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Dietary data collected from 24-hour dietary recalls are observed with significant measurement errors. In the nonparametric curve estimation literature, much of the effort has been devoted to designing methods that are consistent under contamination by noise, and which have been traditionally applied for analyzing those data. However, some foods such as alcohol or fruits are consumed only episodically, and may not be consumed during the day when the 24-hour recall is administered. These so-called excess zeros make existing nonparametric estimators break down, and new techniques need to be developed for such data. We develop two new consistent semiparametric estimators of the distribution of such episodically consumed food data, making parametric assumptions only on some less important parts of the model. We establish its theoretical properties and illustrate the good performance of our fully data-driven method in simulated and real data. Supplementary materials for this article are available online.

Model averaging generally provides better predictions than model selection, but the existing model averaging methods cannot lead to parsimonious models. Parsimony is an especially important property when the number of parameters is large. To achieve a parsimonious model averaging coefficient estimator, we suggest a novel criterion for choosing weights. Asymptotic properties are derived in two practical scenarios: (i) one or more correct models exist in the candidate model set and (ii) all candidate models are misspecified. Under the former scenario, it is proved that our method can put the weight one to the smallest correct model and the resulting model averaging estimators of coefficients have many zeros and thus lead to a parsimonious model. The asymptotic distribution of the estimators is also provided. Under the latter scenario, prediction is mainly focused on and we prove that the proposed procedure is asymptotically optimal in the sense that its squared prediction loss and risk are asymptotically identical to those of the best—but infeasible—model averaging estimator. Numerical analysis shows the promise of the proposed procedure over existing model averaging and selection methods.

© 2017 American Statistical Association. We consider a problem motivated by issues in nutritional epidemiology, across diseases and populations. In this area, it is becoming increasingly common for diseases to be modeled by a single diet score, such as the Healthy Eating Index, the Mediterranean Diet Score, etc. For each disease and for each population, a partially linear single-index model is fit. The partially linear aspect of the problem is allowed to differ in each population and disease. However, and crucially, the single-index itself, having to do with the diet score, is common to all diseases and populations, and the nonparametrically estimated functions of the single-index are the same up to a scale parameter. Using B-splines with an increasing number of knots, we develop a method to solve the problem, and display its asymptotic theory. An application to the NIH-AARP Study of Diet and Health is described, where we show the advantages of using multiple diseases and populations simultaneously rather than one at a time in understanding the effect of increased Milk consumption. Simulations illustrate the properties of the methods. Supplementary materials for this article are available online.

© 2017 American Statistical Association. Prediction precision is arguably the most relevant criterion of a model in practice and is often a sought after property. A common difficulty with covariates measured with errors is the impossibility of performing prediction evaluation on the data even if a model is completely given without any unknown parameters. We bypass this inherent difficulty by using special properties on moment relations in linear regression models with measurement errors. The end product is a model selection procedure that achieves the same optimality properties that are achieved in classical linear regression models without covariate measurement error. Asymptotically, the procedure selects the model with the minimum prediction error in general, and selects the smallest correct model if the regression relation is indeed linear. Our model selection procedure is useful in prediction when future covariates without measurement error become available, for example, due to improved technology or better management and design of data collection procedures. Supplementary materials for this article are available online.

© 2015, © American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. Recent years have seen active developments of various penalized regression methods, such as LASSO and elastic net, to analyze high-dimensional data. In these approaches, the direction and length of the regression coefficients are determined simultaneously. Due to the introduction of penalties, the length of the estimates can be far from being optimal for accurate predictions. We introduce a new framework, regression by projection, and its sparse version to analyze high-dimensional data. The unique nature of this framework is that the directions of the regression coefficients are inferred first, and the lengths and the tuning parameters are determined by a cross-validation procedure to achieve the largest prediction accuracy. We provide a theoretical result for simultaneous model selection consistency and parameter estimation consistency of our method in high dimension. This new framework is then generalized such that it can be applied to principal components analysis, partial least squares, and canonical correlation analysis. We also adapt this framework for discriminant analysis. Compared with the existing methods, where there is relatively little control of the dependency among the sparse components, our method can control the relationships among the components. We present efficient algorithms and related theory for solving the sparse regression by projection problem. Based on extensive simulations and real data analysis, we demonstrate that our method achieves good predictive performance and variable selection in the regression setting, and the ability to control relationships between the sparse components leads to more accurate classification. In supplementary materials available online, the details of the algorithms and theoretical proofs, and R codes for all simulation studies are provided.

© 2014, © 2014 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. We consider the problem of estimating the density of a random variable when precise measurements on the variable are not available, but replicated proxies contaminated with measurement error are available for sufficiently many subjects. Under the assumption of additive measurement errors this reduces to a problem of deconvolution of densities. Deconvolution methods often make restrictive and unrealistic assumptions about the density of interest and the distribution of measurement errors, for example, normality and homoscedasticity and thus independence from the variable of interest. This article relaxes these assumptions and introduces novel Bayesian semiparametric methodology based on Dirichlet process mixture models for robust deconvolution of densities in the presence of conditionally heteroscedastic measurement errors. In particular, the models can adapt to asymmetry, heavy tails, and multimodality. In simulation experiments, we show that our methods vastly outperform a recent Bayesian approach based on estimating the densities via mixtures of splines. We apply our methods to data from nutritional epidemiology. Even in the special case when the measurement errors are homoscedastic, our methodology is novel and dominates other methods that have been proposed previously. Additional simulation results, instructions on getting access to the dataset and R programs implementing our methods are included as part of online supplementary materials.

We describe a new approach to analyze chirp syllables of free-tailed bats from two regions of Texas in which they are predominant: Austin and College Station. Our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps. The data are analyzed by modeling spectrograms of the chirps as responses in a Bayesian functional mixed model. Given the variable chirp lengths, we compute the spectrograms on a relative time scale interpretable as the relative chirp position, using a variable window overlap based on chirp length. We use 2D wavelet transforms to capture correlation within the spectrogram in our modeling and obtain adaptive regularization of the estimates and inference for the regions-specific spectrograms. Our model includes random effect spectrograms at the bat level to account for correlation among chirps from the same bat, and to assess relative variability in chirp spectrograms within and between bats. The modeling of spectrograms using functional mixed models is a general approach for the analysis of replicated nonstationary time series, such as our acoustical signals, to relate aspects of the signals to various predictors, while accounting for between-signal structure. This can be done on raw spectrograms when all signals are of the same length, and can be done using spectrograms defined on a relative time scale for signals of variable length in settings where the idea of defining correspondence across signals based on relative position is sensible.

In simulation studies for discriminant analysis, misclassification errors are often computed using the Monte Carlo method, by testing a classifier on large samples generated from known populations. Although large samples are expected to behave closely to the underlying distributions, they may not do so in a small interval or region, and thus may lead to unexpected results. We demonstrate with an example that the LDA misclassification error computed via the Monte Carlo method may often be smaller than the Bayes error. We give a rigorous explanation and recommend a method to properly compute misclassification errors.

This article describes a class of heteroscedastic generalized linear regression models in which a subset of the regression parameters are rescaled nonparametrically, and develops efficient semiparametric inferences for the parametric components of the models. Such models provide a means to adapt for heterogeneity in the data due to varying exposures, varying levels of aggregation, and so on. The class of models considered includes generalized partially linear models and nonparametrically scaled link function models as special cases. We present an algorithm to estimate the scale function nonparametrically, and obtain asymptotic distribution theory for regression parameter estimates. In particular, we establish that the asymptotic covariance of the semiparametric estimator for the parametric part of the model achieves the semiparametric lower bound. We also describe bootstrap-based goodness-of-scale test. We illustrate the methodology with simulations, published data, and data from collaborative research on ultrasound safety.

Penalized splines have become an increasingly popular tool for nonparametric smoothing because of their use of low-rank spline bases, which makes computations tractable while maintaining accuracy as good as smoothing splines. This article extends penalized spline methodology by both modeling the variance function nonparametrically and using a spatially adaptive smoothing parameter. This combination is needed for satisfactory inference and can be implemented effectively by Bayesian MCMC. The variance process controlling the spatially adaptive shrinkage of the mean and the variance of the heteroscedastic error process are modeled as log-penalized splines. We discuss the choice of priors and extensions of the methodology, in particular, to multivariate smoothing. A fully Bayesian approach provides the joint posterior distribution of all parameters, in particular, of the error standard deviation and penalty functions. MATLAB, C, and FORTRAN programs implementing our methodology are publicly available.

The Wang–Landau (WL) algorithm is an adaptive Markov chain Monte Carlo algorithm used to calculate the spectral density for a physical system. A remarkable feature of the WL algorithm is that it is not trapped by local energy minima, which is very important for systems with rugged energy landscapes. This feature has led to many successful applications of the algorithm in statistical physics and biophysics; however, there does not exist rigorous theory to support its convergence, and the estimates produced by the algorithm can reach only a limited statistical accuracy. In this article we propose the stochastic approximation Monte Carlo (SAMC) algorithm, which overcomes the shortcomings of the WL algorithm. We establish a theorem concerning its convergence. The estimates produced by SAMC can be improved continuously as the simulation proceeds. SAMC also extends applications of the WL algorithm to continuum systems. The potential uses of SAMC in statistics are discussed through two classes of applications, i...

There is a large and increasing literature in methods of estimation for spatial data with binary responses. The goal of this article is to describe some of these methods for the autologistic spatial model, and to discuss computational issues associated with them. The main way we do this is via illustration using a spatial epidemiology data set involving liver cancer. We first demonstrate why maximum likelihood is not currently feasible as a method of estimation in the spatial setting with binary data using the autologistic model. We then discuss alternative methods, including pseudo likelihood, generalized pseudo likelihood, and Monte Carlo maximum likelihood estimators. We describe their asymptotic efficiencies and the computational effort required to compute them. These three methods are applied to the data set and compared in a simulation experiment.

In this article we study penalized regression splines (P-splines), which are low-order basis splines with a penalty to avoid undersmoothing. Such P-splines are typically not spatially adaptive, and hence can have trouble when functions are varying rapidly. Our approach is to model the penalty parameter inherent in the P-spline method as a heteroscedastic regression function. We develop a full Bayesian hierarchical structure to do this and use Markov chain Monte Carlo techniques for drawing random samples from the posterior for inference. The advantage of using a Bayesian approach to P-splines is that it allows for simultaneous estimation of the smooth functions and the underlying penalty curve in addition to providing uncertainty intervals of the estimated curve. The Bayesian credible intervals obtained for the estimated curve are shown to have pointwise coverage probabilities close to nominal. The method is extended to additive models with simultaneous spline-based penalty functions for the unknown funct...

We consider marginal generalized semiparametric partially linear models for clustered data. Lin and Carroll derived the semiparametric efficient score function for this problem in the multivariate Gaussian case, but they were unable to construct a semiparametric efficient estimator that actually achieved the semiparametric information bound. Here we propose such an estimator and generalize the work to marginal generalized partially linear models. We investigate asymptotic relative efficiencies of the estimators that ignore the within-cluster correlation structure either in nonparametric curve estimation or throughout. We evaluate the finite-sample performance of these estimators through simulations and illustrate it using a longitudinal CD4 cell count dataset. Both theoretical and numerical results indicate that properly taking into account the within-subject correlation among the responses can substantially improve efficiency.

The partially linear model Y = XTβ + ν(Z) + ϵ has been studied extensively when data are completely observed. In this article, we consider the case where the covariate X is sometimes missing, with missingness probability π depending on (Y, Z). New methods are developed for estimating β and ν(·). Our methods are shown to outperform asymptotically methods based only on the complete data. Asymptotic efficiency is discussed, and the semiparametric efficient score function is derived. Justification of the use of the nonparametric bootstrap in this context is sketched. The proposed estimators are extended to a working independence analysis of longitudinal/clustered data and applied to analyze an AIDS clinical trial dataset. The results of a simulation experiment are also given to illustrate our approach.

We propose a modification of local polynomial time series regression estimators that improves efficiency when the innovation process is autocorrelated. The procedure is based on a pre-whitening transformation of the dependent variable that must be estimated from the data. We establish the asymptotic distribution of our estimator under weak dependence conditions. We show that the proposed estimation procedure is more efficient than the conventional local polynomial method. We also provide simulation evidence to suggest that gains can be achieved in moderate-sized samples.

We consider nonparametric regression in a longitudinal marginal model of generalized estimating equation (GEE) type with a time-varying covariate in the situation where the number of observations per subject is finite and the number of subjects is large. In such models, the basic shape of the regression function is affected only by the covariate values and not otherwise by the ordering of the observations. Two methods of estimating the nonparametric function can be considered: kernel methods and spline methods. Recently, surprising evidence has emerged suggesting that for kernel methods previously proposed in the literature, it is generally asymptotically preferable to ignore the correlation structure in our marginal model and instead assume that the data are independent, that is, working independence in the GEE jargon. As seen through equivalent kernel results, in univariate independent data problems splines and kernels have similar behavior; smoothing splines are equivalent to kernel regression with a s...

In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely difficult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this article we describe Bayesian approaches to modeling a flexible 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 first, called the iterative conditional modes (ICM), is only partially Bayesian. ICM uses a componentwise maximization routine to find 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 (MCMC) techniques to generate observations from the joint posterior distribution. Use of the MCMC approach has the advantage that interval estimates that directly model and ...

The sandwich estimator, also known as robust covariance matrix estimator, heteroscedasticity-consistent covariance matrix estimate, or empirical covariance matrix estimator, has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations. Its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified. Surprisingly though, there has been little discussion of properties of the sandwich method other than consistency. We investigate the sandwich estimator in quasi-likelihood models asymptotically, and in the linear case analytically. We show that under certain circumstances when the quasi-likelihood model is correct, the sandwich estimate is often far more variable than the usual parametric variance estimate. The increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p...

We consider estimation in a semiparametric generalized linear model for clustered data using estimating equations. Our results apply to the case where the number of observations per cluster is finite, whereas the number of clusters is large. The mean of the outcome variable μ is of the form g(μ) = XTβ + θ(T), where g(·) is a link function, X and T are covariates, β is an unknown parameter vector, and θ(t) is an unknown smooth function. Kernel estimating equations proposed previously in the literature are used to estimate the infinite-dimensional nonparametric function θ(t), and a profile-based estimating equation is used to estimate the finite-dimensional parameter vector β. We show that for clustered data, this conventional profile-kernel method often fails to yield a √n-consistent estimator of β along with appropriate inference unless working independence is assumed or θ(t) is artificially undersmoothed, in which case asymptotic inference is possible. To gain insight into these results, we derive the se...

An important problem in studying the etiology of colon cancer is understanding the relationship between DNA adduct levels (broadly, DNA damage) in cells within colonic crypts in distal and proximal parts of the colon, following treatment with a carcinogen and different types of diet. In particular, it is important to understand whether rats who have elevated adduct levels in particular positions in distal region crypts also have elevated levels in the same positions of the crypts in proximal regions, and whether this relationship depends on diet. We cast this problem as estimating the correlation function of two responses as a function of a covariate for studies where both responses are measured on the same experimental units but not the same subsampling units. Parametric and nonparametric methods are developed and applied to a dataset from an ongoing study, leading to potentially important and surprising biological results. Theoretical calculations suggest that the nonparametric method, based on nonparam...

We consider local polynomial kernel regression with a single covariate for clustered data using estimating equations. We assume that at most m < ∞ observations are available on each cluster. In the case of random regressors, with no measurement error in the predictor, we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent. In the further special case of longitudinal data on individuals with fixed common observation times, we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average. We also consider what happens when the predictor is measured with error. Using the SIMEX approach to correct for measurement error, we construct an asymptotic theory for both the pooled and the weighted average estimators. Surprisingly, for the same amount of smoothing, the weighted a...

Large epidemiological studies often require short food frequency questionnaires (FFQ) to minimize the respondent burden or to control for confounding from dietary factors. In this analysis, we compared the extensively used National Cancer Institute-Block 60- and 100-item FFQs with one another and with usual intake as estimated from 12 days of validation data. The analysis focused on nutrients from different aspects of the diet, including energy, fat, saturated fat, beta-carotene, dietary fiber, and vitamin C. By use of an errors-in-variables analysis, the correlations of usual intake with the two types of FFQs for these nutrients were not different. Attenuation coefficients, a measure of misclassification error, for both FFQs were of similar magnitude and indicated that substantial attenuation of logistic regression coefficients would result from either FFQ. Our results confirm previous analyses describing the validity and utility of the 60-item FFQ (Epidemiology 1, 58-64, 1990) and indicate that it is essentially equivalent to the 100-item FFQ for epidemiological analyses of major nutrients.

Estimating equations have found wide popularity recently in parametric problems, yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula. Motivated by a problem in nutritional epidemiology, we use estimating equations to derive nonparametric estimators of a “parameter” depending on a predictor. The nonparametric component is estimated via local polynomials with loess or kernel weighting; asymptotic theory is derived for the latter. In keeping with the estimating equation paradigm, variances of the nonparametric function estimate are estimated using the sandwich method, in an automatic fashion, without the need (typical in the literature) to derive asymptotic formulas and plug-in an estimate of a density function. The same philosophy is used in estimating the bias of the nonparametric function; that is, an empirical method is used without deriving asymptotic theory on a case-by-case basis. The methods are applied to a series of examples. The applicatio...

We consider generalized linear mixed models (GLMMs) for clustered data when one of the predictors is measured with error. When the measurement error is additive and normally distributed and the error-prone predictor is itself normally distributed, we show that the observed data also follow a GLMM but with a different fixed effects structure from the original model, a different and more complex random effects structure, and restrictions on the parameters. This characterization enables us to compute the biases that result in common GLMMs when one ignores measurement error. For instance, in one common situation the biases in parameter estimates become larger as the number of observations within a cluster increases, both for regression coefficients and for variance components. Parameter estimation is described using the SIMEX method, a relatively new functional method that makes no assumptions about the structure of the unobservable predictors. Simulations and an example illustrate the results.

We address the following question: What is the pattern of human exposure to ozone in Harris County (Houston) since 1980? While there has been considerable research on characterizing ozone measured at fixed monitoring stations, little is known about ozone away from the monitoring stations, and whether areas of higher ozone correspond to areas of high population density. To address this question, we build a spatial-temporal model for hourly ozone levels that predicts ozone at any location in Harris County at any time between 1980 and 1993. Along with building the model, we develop a fast model-fitting method that can cope with the massive amounts of available data and takes into account the substantial number of missing observations. Having built the model, we combine it with census tract information, focusing on young children. We conclude that the highest ozone levels occur at locations with relatively small populations of young children. Using various measures of exposure, we estimate that expos...

Methods are devised for estimating the parameters of a prospective logistic model in a case-control study with dichotomous response D that depends on a covariate X. For a portion of the sample, both the gold standard X and a surrogate covariate W are available; however, for the greater portion of the data, only the surrogate covariate W is available. By using a mixture model, the relationship between the true covariate and the response can be modeled appropriately for both types of data. The likelihood depends on the marginal distribution of X and the measurement error density (W|X, D). The latter is modeled parametrically based on the validation sample. The marginal distribution of the true covariate is modeled using a nonparametric mixture distribution. In this way we can improve the efficiency and reduce the bias of the parameter estimates. The results also apply when there is no validation data provided the error distribution is known or estimated from an independent data source. Many of the ...

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.

Orthogonal regression is one of the standard linear regression methods to correct for the effects of measurement error in predictors. We argue that orthogonal regression is often misused in errors-in-variables linear regression because of a failure to account for equation errors. The typical result is to overcorrect for measurement error, that is, overestimate the slope, because equation error is ignored. The use of orthogonal regression must include a careful assessment of equation error, and not merely the usual (often informal) estimation of the ratio of measurement error variances. There are rarer instances, for example, an example from geology discussed here, where the use of orthogonal regression without proper attention to modeling may lead to either overcorrection or undercorrection, depending on the relative sizes of the variances involved. Thus our main point, which does not seem to be widely appreciated, is that orthogonal regression, just like any measurement error analysis, requires ...

Simpson, Carroll, Zhou and Guth (1996) developed an ordinal response regression approach to meta-analysis of data from diverse toxicology studies, applying the methodology to a database of acute inhalation studies of tetra-chloroethylene. We present an alternative analysis of the same data, with two major differences: (1) interval censored scores are assigned worst-case values, e.g., a score known to be in the interval [0,1] is set equal to 1; and (2) the response is reduced to a binary response (adverse, nonadverse). We explore the stability of the analysis by varying a robustness parameter and graphing the curves traced out by the estimates and confidence intervals.

In a classical case-control study, Prentice and Pyke proposed to ignore the study design and instead base estimation and inference on a random sampling (i.e., prospective) formulation. We generalize this prospective formulation of case-control studies to include multiplicative models, stratification, missing data, measurement error, robustness, and other examples. The resulting estimators, which ignore the case-control study aspect and instead are based on a random-sampling formulation, are typically consistent for nonintercept parameters and are asymptotically normally distributed. We derive the resulting asymptotic covariance matrix of the parameter estimates. The covariance matrix obtained by ignoring the case-control sampling scheme and using prospective formulas instead is shown to be at worst asymptotically conservative and asymptotically correct in a variety of problems; a simple sufficient condition guaranteeing the latter is obtained.

A correction method is proposed for models including the generalized linear model when the covariate is measured with error. The method requires a separate validation data set that consists of the surrogate W and the true covariate X or an unbiased estimate X⊃ of X. We do not require the classical additive measurement error model in which the surrogate is unbiased for the true covariates. We first obtain an estimate of E(X|W) by using nonparametric kernel regression of X or X⊃ on W based on the validation data. Then we perform a standard analysis with the unknown X replaced by the estimate of E(X|W). The asymptotic distribution of the resulting regression parameter estimator is obtained. Generalizations to include components of X measured without error are also discussed.

In problems where the recorded signal comprises two or more bands (i.e., wavelengths), it is often the case that the bands are not perfectly aligned in the focal plane. Such misregistration may be caused by atmospheric or oceanographic effects, which refract different bands to differing degrees, or it may result from features of the design of the recording equipment. This article develops two methods for estimating the lag, or the amount by which the bands are out of alignment on a pixel grid, in cases where the recorded data are obtained by pixel averaging. Our two techniques are applicable directly to the signal domain and are based on penalized least squares (or errors in variables) and maximum cross-covariance. They use mathematical interpolation to accommodate the effect of pixel averaging. We introduce a concise and tractable asymptotic model for comparing the performance of the two techniques. The model predicts that the techniques should perform similarly. This is borne out by simulation ...

In heteroscedastic regression models assumptions about the error distribution determine the method of consistent estimation of parameters. For example, consider the case where the model specifies the regression and dispersion functions for the data but robustness is of concern and one wishes to use least absolute error regressions. Except in certain special circumstances, parameter estimates obtained in this way are inconsistent. In this article we expand the heteroscedastic model so that all of the common methods yield consistent estimates of the major model parameters. Asymptotic theory shows the extent to which standard results on the effect of estimating regression and dispersion parameters carry over into this setting. Careful attention is given to the question of when one can adapt for heteroscedasticity when estimating the regression parameters. We find that in many cases such adaption is not possible. This complicates inference about the regression parameters but does not lead to intracta...

We devise methods for estimating the parameters of a prospective logistic model with dichotomous response D and arbitrary covariates X from case-control data when these covariates are measured with error. We suppose that some fraction of the cases and controls provide only the error-prone covariate measurements, W (the “incomplete” or “reduced” data), whereas some of the cases and controls provide measurements on X and W (the “complete” data). We assume a measurement error density with a finite set of parameters α, namely fw|xD(w|x, d, α), and nondifferential error is treated as a special case of this model, fw|x(w|X, α). Our algorithm estimates both the logistic parameters and α from a pseudolikelihood. Because empirical distribution functions are used in place of needed distributions in the pseudolikelihoods, the required asymptotic theory is more elaborate than for pseudolikelihoods based on substitution for a finite number of nuisance parameters. We also examine computationally simpler methods under t...

A general nonlinear regression problem is considered with measurement error in the predictors. We assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function. Instead of observing the predictor, we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance. We identify an important transformation of the surrogate variable. Using this transformed variable, we show that if one proceeds with the usual analysis ignoring measurement error, then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter, up to a constant of proportionality. We derive the asymptotic distribution of the estimates. A simulation study is conducted applying sliced inverse regression in this context.

The folklore on one-step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n –1/4, where n is the sample size. We investigate the extent to which this folklore is valid for one-step GM estimators and their associated standard errors in linear regression. We find that one-step GM estimates based on Newton-Raphson or Scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points. Moreover, these estimators have bounded influence functions, and their standard errors can have high breakdown points. The folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic....

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.

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

In weighted least squares, it is typical that the weights are unknown and must be estimated. Most packages provide standard errors assuming that the weights are known. This is fine for sufficiently large sample sizes, but what about for small-to-moderate sample sizes? The investigation of this article into the effect of estimating weights proceeds under the assumption typical in practice—that one has a parametric model for the variance function. In this context, generalized least squares consists of (a) an initial estimate of the regression parameter, (b) a method for estimating the variance function, and (c) the number of iterations in reweighted least squares. By means of expansions for the covariance, it is shown that each of (a)—(c) can matter in problems of small to moderate size. A few conclusions may be of practical interest. First, estimated standard errors assuming that the weights are known can be too small in practice. The investigation indicates that a simple bootstrap operation resul...

Andrews et al (1972) carried out an extensive Monte Carlo study of robust estimators of location. Their conclusions were that the hampel and the skipped estimates, as classes, seemed to be preferable to some of the other currently fashionable estimators. The present study extends this work to include estimators not previously examined. The estimators are compared over short-tailed as well as long-tailed alternatives and also over some dependent data generated by first-order autoregressive schemes. The conclusions of the present study are threefold. First, from our limited study, none of the so-called robust estimators are very efficient over short-tailed situations. More work seems to be necessary in this situation. Second, none of the estimators perform very well in dependent data situations, particularly when the correlation is large and positive. This seems to be a rather pressing problem. Finally, for long-tailed alternatives, the hampel estimators and Hogg-type adaptive versions of the hampels are th...