 1 and Cédric Heuchenne 2 1 QuantOM, HEC-Management School of
More informationBootstrap of residual processes in regression: to smooth or not to smooth?
Bootstrap of residual processes in regression: to smooth or not to smooth? arxiv:1712.02685v1 [math.st] 7 Dec 2017 Natalie Neumeyer Ingrid Van Keilegom December 8, 2017 Abstract In this paper we consider
More informationModelling Non-linear and Non-stationary Time Series
Modelling Non-linear and Non-stationary Time Series Chapter 2: Non-parametric methods Henrik Madsen Advanced Time Series Analysis September 206 Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September
More informationEstimation of the Bivariate and Marginal Distributions with Censored Data
Estimation of the Bivariate and Marginal Distributions with Censored Data Michael Akritas and Ingrid Van Keilegom Penn State University and Eindhoven University of Technology May 22, 2 Abstract Two new
More informationLikelihood ratio confidence bands in nonparametric regression with censored data
Likelihood ratio confidence bands in nonparametric regression with censored data Gang Li University of California at Los Angeles Department of Biostatistics Ingrid Van Keilegom Eindhoven University of
More informationNonparametric Estimation of Regression Functions In the Presence of Irrelevant Regressors
Nonparametric Estimation of Regression Functions In the Presence of Irrelevant Regressors Peter Hall, Qi Li, Jeff Racine 1 Introduction Nonparametric techniques robust to functional form specification.
More informationI N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S (I S B A)
I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S (I S B A) UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 153 ESTIMATION
More informationUNIVERSITÄT POTSDAM Institut für Mathematik
UNIVERSITÄT POTSDAM Institut für Mathematik Testing the Acceleration Function in Life Time Models Hannelore Liero Matthias Liero Mathematische Statistik und Wahrscheinlichkeitstheorie Universität Potsdam
More informationTESTING FOR THE EQUALITY OF k REGRESSION CURVES
Statistica Sinica 17(2007, 1115-1137 TESTNG FOR THE EQUALTY OF k REGRESSON CURVES Juan Carlos Pardo-Fernández, ngrid Van Keilegom and Wenceslao González-Manteiga Universidade de Vigo, Université catholique
More informationMinimum Hellinger Distance Estimation in a. Semiparametric Mixture Model
Minimum Hellinger Distance Estimation in a Semiparametric Mixture Model Sijia Xiang 1, Weixin Yao 1, and Jingjing Wu 2 1 Department of Statistics, Kansas State University, Manhattan, Kansas, USA 66506-0802.
More informationKernel density estimation with doubly truncated data
Kernel density estimation with doubly truncated data Jacobo de Uña-Álvarez jacobo@uvigo.es http://webs.uvigo.es/jacobo (joint with Carla Moreira) Department of Statistics and OR University of Vigo Spain
More information41903: Introduction to Nonparametrics
41903: Notes 5 Introduction Nonparametrics fundamentally about fitting flexible models: want model that is flexible enough to accommodate important patterns but not so flexible it overspecializes to specific
More informationLecture 3. Truncation, length-bias and prevalence sampling
Lecture 3. Truncation, length-bias and prevalence sampling 3.1 Prevalent sampling Statistical techniques for truncated data have been integrated into survival analysis in last two decades. Truncation in
More informationNonparametric Model Construction
Nonparametric Model Construction Chapters 4 and 12 Stat 477 - Loss Models Chapters 4 and 12 (Stat 477) Nonparametric Model Construction Brian Hartman - BYU 1 / 28 Types of data Types of data For non-life
More informationSemiparametric modeling and estimation of the dispersion function in regression
Semiparametric modeling and estimation of the dispersion function in regression Ingrid Van Keilegom Lan Wang September 4, 2008 Abstract Modeling heteroscedasticity in semiparametric regression can improve
More informationNonparametric Methods
Nonparametric Methods Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania July 28, 2009 Michael R. Roberts Nonparametric Methods 1/42 Overview Great for data analysis
More informationModel-free prediction intervals for regression and autoregression. Dimitris N. Politis University of California, San Diego
Model-free prediction intervals for regression and autoregression Dimitris N. Politis University of California, San Diego To explain or to predict? Models are indispensable for exploring/utilizing relationships
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationRegression Discontinuity Design Econometric Issues
Regression Discontinuity Design Econometric Issues Brian P. McCall University of Michigan Texas Schools Project, University of Texas, Dallas November 20, 2009 1 Regression Discontinuity Design Introduction
More informationLocal Polynomial Regression
VI Local Polynomial Regression (1) Global polynomial regression We observe random pairs (X 1, Y 1 ),, (X n, Y n ) where (X 1, Y 1 ),, (X n, Y n ) iid (X, Y ). We want to estimate m(x) = E(Y X = x) based
More informationA GENERALIZED ADDITIVE REGRESSION MODEL FOR SURVIVAL TIMES 1. By Thomas H. Scheike University of Copenhagen
The Annals of Statistics 21, Vol. 29, No. 5, 1344 136 A GENERALIZED ADDITIVE REGRESSION MODEL FOR SURVIVAL TIMES 1 By Thomas H. Scheike University of Copenhagen We present a non-parametric survival model
More informationAdditive Isotonic Regression
Additive Isotonic Regression Enno Mammen and Kyusang Yu 11. July 2006 INTRODUCTION: We have i.i.d. random vectors (Y 1, X 1 ),..., (Y n, X n ) with X i = (X1 i,..., X d i ) and we consider the additive
More informationNONPARAMETRIC ADJUSTMENT FOR MEASUREMENT ERROR IN TIME TO EVENT DATA: APPLICATION TO RISK PREDICTION MODELS
BIRS 2016 1 NONPARAMETRIC ADJUSTMENT FOR MEASUREMENT ERROR IN TIME TO EVENT DATA: APPLICATION TO RISK PREDICTION MODELS Malka Gorfine Tel Aviv University, Israel Joint work with Danielle Braun and Giovanni
More informationNonparametric Regression
Nonparametric Regression Econ 674 Purdue University April 8, 2009 Justin L. Tobias (Purdue) Nonparametric Regression April 8, 2009 1 / 31 Consider the univariate nonparametric regression model: where y
More informationNONPARAMETRIC CONFIDENCE INTERVALS FOR MONOTONE FUNCTIONS. By Piet Groeneboom and Geurt Jongbloed Delft University of Technology
NONPARAMETRIC CONFIDENCE INTERVALS FOR MONOTONE FUNCTIONS By Piet Groeneboom and Geurt Jongbloed Delft University of Technology We study nonparametric isotonic confidence intervals for monotone functions.
More informationNonparametric Function Estimation with Infinite-Order Kernels
Nonparametric Function Estimation with Infinite-Order Kernels Arthur Berg Department of Statistics, University of Florida March 15, 2008 Kernel Density Estimation (IID Case) Let X 1,..., X n iid density
More informationPractice Exam 1. (A) (B) (C) (D) (E) You are given the following data on loss sizes:
Practice Exam 1 1. Losses for an insurance coverage have the following cumulative distribution function: F(0) = 0 F(1,000) = 0.2 F(5,000) = 0.4 F(10,000) = 0.9 F(100,000) = 1 with linear interpolation
More informationWell-developed and understood properties
1 INTRODUCTION TO LINEAR MODELS 1 THE CLASSICAL LINEAR MODEL Most commonly used statistical models Flexible models Well-developed and understood properties Ease of interpretation Building block for more
More informationIntroduction. Linear Regression. coefficient estimates for the wage equation: E(Y X) = X 1 β X d β d = X β
Introduction - Introduction -2 Introduction Linear Regression E(Y X) = X β +...+X d β d = X β Example: Wage equation Y = log wages, X = schooling (measured in years), labor market experience (measured
More informationNonparametric Inference via Bootstrapping the Debiased Estimator
Nonparametric Inference via Bootstrapping the Debiased Estimator Yen-Chi Chen Department of Statistics, University of Washington ICSA-Canada Chapter Symposium 2017 1 / 21 Problem Setup Let X 1,, X n be
More informationAFT Models and Empirical Likelihood
AFT Models and Empirical Likelihood Mai Zhou Department of Statistics, University of Kentucky Collaborators: Gang Li (UCLA); A. Bathke; M. Kim (Kentucky) Accelerated Failure Time (AFT) models: Y = log(t
More informationECO Class 6 Nonparametric Econometrics
ECO 523 - Class 6 Nonparametric Econometrics Carolina Caetano Contents 1 Nonparametric instrumental variable regression 1 2 Nonparametric Estimation of Average Treatment Effects 3 2.1 Asymptotic results................................
More informationDensity estimation Nonparametric conditional mean estimation Semiparametric conditional mean estimation. Nonparametrics. Gabriel Montes-Rojas
0 0 5 Motivation: Regression discontinuity (Angrist&Pischke) Outcome.5 1 1.5 A. Linear E[Y 0i X i] 0.2.4.6.8 1 X Outcome.5 1 1.5 B. Nonlinear E[Y 0i X i] i 0.2.4.6.8 1 X utcome.5 1 1.5 C. Nonlinearity
More informationQuantile Regression for Residual Life and Empirical Likelihood
Quantile Regression for Residual Life and Empirical Likelihood Mai Zhou email: mai@ms.uky.edu Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA Jong-Hyeon Jeong email: jeong@nsabp.pitt.edu
More informationNonparametric Regression Härdle, Müller, Sperlich, Werwarz, 1995, Nonparametric and Semiparametric Models, An Introduction
Härdle, Müller, Sperlich, Werwarz, 1995, Nonparametric and Semiparametric Models, An Introduction Tine Buch-Kromann Univariate Kernel Regression The relationship between two variables, X and Y where m(
More informationIndividualized Treatment Effects with Censored Data via Nonparametric Accelerated Failure Time Models
Individualized Treatment Effects with Censored Data via Nonparametric Accelerated Failure Time Models Nicholas C. Henderson Thomas A. Louis Gary Rosner Ravi Varadhan Johns Hopkins University July 31, 2018
More informationSTAT Section 2.1: Basic Inference. Basic Definitions
STAT 518 --- Section 2.1: Basic Inference Basic Definitions Population: The collection of all the individuals of interest. This collection may be or even. Sample: A collection of elements of the population.
More informationOn Fractile Transformation of Covariates in Regression 1
1 / 13 On Fractile Transformation of Covariates in Regression 1 Bodhisattva Sen Department of Statistics Columbia University, New York ERCIM 10 11 December, 2010 1 Joint work with Probal Chaudhuri, Indian
More informationEfficient Estimation for the Partially Linear Models with Random Effects
A^VÇÚO 1 33 ò 1 5 Ï 2017 c 10 Chinese Journal of Applied Probability and Statistics Oct., 2017, Vol. 33, No. 5, pp. 529-537 doi: 10.3969/j.issn.1001-4268.2017.05.009 Efficient Estimation for the Partially
More informationBickel Rosenblatt test
University of Latvia 28.05.2011. A classical Let X 1,..., X n be i.i.d. random variables with a continuous probability density function f. Consider a simple hypothesis H 0 : f = f 0 with a significance
More informationDEPARTMENT MATHEMATIK ARBEITSBEREICH MATHEMATISCHE STATISTIK UND STOCHASTISCHE PROZESSE
Estimating the error distribution in nonparametric multiple regression with applications to model testing Natalie Neumeyer & Ingrid Van Keilegom Preprint No. 2008-01 July 2008 DEPARTMENT MATHEMATIK ARBEITSBEREICH
More informationRandom fraction of a biased sample
Random fraction of a biased sample old models and a new one Statistics Seminar Salatiga Geurt Jongbloed TU Delft & EURANDOM work in progress with Kimberly McGarrity and Jilt Sietsma September 2, 212 1
More informationChapter 3: Maximum Likelihood Theory
Chapter 3: Maximum Likelihood Theory Florian Pelgrin HEC September-December, 2010 Florian Pelgrin (HEC) Maximum Likelihood Theory September-December, 2010 1 / 40 1 Introduction Example 2 Maximum likelihood
More informationSmooth nonparametric estimation of a quantile function under right censoring using beta kernels
Smooth nonparametric estimation of a quantile function under right censoring using beta kernels Chanseok Park 1 Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 Short Title: Smooth
More informationTests of independence for censored bivariate failure time data
Tests of independence for censored bivariate failure time data Abstract Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of χ
More informationPractical Bayesian Optimization of Machine Learning. Learning Algorithms
Practical Bayesian Optimization of Machine Learning Algorithms CS 294 University of California, Berkeley Tuesday, April 20, 2016 Motivation Machine Learning Algorithms (MLA s) have hyperparameters that
More informationSimple Linear Regression (Part 3)
Chapter 1 Simple Linear Regression (Part 3) 1 Write an Estimated model Statisticians/Econometricians usually write an estimated model together with some inference statistics, the following are some formats
More informationInference for Non-stationary Time Series Auto Regression. By Zhou Zhou 1 University of Toronto January 21, Abstract
Inference for Non-stationary Time Series Auto Regression By Zhou Zhou 1 University of Toronto January 21, 2013 Abstract The paper considers simultaneous inference for a class of non-stationary autoregressive
More informationNonparametric Regression. Badr Missaoui
Badr Missaoui Outline Kernel and local polynomial regression. Penalized regression. We are given n pairs of observations (X 1, Y 1 ),...,(X n, Y n ) where Y i = r(x i ) + ε i, i = 1,..., n and r(x) = E(Y
More informationSTAT 6385 Survey of Nonparametric Statistics. Order Statistics, EDF and Censoring
STAT 6385 Survey of Nonparametric Statistics Order Statistics, EDF and Censoring Quantile Function A quantile (or a percentile) of a distribution is that value of X such that a specific percentage of the
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationNADARAYA WATSON ESTIMATE JAN 10, 2006: version 2. Y ik ( x i
NADARAYA WATSON ESTIMATE JAN 0, 2006: version 2 DATA: (x i, Y i, i =,..., n. ESTIMATE E(Y x = m(x by n i= ˆm (x = Y ik ( x i x n i= K ( x i x EXAMPLES OF K: K(u = I{ u c} (uniform or box kernel K(u = u
More informationSemiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data
Electronic Journal of Statistics Vol. 4 (2010) 133 160 ISSN: 1935-7524 DOI: 10.1214/09-EJS547 Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data
More informationEstimation of the Conditional Variance in Paired Experiments
Estimation of the Conditional Variance in Paired Experiments Alberto Abadie & Guido W. Imbens Harvard University and BER June 008 Abstract In paired randomized experiments units are grouped in pairs, often
More informationGoodness-of-fit tests for the cure rate in a mixture cure model
Biometrika (217), 13, 1, pp. 1 7 Printed in Great Britain Advance Access publication on 31 July 216 Goodness-of-fit tests for the cure rate in a mixture cure model BY U.U. MÜLLER Department of Statistics,
More informationNew Local Estimation Procedure for Nonparametric Regression Function of Longitudinal Data
ew Local Estimation Procedure for onparametric Regression Function of Longitudinal Data Weixin Yao and Runze Li The Pennsylvania State University Technical Report Series #0-03 College of Health and Human
More informationGenerated Covariates in Nonparametric Estimation: A Short Review.
Generated Covariates in Nonparametric Estimation: A Short Review. Enno Mammen, Christoph Rothe, and Melanie Schienle Abstract In many applications, covariates are not observed but have to be estimated
More informationA Review on Empirical Likelihood Methods for Regression
Noname manuscript No. (will be inserted by the editor) A Review on Empirical Likelihood Methods for Regression Song Xi Chen Ingrid Van Keilegom Received: date / Accepted: date Abstract We provide a review
More informationEstimation theory. Parametric estimation. Properties of estimators. Minimum variance estimator. Cramer-Rao bound. Maximum likelihood estimators
Estimation theory Parametric estimation Properties of estimators Minimum variance estimator Cramer-Rao bound Maximum likelihood estimators Confidence intervals Bayesian estimation 1 Random Variables Let
More informationNonparametric Predictive Inference (An Introduction)
Nonparametric Predictive Inference (An Introduction) SIPTA Summer School 200 Durham University 3 September 200 Hill s assumption A (n) (Hill, 968) Hill s assumption A (n) (Hill, 968) X,..., X n, X n+ are
More informationFinite Sample Performance of Semiparametric Binary Choice Estimators
University of Colorado, Boulder CU Scholar Undergraduate Honors Theses Honors Program Spring 2012 Finite Sample Performance of Semiparametric Binary Choice Estimators Sean Grover University of Colorado
More informationNonparametric two-sample tests of longitudinal data in the presence of a terminal event
Nonparametric two-sample tests of longitudinal data in the presence of a terminal event Jinheum Kim 1, Yang-Jin Kim, 2 & Chung Mo Nam 3 1 Department of Applied Statistics, University of Suwon, 2 Department
More informationLeast Squares Model Averaging. Bruce E. Hansen University of Wisconsin. January 2006 Revised: August 2006
Least Squares Model Averaging Bruce E. Hansen University of Wisconsin January 2006 Revised: August 2006 Introduction This paper developes a model averaging estimator for linear regression. Model averaging
More informationBayesian Nonparametric Accelerated Failure Time Models for Analyzing Heterogeneous Treatment Effects
Bayesian Nonparametric Accelerated Failure Time Models for Analyzing Heterogeneous Treatment Effects Nicholas C. Henderson Thomas A. Louis Gary Rosner Ravi Varadhan Johns Hopkins University September 28,
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO
UNIVERSITY OF CALIFORNIA, SAN DIEGO Estimation of the primary hazard ratio in the presence of a secondary covariate with non-proportional hazards An undergraduate honors thesis submitted to the Department
More informationLocal linear multiple regression with variable. bandwidth in the presence of heteroscedasticity
Local linear multiple regression with variable bandwidth in the presence of heteroscedasticity Azhong Ye 1 Rob J Hyndman 2 Zinai Li 3 23 January 2006 Abstract: We present local linear estimator with variable
More informationIV Quantile Regression for Group-level Treatments, with an Application to the Distributional Effects of Trade
IV Quantile Regression for Group-level Treatments, with an Application to the Distributional Effects of Trade Denis Chetverikov Brad Larsen Christopher Palmer UCLA, Stanford and NBER, UC Berkeley September
More informationUniform Convergence Rates for Nonparametric Estimation
Uniform Convergence Rates for Nonparametric Estimation Bruce E. Hansen University of Wisconsin www.ssc.wisc.edu/~bansen October 2004 Preliminary and Incomplete Abstract Tis paper presents a set of rate
More informationNONPARAMETRIC DENSITY ESTIMATION WITH RESPECT TO THE LINEX LOSS FUNCTION
NONPARAMETRIC DENSITY ESTIMATION WITH RESPECT TO THE LINEX LOSS FUNCTION R. HASHEMI, S. REZAEI AND L. AMIRI Department of Statistics, Faculty of Science, Razi University, 67149, Kermanshah, Iran. ABSTRACT
More informationMinimax Rate of Convergence for an Estimator of the Functional Component in a Semiparametric Multivariate Partially Linear Model.
Minimax Rate of Convergence for an Estimator of the Functional Component in a Semiparametric Multivariate Partially Linear Model By Michael Levine Purdue University Technical Report #14-03 Department of
More informationA Monte Carlo Comparison of Various Semiparametric Type-3 Tobit Estimators
ANNALS OF ECONOMICS AND FINANCE 4, 125 136 (2003) A Monte Carlo Comparison of Various Semiparametric Type-3 Tobit Estimators Insik Min Department of Economics, Texas A&M University E-mail: i0m5376@neo.tamu.edu
More informationTest for Discontinuities in Nonparametric Regression
Communications of the Korean Statistical Society Vol. 15, No. 5, 2008, pp. 709 717 Test for Discontinuities in Nonparametric Regression Dongryeon Park 1) Abstract The difference of two one-sided kernel
More informationSpline Density Estimation and Inference with Model-Based Penalities
Spline Density Estimation and Inference with Model-Based Penalities December 7, 016 Abstract In this paper we propose model-based penalties for smoothing spline density estimation and inference. These
More informationAdaptive test of conditional moment inequalities
Adaptive test of conditional moment inequalities Denis Chetverikov The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP36/12 Adaptive Test of Conditional Moment Inequalities
More informationNon-parametric Inference and Resampling
Non-parametric Inference and Resampling Exercises by David Wozabal (Last update 3. Juni 2013) 1 Basic Facts about Rank and Order Statistics 1.1 10 students were asked about the amount of time they spend
More informationInference on distributions and quantiles using a finite-sample Dirichlet process
Dirichlet IDEAL Theory/methods Simulations Inference on distributions and quantiles using a finite-sample Dirichlet process David M. Kaplan University of Missouri Matt Goldman UC San Diego Midwest Econometrics
More informationECON 4160, Autumn term Lecture 1
ECON 4160, Autumn term 2017. Lecture 1 a) Maximum Likelihood based inference. b) The bivariate normal model Ragnar Nymoen University of Oslo 24 August 2017 1 / 54 Principles of inference I Ordinary least
More informationOn the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models
On the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models Thomas Kneib Department of Mathematics Carl von Ossietzky University Oldenburg Sonja Greven Department of
More informationPricing and Risk Analysis of a Long-Term Care Insurance Contract in a non-markov Multi-State Model
Pricing and Risk Analysis of a Long-Term Care Insurance Contract in a non-markov Multi-State Model Quentin Guibert Univ Lyon, Université Claude Bernard Lyon 1, ISFA, Laboratoire SAF EA2429, F-69366, Lyon,
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 02: Overview Continued
Introduction to Empirical Processes and Semiparametric Inference Lecture 02: Overview Continued Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and Operations Research
More informationIntroduction to Econometrics Final Examination Fall 2006 Answer Sheet
Introduction to Econometrics Final Examination Fall 2006 Answer Sheet Please answer all of the questions and show your work. If you think a question is ambiguous, clearly state how you interpret it before
More informationFundamental concepts of functional data analysis
Fundamental concepts of functional data analysis Department of Statistics, Colorado State University Examples of functional data 0 1440 2880 4320 5760 7200 8640 10080 Time in minutes The horizontal component
More informationStatistica Sinica Preprint No: SS R4
Statistica Sinica Preprint No: SS-13-173R4 Title The independence process in conditional quantile location-scale models and an application to testing for monotonicity Manuscript ID SS-13-173R4 URL http://www.stat.sinica.edu.tw/statistica/
More informationApplied Econometrics (MSc.) Lecture 3 Instrumental Variables
Applied Econometrics (MSc.) Lecture 3 Instrumental Variables Estimation - Theory Department of Economics University of Gothenburg December 4, 2014 1/28 Why IV estimation? So far, in OLS, we assumed independence.
More informationSingle-level Models for Binary Responses
Single-level Models for Binary Responses Distribution of Binary Data y i response for individual i (i = 1,..., n), coded 0 or 1 Denote by r the number in the sample with y = 1 Mean and variance E(y) =
More informationEcon 582 Nonparametric Regression
Econ 582 Nonparametric Regression Eric Zivot May 28, 2013 Nonparametric Regression Sofarwehaveonlyconsideredlinearregressionmodels = x 0 β + [ x ]=0 [ x = x] =x 0 β = [ x = x] [ x = x] x = β The assume
More informationStatistical Properties of Numerical Derivatives
Statistical Properties of Numerical Derivatives Han Hong, Aprajit Mahajan, and Denis Nekipelov Stanford University and UC Berkeley November 2010 1 / 63 Motivation Introduction Many models have objective
More informationSize and Shape of Confidence Regions from Extended Empirical Likelihood Tests
Biometrika (2014),,, pp. 1 13 C 2014 Biometrika Trust Printed in Great Britain Size and Shape of Confidence Regions from Extended Empirical Likelihood Tests BY M. ZHOU Department of Statistics, University
More informationNonparametric estimation of extreme risks from heavy-tailed distributions
Nonparametric estimation of extreme risks from heavy-tailed distributions Laurent GARDES joint work with Jonathan EL METHNI & Stéphane GIRARD December 2013 1 Introduction to risk measures 2 Introduction
More informationUniversidade de Vigo
Universidade de Vigo Nonparametric regression with doubly truncated data Carla Moreira, Jacobo de Uña-Álvarez and Luís Meira-Machado Report 12/04 Discussion Papers in Statistics and Operation Research
More informationLocal Linear Estimation with Censored Data
Introduction Main Result Comparison with Related Works June 6, 2011 Introduction Main Result Comparison with Related Works Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression
More informationBootstrap, Jackknife and other resampling methods
Bootstrap, Jackknife and other resampling methods Part III: Parametric Bootstrap Rozenn Dahyot Room 128, Department of Statistics Trinity College Dublin, Ireland dahyot@mee.tcd.ie 2005 R. Dahyot (TCD)
More informationECON 721: Lecture Notes on Nonparametric Density and Regression Estimation. Petra E. Todd
ECON 721: Lecture Notes on Nonparametric Density and Regression Estimation Petra E. Todd Fall, 2014 2 Contents 1 Review of Stochastic Order Symbols 1 2 Nonparametric Density Estimation 3 2.1 Histogram
More informationWeb Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D.
Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Ruppert A. EMPIRICAL ESTIMATE OF THE KERNEL MIXTURE Here we
More informationGOODNESS-OF-FIT TEST FOR RANDOMLY CENSORED DATA BASED ON MAXIMUM CORRELATION. Ewa Strzalkowska-Kominiak and Aurea Grané (1)
Working Paper 4-2 Statistics and Econometrics Series (4) July 24 Departamento de Estadística Universidad Carlos III de Madrid Calle Madrid, 26 2893 Getafe (Spain) Fax (34) 9 624-98-49 GOODNESS-OF-FIT TEST
More informationLinear Regression Model. Badr Missaoui
Linear Regression Model Badr Missaoui Introduction What is this course about? It is a course on applied statistics. It comprises 2 hours lectures each week and 1 hour lab sessions/tutorials. We will focus
More informationSupplementary Materials for Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach
Supplementary Materials for Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach Part A: Figures and tables Figure 2: An illustration of the sampling procedure to generate a surrogate
More informationTesting for Rank Invariance or Similarity in Program Evaluation: The Effect of Training on Earnings Revisited
Testing for Rank Invariance or Similarity in Program Evaluation: The Effect of Training on Earnings Revisited Yingying Dong and Shu Shen UC Irvine and UC Davis Sept 2015 @ Chicago 1 / 37 Dong, Shen Testing
More informationRegression model with left truncated data
Regression model with left truncated data School of Mathematics Sang Hui-yan 00001107 October 13, 2003 Abstract For linear regression model with truncated data, Bhattacharya, Chernoff, and Yang(1983)(BCY)
More informationMultivariate random variables
Multivariate random variables DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Joint distributions Tool to characterize several
More information