Leonard A. Stefanski

Simulation-Extrapolation Estimation in Parametric Measurement Error Models

We describe a simulation-based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated. The method entails adding add...

Deconvolving kernel density estimators

This paper considers estimation of a continuous bounded probability density when observations from the density are contaminated by additive measurement errors having a known distribution. Properties of the estimator obtained by deconvolving a kernel estimator of the observed data are investigated. When the kernel used is sufficiently smooth the deconvolved estimator is shown to be pointwise consistent and bounds on its integrated mean squared error are derived. Very weak assumptions are made on the measurement-error density thereby permitting a comparison of the effects of different types of measurement error on the deconvolved estimator

Approximate Quasi-likelihood Estimation in Models with Surrogate Predictors

Abstract We consider quasi-likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error. We review and extend four approaches to estimation in this problem, all of them based on small measurement error approximations. A taxonomy of the data sets likely to be available in measurement error studies is developed. An asymptotic theory based on this taxonomy is obtained and includes measurement error and Berkson error models as special cases.

Simulation-Extrapolation: The Measurement Error Jackknife

Abstract This article provides theoretical support for our simulation-based estimation procedure, SIMEX, for measurement error models. We do so by establishing a strong relationship between SIMEX estimation and jackknife estimation. A result of our investigation is the identification of a variance estimation method for SIMEX that parallels jackknife variance estimation. Data from the Framingham Heart Study are used to illustrate the variance estimation procedure in logistic regression measurement error models.

The calculus of M-estimation

Since the seminal papers by Huber in the 1960s, M-estimation methods (also known as estimating equation methods) have been increasingly important for asymptotic analysis and approximate inference. This article illustrates the breadth and generality of the M-estimation approach, thereby facilitating its use inpractice and in the classroom as a unifying approach to the study of large-sample inference.

Conditionally Unbiased Bounded-Influence Estimation in General Regression Models, with Applications to Generalized Linear Models

Abstract In this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied. A subclass of the class of M estimators is defined by imposing the restriction that the score function must be conditionally unbiased, given x. Within this class of conditionally Fisher-consistent estimators, optimal bounded-influence estimators of regression parameters are identified, and their asymptotic properties are studied. The estimators studied in this article and the efficient bounded-influence estimators studied by Stefanski, Carroll, and Ruppert (1986) depend on an auxiliary centering constant and nuisance matrix. The centering constant can be given explicitly for the conditionally Fisher-consistent estimators, and thus they are easier to compute than the estimators studied by Stefanski et al. (1986). In addition, estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally Fisher-consistent estimators; the sam...

Unbiased estimation of a nonlinear function a normal mean with application to measurement err oorf models

Let W be a normal random variable with mean μand known variance σ2. Conditions on the function f(·) are given under which there exists an unbiased estimator, f(W), of f(μ) for all real μ. In particular it is shown that f(·) must be an entire function over the complex plane. Infinite series solutions for F(·) are obtained which are shown to be valid under growth conditions of the derivatives, fk( ·), of f(·). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurement-error 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.

Asymptotics for the SIMEX Estimator in Nonlinear Measurement Error Models

Abstract Cook and Stefanski have described a computer-intensive method, the SIMEX method, for approximately consistent estimation in regression problems with additive measurement error. In this article we derive the asymptotic distribution of their estimators and show how to compute estimated standard errors. These standard error estimators can either be used alone or as prepivoting devices in a bootstrap analysis. We also give theoretical justification to some of the phenomena observed by Cook and Stefanski in their simulations.

Rates of convergence of some estimators in a class of deconvolution problems

This paper studies the problem of estimating the density of U when only independent copies of X = U + Z is observable where Z is an independent measurement error. Convergence rates of a family of deconvolved kernel density estimators are obtained under different assumptions on the density of Z.

Controlling Variable Selection by the Addition of Pseudovariables

We propose a new approach to variable selection designed to control the false selection rate (FSR), defined as the proportion of uninformative variables included in selected models. The method works by adding a known number of pseudovariables to the real dataset, running a variable selection procedure, and monitoring the proportion of pseudovariables falsely selected. Information obtained from bootstrap-like replications of this process is used to estimate the proportion of falsely selected real variables and to tune the selection procedure to control the FSR.