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44
Variance Function Estimation in . . .
, 1986
"... Heteroscedastic regression models are used to analyze data in a variety of fields, including economics, engineering and the biological and physical sciences. Often, the heteroscedasticity is modeled as a function of the regression and other structural parameters. Standard asymptotic theory implies t ..."
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Cited by 121 (9 self)
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Heteroscedastic regression models are used to analyze data in a variety of fields, including economics, engineering and the biological and physical sciences. Often, the heteroscedasticity is modeled as a function of the regression and other structural parameters. Standard asymptotic theory implies that under reasonable conditions, how one estimates the variance function, in particular the structural parameters, has no effect on the first order properties of the estimates of the regression parameters; however, it has been noted in practice that how one estimates the variance function does matter. Furthermore. in some settings. estimation of the variance function is of independent interest or plays an important role in the properties of estimates of other quantities besides the regression parameters. We develop a general theory for variance function estimation in regression which includes most methods in common use. In particular.
Nonparametric regression with errors in variables
 Annals of Statistics
, 1993
"... The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction ofa new class of kernel estimators. It is shown that optima/local and global rates of convergence of these kernel estimators can be ..."
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Cited by 84 (1 self)
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The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction ofa new class of kernel estimators. It is shown that optima/local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth. It is also shown that these results hold uniformly over a class of joint distributions of the response and the covariates, which includes ordinary smooth regression functions as well as covariates with distributions satisfying regularity conditions. Furthermore, to achieve optimality, we show that the convergence rates of all nonparametric estimators have a lower bound possessed by the kernel estimators. oAbbreviated title. Errorinvariable regression AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62J99. Key words and phrases. Nonparametric regression; Kernel estimator; Errors in variables; Optimal rates
Bayesian Smoothing and Regression Splines for Measurement Error Problems
 Journal of the American Statistical Association
, 2001
"... In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely dicult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this paper we describe Bayes ..."
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Cited by 41 (7 self)
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In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely dicult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this paper we describe Bayesian approaches to modeling a exible regression function when the predictor variable is measured with error. The regression function is modeled with smoothing splines and regression P{splines. Two methods are described for exploration of the posterior. The rst is called iterative conditional modes (ICM) and is only partially Bayesian. ICM uses a componentwise maximization routine to nd the mode of the posterior. It also serves to create starting values for the second method, which is fully Bayesian and uses Markov chain Monte Carlo techniques to generate observations from the joint posterior distribution. Using the MCMC approach has the advantage that interval estimates that directly model and adjust for the measurement error are easily calculated. We provide simulations with several nonlinear regression functions and provide an illustrative example. Our simulations indicate that the frequentist mean squared error properties of the fully Bayesian method are better than those of ICM and also of previously proposed frequentist methods, at least in the examples we have studied. KEY WORDS: Bayesian methods; Eciency; Errors in variables; Functional method; Generalized linear models; Kernel regression; Measurement error; Nonparametric regression; P{splines; Regression Splines; SIMEX; Smoothing Splines; Structural modeling. Short title. Nonparametric Regression with Measurement Error Author Aliations Scott M. Berry (Email: scott@berryconsultants.com) is Statistical Scientist,...
SimulationExtrapolation: The Measurement Error Jackknife
 Journal of the American Statistical Association
, 1995
"... by ..."
Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models
 Communications in Statistics, Series A
, 1989
"... generalized linear model; Hestimation; measurement error; structural IIIOdels; unbiased estimation. Let Wbe a normal random variable with mean ~ variance 0 2 • and known Conditions on the function fee) are given under which there ' exists an unbiased estimator, T(W), of f(~) real~. for all In& ..."
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Cited by 32 (1 self)
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generalized linear model; Hestimation; measurement error; structural IIIOdels; unbiased estimation. Let Wbe a normal random variable with mean ~ variance 0 2 • and known Conditions on the function fee) are given under which there ' exists an unbiased estimator, T(W), of f(~) real~. for all In'particular it is shown that f(e) must De an entire function over the complex plane. Infinite series solutions for fee) are obtained which are shown to be valid under growth conditions of the derivatives, f(k)(e), of fee). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurementerror models as a means of finding unbiased score functions when measurement error is normally distributed. Relative efficiencies comparing the proposed method to the use of conditional scores (Stefanski and Carroll, 1987) are given for the Poisson regression model with canonical link. 1.1 The Estimation Problem 1.
Corrections for Exposure Measurement Error in Logistic Regression Models With an Application to Nutritional Data
, 1994
"... this paper is to show how the method of Rosner et al. may be extended by including a second term in the Taylor series approximation. The resulting estimates can then be compared with one another. The two approximations are derived under less restrictive assumptions than those used by Rosner et al. I ..."
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Cited by 14 (0 self)
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this paper is to show how the method of Rosner et al. may be extended by including a second term in the Taylor series approximation. The resulting estimates can then be compared with one another. The two approximations are derived under less restrictive assumptions than those used by Rosner et al. In particular, the distribution of the measurement error need not be specified and the disease under study does not have to be rare. The formulae for the corrected estimates and their standard errors are essentially similar for the two approximations. When the measurement error model is linear, the standard errors include a component which reflects the variability of the measurement error parameters estimated from the validation data. We apply the two approximate correction methods to baseline data from an epidemiological cohort study. A logistic model of the effect of nutrient intake on the ratio of serum HDL cholesterol to total cholesterol is used as an example. The two correction methods lead to very similar estimates, both of which are markedly different from the naive logistic estimates. The approach of Rosner et al. together with the extension is described in the next section. There we also compare the corrections to others proposed in the literature, and describe the software used in the computations. The example is presented in section 3, where we also examine the assumptions of the correction methods. Section 4 gives some conclusions. 2. APPROXIMATE MODELS Suppose a dichotomous response variable Y (disease status, for example) depends on predictor vectors X (p 1 \Theta 1) and U (p 2 \Theta 1) through a logistic model P(Y = 1jx; u) = F (ff + fi
Semiparametric quasilikelihood and variance function estimation in measurement error models
 Journal of Econometrics
, 1993
"... We consider a quasilikelihood/variance function model when a predictor X is measured with error and a surrogate W is observed. When in addition to a primary data set containing (Y, W) a validation data set exists for which (X, W) is observed, we can (i) estimate the first and second moments of the r ..."
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Cited by 13 (0 self)
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We consider a quasilikelihood/variance function model when a predictor X is measured with error and a surrogate W is observed. When in addition to a primary data set containing (Y, W) a validation data set exists for which (X, W) is observed, we can (i) estimate the first and second moments of the response Y given W by kernel regression; (ii) use quasilikelihood and variance function techniques to estimate the regression parameters as well as variance structure parameters. The estimators are shown to be asymptotically normally distributed, with asymptotic variance depending on the size of the validation data set and not on the bandwith used in the kernel estimates. A more refined analysis of the asymptotic covariance shows that the optimal bandwidth converges to zero at the rate n‘13. 1.
On closed form semiparametric estimators for measurement error models
 Statist. Sin
, 2006
"... Abstract: We examine the locally efficient semiparametric estimator proposed by Tsiatis and Ma (2004) in the situation when a sufficient and complete statistic exists. We derive a closed form solution and show that when implemented in generalized linear models with normal measurement error, this est ..."
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Cited by 8 (7 self)
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Abstract: We examine the locally efficient semiparametric estimator proposed by Tsiatis and Ma (2004) in the situation when a sufficient and complete statistic exists. We derive a closed form solution and show that when implemented in generalized linear models with normal measurement error, this estimator is equivalent to the efficient score estimator in Stefanski and Carroll (1987). We also demonstrate how other consistent semiparametric estimators naturally emerge. The method is used in an extension of the usual generalized linear models. Key words and phrases: Measurement error models, semiparametric estimator. 1.
NONPARAMETRIC FUNCTION ESTIMATION INVOLVING ERRORSINVARIABLES
, 1990
"... We examine the effect oferrors in covariates in rionparametric function estimation. These functions include densities, regressions and conditional quantiles. To estimate these functions, we use the idea of deconvoluting kernels in conjunction with the ordinary kernel methods. We also discuss a new c ..."
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Cited by 7 (1 self)
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We examine the effect oferrors in covariates in rionparametric function estimation. These functions include densities, regressions and conditional quantiles. To estimate these functions, we use the idea of deconvoluting kernels in conjunction with the ordinary kernel methods. We also discuss a new class of function estimators based on local polynomials. oAbbreviated title. Errorinvariable regression AMS 1980 6ubject cla66ification. Primary 62G20. Secondary 62G05, 62J99. Key wom6 and phra6e6. Nonparametric regression; Deconvolution; Kernel estimator;
2011): “Method of moments estimation and identifiability of semiparametric nonlinear errorsinvariables models
 Journal of Econometrics
"... This paper deals with a nonlinear errorsinvariables model where the distributions of the unobserved predictor variables and of the measurement errors are nonparametric. Using the instrumental variable approach, we propose method of moments estimators for the unknown parameters and simulationbased ..."
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Cited by 6 (2 self)
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This paper deals with a nonlinear errorsinvariables model where the distributions of the unobserved predictor variables and of the measurement errors are nonparametric. Using the instrumental variable approach, we propose method of moments estimators for the unknown parameters and simulationbased estimators to overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals. Both estimators are consistent and asymptotically normally distributed under fairly general regularity conditions. Moreover, rootn consistent semiparametric estimators and a rank condition for model identifiability are derived using the combined methods of nonparametric technique and Fourier deconvolution.