Asymptotic optimality of the quasi-score estimator in a class of linear score estimators

We prove that the quasi-score estimator in a mean-variance model is optimal in the class of (unbiased) linear score estimators, in the sense that the difference of the asymptotic covariance matrices of the linear score and quasi-score estimator is positive semi-definite. We also give conditions under which this difference is zero or under which it is positive definite. This result can be applied to measurement error models where it implies that the quasi-score estimator is asymptotically more efficient than the corrected score estimator

Relative Efficiency of Maximum Likelihood and Other Estimators in a Nonlinear Regression Model with Small Measurement Errors

We compare the asymptotic covariance matrix of the ML estimator in a nonlinear measurement error model to the asymptotic covariance matrices of the CS and SQS estimators studied in Kukush et al (2002). For small measurement error variances they are equal up to the order of the measurement error variance and thus nearly equally efficient

Comparing the efficiency of structural and functional methods in measurement error models

The paper is a survey of recent investigations by the authors and others into the relative efficiencies of structural and functional estimators of the regression parameters in a measurement error model. While structural methods, in particular the quasi-score (QS) method, take advantage of the knowledge of the regressor distribution (if available), functional methods, in particular the corrected score (CS) method, discards such knowledge and works even if such knowledge is not available. Among other results, it has been shown that QS is more efficient than CS as long as the regressor distribution is completely known. However, if nuisance parameters in the regressor distribution have to be estimated, this is no more true in general. But by modifying the QS method, the adverse effect of the nuisance parameters can be overcome. For small measurement errors, the efficiencies of QS and CS become almost indistinguishable, whether nuisance parameters are present or not. QS is (asymptotically) biased if the regressor distribution has been misspecified, while CS is always consistent and thus more robust than QS

Quasi Score is more efficient than Corrected Score in a general nonlinear measurement error model

We compare two consistent estimators of the parameter vector beta of a general exponential family measurement error model with respect to their relative efficiency. The quasi score (QS) estimator uses the distribution of the regressor, the corrected score (CS) estimator does not make use of this distribution and is therefore more robust. However, if the regressor distribution is known, QS is asymptotically more efficient than CS. In some cases it is, in fact, even strictly more efficient, in the sense that the difference of the asymptotic covariance matrices of CS and QS is positive definite

Comparison of three estimators in a polynomial regression with measurement errors

In a polynomial regression with measurement errors in the covariate, which is supposed to be normally distributed, one has (at least) three ways to estimate the unknown regression parameters: one can apply ordinary least squares (OLS) to the model without regard of the measurement error or one can correct for the measurement error, either by correcting the estimating equation (ALS) or by correcting the mean and variance functions of the dependent variable, which is done by conditioning on the observable, error ridden, counter part of the covariate (SLS). While OLS is biased the other two estimators are consistent. Their asymptotic covariance matrices can be compared to each other, in particular for the case of a small measurement error variance

Hypothesis testing of the drift parameter sign for fractional Ornstein-Uhlenbeck process

We consider the fractional Ornstein-Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$. We also study the estimators for drift parameter for continuous and discrete observations and prove their strong consistency for all $H\in(0,1)$.Comment: 15 page

Optimality of the quasi-score estimator in a mean-variance model with applications to measurement error models

We consider a regression of $y$ on $x$ given by a pair of mean and variance functions with a parameter vector $\theta$ to be estimated that also appears in the distribution of the regressor variable $x$. The estimation of $\theta$ is based on an extended quasi score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-$y$ unbiased estimating functions. Of special interest is the case where the distribution of $x$ depends only on a subvector $\alpha$ of $\theta$, which may be considered a nuisance parameter. In general, $\alpha$ must be estimated simultaneously together with the rest of $\theta$, but there are cases where $\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.We also study a number of special measurement error models in greater detail

Optimality of Quasi-Score in the multivariate mean-variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean-variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ``extended'' in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework

Quasi Score is more efficient than Corrected Score in a polynomial measurement error model

We consider a polynomial regression model, where the covariate is measured with Gaussian errors. The measurement error variance is supposed to be known. The covariate is normally distributed with known mean and variance. Quasi Score (QS) and Corrected Score (CS) are two consistent estimation methods, where the first makes use of the distribution of the covariate (structural method), while the latter does not (functional method). It may therefore be surmised that the former method is (asymptotically) more efficient than the latter one. This can, indeed, be proved for the regression parameters. We do this by introducing a third, so-called Simple Score (SS), estimator, the efficiency of which turns out to be intermediate between QS and CS. When one includes structural and functional estimators for the variance of the error in the equation, SS is still more efficient than CS. When the mean and variance of the covariate are not known and have to be estimated as well, one can still maintain that QS is more efficient than SS for the regression parameters