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We consider the partially linear model relating a response Y to predictors (X,T) with mean function XT β + g(T) when the T’s are measured with additive error. We derive an estimator of β by modification local-likelihood method. The resulting estimator of β is shown to be asymptotically normal.We consider the partially linear model relating a response Y to predictors (X,T) with mean function XT β + g(T) when the T’s are measured with additive error. We derive an estimator of β by modification local-likelihood method. The resulting estimator of β is shown to be asymptotically normal.

This work deals with a generalization of the Total Least Squares method in the context of the functional linear model. We first propose a smoothing splines estimator of the functional coefficient of the model without noise in the covariates and we obtain an asymptotic result for this estimator. Then, we adapt this estimator to the case where the covariates are noisy and we also derive an upper bound for the convergence speed. Our estimation procedure is evaluated by means of simulations.

Parametric approaches based on a priori models of the speech are often used in the framework of speech enhancement using a single microphone. When the speech is modeled by means of a stationary autoregressive (AR) process, a frame-by-frame approach is usually considered. However, it requires the unbiased estimations of the autoregressive parameters and of the noise variances for the subsequent implementation of a filter (Kalman, H-infinity, etc.). The purpose of this paper is twofold. Firstly, we propose to view the AR parameter estimation as an errors-in-variables issue. Secondly, we implement an optimal smoothing procedure based on a constrained minimum variance estimation of the signal. Then, we test the procedure based on both steps in the field of speech enhancement.

This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions., Published at http://dx.doi.org/10.1214/009053604000000670 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider the regression model with errors-in-variables where we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f(X)+\xi, \; Z=X+\sigma\varepsilon$, involving independent and unobserved random variables $X,\xi,\varepsilon$. The density $g$ of $X$ is unknown, whereas the density of $\sigma\varepsilon$ is completely known. Using the observations $(Y_i, Z_i)$, $i=1,\cdots,n$, we propose an estimator of the regression function $f$, built as the ratio of two penalized minimum contrast estimators of $\ell=fg$ and $g$, without any prior knowledge on their smoothness. We prove that its $\mathbb{L}_2$-risk on a compact set is bounded by the sum of the two $\mathbb{L}_2(\mathbb{R})$-risks of the estimators of $\ell$ and $g$, and give the rate of convergence of such estimators for various smoothness classes for $\ell$ and $g$, when the errors $\varepsilon$ are either ordinary smooth or super smooth. The resulting rate is optimal in a minimax sense in all cases where lower bounds are available.

This thesis concerns modeling and control of bilinear systems (BLS). BLS are linear but not jointly linear in state and control. In the first part of the thesis, a background to BLS and their applications to modeling and control is given. The second part, and likewise the principal theme of this thesis, is dedicated to theoretical aspects of identification, modeling and control of mainly BLS, but also linear systems. In the last part of the thesis, applications of bilinear and linear modeling and control to the activated sludge process (ASP) are given.

The following identification problem is considered: Minimize the /spl lscr//sub 2/ norm of the difference between a given time series and an approximating one under the constraint that the approximating time series is a trajectory of a linear time invariant system of a fixed complexity. The complexity is measured by the input dimension and the maximum lag. The question leads to a problem that is known as the global total least squares problem and alternatively can be viewed as maximum likelihood identification in the errors-in-variables setup. Multiple time series and latent variables can be considered in the same setting. Special cases of the problem are autonomous system identification, approximate realization, and finite time optimal /spl lscr//sub 2/ model reduction. The identification problem is related to the structured total least squares problem. This paper presents an efficient software package that implements the theory. The proposed method and software are tested on data sets from the database for the identification of systems DAISY.

SIMEX was introduced by Cook and Stefanski (1994) as a simulation type estimator in errors-in-variables models. The idea of the SIMEX procedure is to compensate for the effect of the measurement errors while still using naive regression estimators. Polzehl and Zwanzig (2004) defined a symmetrized version of this estimator. In this paper we establish some results relating these two simulation-extrapolation-type estimators to well known consistent estimators like the total least squares estimator (TLS) and the moment estimator (MME) in the context of errors-in-variables models. We further introduce an adaptive SIMEX (ASIMEX), which is calculated like SIMEX, but based on an estimated variance. The main result of this paper is that SYMEX, ASIMEX are equivalent to TLS. Additionally we see that SIMEX is equivalent to the moment estimator.

It is well known that consistent estimators of errors-in-variables models require knowledge of the ratio of error variances. What is not well known is that a Joint Least Squares estimator is robust to a wide misspecification of that ratio. Through a series of Monte Carlo experiments we show that an easy-to-implement estimator produces estimates that are nearly unbiased for a wide range of the ratio of error variances. These MC analyses encompass linear and nonlinear specifications and also a system on nonlinear equations where all the variables are measured with errors.

The consequences of errors in trade classification are potentially worse than documented in existing empirical research. This is demonstrated by the use of a formal model of classification errors in a generic regression-type microstructure model. The bias is a function of the probability of trade-reversal in addition to the probability of an error. These parameters depend on stock and trade characteristics in addition to trading procedures and trade reporting standards. The bias is highly sensitive to the background variables, thus causing concern about the validity of empirical studies applying possibly erroneous classification methods without controlling for such effects. The theory, outlined in the paper, predicts that given empirical evidence on error rates, effective spreads must realistically be expected to be downward biased by more than 50%. However, the bias one can observe from using the TORQ database is less severe and has the opposite sign. This is due to special features of the NYSE trading process which may not carry over to other markets. This research also emphasizes the need for proper adjustment of classification error bias. Therefore, the paper proposes a GMM estimator for improved estimation. Simulation evidence indicates that in medium and larger sized samples the method is capable of removing virtually all the bias in market quality statistics like e¤ective, realized, and adverse selection spreads. This is empirically verified in an application to data from TORQ.

Techniques for the analysis of spatial data have, to date, tended to ignore any effect caused by error in specifying the spatial locations at which measurements are recorded. This paper reviews the methods for adjusting spatial inference in the presence of data-location error, particularly for data that have a continuous spatial index (geostatistical data). New kriging equations are developed and evaluated based on a simulation experiment. They are also applied to remote-sensing data from the Total Ozone Mapping Spectrometer instrument on the Nimbus-7 satellite, where the location error is caused by assignment of the data to their nearest grid-cell centers. The remote-sensing data measure total column ozone (TCO), which is important for protecting the Earth's surface from ultraviolet and other radiation.

In this paper we develop a simple maximum likelihood estimator for probit models where the regressors have measurement error. We first assume precise information about the reliability ratios (or, equivalently, the proxy correlations) of the regressors. We then show how reasonable bounds for the parameter estimates can be obtained when only imprecise information is available. The analysis is also extended to situations where the measurement error has non-zero mean and is correlated with the true values of the regressors. An extensive simulation study shows that the estimator works very well, even in quite small samples. Finally the method is applied to data explaining sick leave in Sweden.

The axiomatic literature on technical efficiency measurement has drawn attention to the indication problem of the Debreu-Farrell (DF) measure. We follow a shadow price approach to preserve the DF benchmark while reconciling it with the Koopmans efficiency characterization. First, we define a set of Koopmans efficient references that can be rationalized in a similar way as the DF projection. The indication problem is then captured using a measure of implicit allocative or mix efficiency, also interpretable as a dominance measure in price space. We consequently present a mix-adjusted DF framework for efficiency measurement in which e.g. the Zieschang (1984) procedure can be

We consider the partially linear model relating a response $Y$ to predictors ($X, T$) with mean function $X^{\top}\beta + g(T)$ when the $X$’s are measured with additive error. The semiparametric likelihood estimate of Severini and Staniswalis leads to biased estimates of both the parameter $\beta$ and the function $g(\cdot)$ when measurement error is ignored. We derive a simple modification of their estimator which is a semiparametric version of the usual parametric correction for attenuation. The resulting estimator of $\beta$ is shown to be consistent and its asymptotic distribution theory is derived. Consistent standard error estimates using sandwich-type ideas are also developed.

This paper develops a procedure for estimating parameters of a cross-sectional stochastic frontier production function when the factors of production suffer from measurement errors. Specifically, we use Fuller's (1987) reliability ratio concept to develop an estimator for the model in Aigner et al (1977). Our Monte-Carlo simulation exercise illustrates the direction and the severity of bias in the estimates of the elasticity parameters and the returns to scale feature of the production function when using the traditional maximum-likelihood estimator (MLE) in presence of measurement errors. In contrast the reliability ratio based estimator consistently estimates these parameters even under extreme degree of measurement errors. Additionally, estimates of firm level technical efficiency are severely biased for traditional MLE compared to reliability ratio estimator, rendering inter-firm efficiency comparisons infeasible. The seriousness of measurement errors in a practical setting is demonstrated by using data for a cross-section of publicly traded U.S. corporations.

Many studies suggest that changes in household economic resources (incomes and expenditure) have little effect on nutrient intakes and child malnutrition in developing countries. This paper examines the impact that errors-in-variables have on inferences about the importance of household incomes to the calorie and protein demands of households. Results are based on a new household survey from Papua New Guinea, with repeated observations on households during the year. These repeated observations allow regression estimates to be corrected for the differing reliabilities of the explanatory variables.

Structural errors-in-variables models with dependent spatial observations are studied. The presence of validation data is assumed. An estimator for regression parameters proposed by Lee and Sepanski [1] is studied. Consistency and asymptotic normality of the estimator are established in the case of increasing domain. Infill asymptotic properties are described. Simulation results are also presented.

The use of periodic excitation signals in identification experiments is advocated. With periodic excitation it is possible to separate the driving signals and the disturbances, which for instance implies that the noise properties can be independently estimated. In the paper a non-parametric noise model, estimated directly from the measured data, is used in a compensation strategy applicable to both least squares and total least squares estimation. The resulting least squares and total least squares methods are applicable in the errors-in-variables situation and give consistent estimates regardless of the noise. The feasibility of the idea is illustrated in a simulation study.

In this paper we prove an existence theorem for a special nonlinear total least squares problem. We show that the optimal parameters of the generalized logistic function exist in the sense of total least squares, provided the data satisfy the Chebyshev's inequality.

General Method of Moments (GMM) estimation of a linear one-equation model using panel data with errors-in-variables is considered. To eliminate fixed individual heterogeneity, the equation is differenced across one or more than one periods and estimated by means of instrumental variables. With non-autocorrelated measurement error, we show that only the one-period and a few two-period differences are essential, i.e. relevant for GMM-estimation. GMM estimation based on all orthogonality conditions on the basis of a generalized inverse formulation is shown to be equivalent to estimation using only the essential orthogonality conditions

Motivated by an example in nutritional epidemiology, we investigate some design and analysis aspects of linear measurement error models with missing surrogate data. The specific problem investigated consists of an initial large sample in which the response (a food frequency questionnaire, FFQ) is observed, and then a smaller calibration study in which replicates of the error prone predictor are observed (food records or recalls, FR). The difference between our analysis and most of the measurement error model literature is that in our study, the selection into the calibration study can depend upon the value of the response. Rationale for this type of design is given. Two major problems are investigated. In the design of a calibration study, one has the option of larger sample sizes and fewer replicates, or smaller sample sizes and more replicates. Somewhat surprisingly, neither strategy is uniformly preferable in cases of practical interest. The answers depend on the instrument used (recalls or records) and the parameters of interest. The second problem investigated is one of analysis. In the usual linear model with no missing data, method of moments estimates and normal-theory maximum likelihood estimates are approximately equivalent, with the former method in most use because it can be calculated easily and explicitly. Both estimates are valid without any distributional assumptions. In contrast, in the missing data problem under consideration, only the moments estimate is distribution-free, but the maximum likelihood estimate has at least 50% greater precision in practical situations when normality obtains. Implications for the design of nutritional calibration studies are discussed.

General Method of Moments (GMM) estimation of a linear one-equation model using panel data with errors-in-variables is considered. To eliminate fixed individual heterogeneity, the equation is differenced across one or more than one periods and estimated by means of instrumental variables. With non-autocorrelated measurement error, we show that only the one-period and a few two-period differences are essential, i.e. relevant for GMM-estimation. GMM estimation based on all orthogonality conditions on the basis of a generalized inverse formulation is shown to be equivalent to estimation using only the essential orthogonality conditions Keywords: Panel Data, Errors-in-Variables, Instrumental Variables, GMM Estimation, Generalized inverse

We consider the problem of estimating quantile regression coefficients in errors-in-variables models. When the error variables for both the response and the manifest variables have a joint distribution that is spherically symmetric but otherwise unknown, the regression quantile estimates based on orthogonal residuals are shown to be consistent and asymptotically normal. We also extend the work to partially linear models when the response is related to some additional covariate.

We consider the partially linear model relating a response Y to predictors (X,T) with mean function XT Ø + g(T) when the T's are measured with additive error. We derive an estimator of Ø by modification local-likelihood method. The resulting estimator of Ø is shown to be asymptotically normal.We consider the partially linear model relating a response Y to predictors (X,T) with mean function XT Ø + g(T) when the T's are measured with additive error. We derive an estimator of Ø by modification local-likelihood method. The resulting estimator of Ø is shown to be asymptotically normal.

In many problems one wants to model the relationship between a response Y and a covariate X. Sometimes it is difficult, expensive, or even impossible to observe X directly, but one can instead observe a substitute variable W which is easier to obtain. By far the most common model for the relationship between the actual covariate of interest X and the substitute W is W = X + U, where the variable U represents measurement error. This assumption of additive measurement error may be unreasonable for certain data sets. We propose a new model, namely h(W) = h(X) + U, where h(.) is a monotone transformation function selected from some family H of monotone functions. The idea of the new model is that, in the correct scale, measurement error is additive. We propose two possible transformation families H. One is based of selecting a transformation which makes the within sample mean and standard deviation of replicated W’s uncorrelated. The second is based on selecting the transformation so that the errors (U’s) fit a prespecified distribution. Transformation families used are the parametric power transformations and a cubic spline family. Several data examples are presented to illustrate the methods.

The problem of constructing confidence sets for the structural errors-in-variables model is considered under the assumption that the variance of the error associated with the covariate is known. Previously proposed confidence sets for this model suffer from the problem that they all have zero confidence levels for any sample size, where the confidence level of a confidence set is defined to be the infimum of coverage probability over the parameter space. In this paper we construct some asymptotically honest confidence sets; that is, the limiting values of their confidence levels are at least as large as the nominal probabilities when the sample size goes to $\infty$. A desirable property of the proposed confidence set for the slope is also established.

Given the data $(p_i, t_i, f_i), $ $i=1, ldots, m, $ we consider the existence problem for the optimal parameters for the exponential function approximating this data in the sense of total least squares. We give sufficient conditions which guarantee the existence of such optimal parameters.

We consider a semiparametric estimation method for general regression models when some of the predictors are measured with error. The technique relies on a kernel regression of the "true" covariate on all the observed covariates and surrogates. This requires a nonparametric regression in as many dimensions as there are covariates and surrogates. The usual theory copes with such higher-dimensional problems by using higher-order kernels, but this is unrealistic for most problems. We show that the usual theory is essentially as good as one can do with this technique. Instead of regression with higher-order kernels, we propose the use of dimension reduction techniques. We assume that the "true" covariate depends only on a linear combination of the observed covariates and surrogates. If this linear combination were known, we could apply the one-dimensional versions of the semiparametric problem, for which standard kernels are applicable. We show that if one can estimate the linear directions at the root-$n$ rate, then asymptotically the resulting estimator of the parameters in the main regression model behaves as if the linear combination were known. Simulations lend some credence to the asymptotic results.

This paper examines panel data modelling with latent variables in analyzing log-linear relations between inputs and output of firms. Our particular focus is on (i) the "increasing returns to scale puzzle" for labour input and (ii) the GMM estimation in the context of errors-in-variables and panel data. The IV's used for the observed log-differenced output are log output (in level form) for other years than those to which the difference(s) refer. Flexible assumptions are made about the second order moments of the errors, the random coefficients, and other latent variables, allowing, inter alia, for arbitrary heteroskedasticity and autocorrelation up to the first order of the errors-in-variables. We compare OLS, 2SLS, and GMM estimates of the average input response elasticity (which in some cases can be interpreted as an average inverse scale elasticity), and investigate whether year specific estimates differ substantially from those obtained when data for all years are combined. The results confirm the "increasing returns to scale puzzle" for labour input (measured in three different ways), but indicate approximately constant returns to scale when we consider the material input response. This indicates non-homotheticity of the production technology.

We examine estimation of a model of producer behavior in the presence of correlated measurement errors in the regressors. Scale economies and price-cost margins are estimated from a set of panel data for manufacturing plants. The paper presents a somewhat new model for estimation of these parameters which is highly flexible but with a simple regression structure. Perhaps the most important contribution of the paper is some new results on deriving parameter bounds for a regression model with errors in variables. In particular, we consider the case where the measurement errors might be correlated. We derive asymptotic standard errors for the parameter bounds. These asymptotic standard errors are compared to bootstrap estimates. Our new results on parameter bounds are applied to the estimation of the model of producer behavior.

This paper examines panel data modelling with latent variables in analyzing log-linear relations between inputs and output of firms. Our particular focus is on (i) the "increasing returns to scale puzzle" for labour input and (ii) the GMM estimation in the context of errors-in-variables and panel data. The IV's used for the observed log-differenced output are log output (in level form) for other years than those to which the difference(s) refer. Flexible assumptions are made about the second order moments of the errors, the random coefficients, and other latent variables, allowing, inter alia, for arbitrary heteroskedasticity and autocorrelation up to the first order of the errors-in-variables. We compare OLS, 2SLS, and GMM estimates of the average input response elasticity (which in some cases can be interpreted as an average inverse scale elasticity), and investigate whether year specific estimates differ substantially from those obtained when data for all years are combined. The results confirm the 'increasing returns to scale puzzle" for labour input (measured in three different ways), but indicate approximately constant returns to scale when we consider the material input response. This indicates non-homotheticity of the production technology.

We examine estimation of a model of producer behavior in the presence of correlated measurement errors in the regressors. Scale economies and price-cost margins are estimated from a set of panel data for manufacturing plants. The paper presents a somewhat new model for estimation of these parameters which is highly flexible but with a simple regression structure. Perhaps the most important contribution of the paper is some new results on deriving parameter bounds for a regression model with errors in variables. In particular, we consider the case where the measurement errors might be correlated. We derive asymptotic standard errors for the parameter bounds. These asymptotic standard errors are compared to bootstrap estimates. Our new results on parameter bounds are applied to the estimation of the model of producer behavior. Norges Forskningsrå

Orthogonal regression $M$-estimates are considered from a bias robust point of view. Their maximum bias over epsilon-contamination neighborhoods is characterized, and maximum bias curves are computed. The most bias robust orthogonal regression $M$-estimate is derived and shown to be a "mode type" estimate; for instance, in the two-dimensional case this estimate can be computed by locating a strip of fixed width covering the maximum number of data points. It will be shown that, although orthogonal regression $M$-estimates with bounded loss function have unbounded influence function, the derivative of their maximum bias curve at zero is finite. Finally, an implicit formula for an upper bound for the breakdown point of all orthogonal regression $M$-estimates is found. The upper bound, which depends on the signal-to-noise ratio, is sharp and attained by the most bias robust estimate.

Consider a generalized linear model with response $Y$ and scalar predictor $X$. Instead of observing $X$, a surrogate $W = X + Z$ is observed, where $Z$ represents measurement error and is independent of $X$ and $Y$. The efficient score test for the absence of association depends on $m(w) = E(X\mid W = w)$ which is generally unknown. Assuming that the distribution of $Z$ is known, asymptotically efficient tests are constructed using nonparametric estimators of $m(w)$. Rates of convergence for the estimator of $m(w)$ are established in the course of proving efficiency of the proposed test.

The usual assumption in the classical errors-in-variables problem of independent measurement errors cannot necessarily be maintained when the data are time series; errors may be strongly serially correlated, possibly containing seasonal effects and trends. When it is possible to identify frequency bands over which the signal-to-noise ratio is large, an approximate solution to the errors-in-variables problem is to omit the remaining frequencies from a time series regression. We draw attention to the danger of “leakage” from the omitted frequencies, and show that the consequent bias can be reduced by means of tapering.