A comparison of asymptotic covariance matrices of three consistent estimators in the Poisson regression model with measurement errors

We consider a Poisson model, where the mean depends on certain covariates in a log-linear way with unknown regression parameters. Some or all of the covariates are measured with errors. The covariates as well as the measurement errors are both jointly normally distributed, and the error covariance matrix is supposed to be known. Three consistent estimators of the parameters - the corrected score, a structural, and the quasi-score estimators - are compared to each other with regard to their relative (asymptotic) efficiencies. The paper extends an earlier result for a scalar covariate

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

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

Comparison of three estimators in Poisson errors-in-variables model with one covariate

A structural errors-in-variables model is investigated, where the response variable follows a Poisson distribution. Assuming the error variance to be known, we consider three consistent estimators and compare their relative efficiencies by means of their asymptotic covariance matrices. The comparison is made for arbitrary error variances. The structural quasi-likelihood (QL) estimator is based on a quasi score function, which is constructed from a conditional mean-variance model. The corrected estimator is based on an error-corrected likelihood score function. The alternative estimator is constructed to remove the asymptotic bias of the naive (i.e., ordinary maximum likelihood) estimator. It is shown that the QL estimator is strictly more efficient than the alternative estimator, and the latter one is strictly more efficient than the corrected estimator

Distance between the fractional Brownian motion and the space of adapted Gaussian martingales

We consider the distance between the fractional Brownian motion defined on the interval [0,1] and the space of Gaussian martingales adapted to the same filtration. As the distance between stochastic processes, we take the maximum over [0,1] of mean-square deviances between the values of the processes. The aim is to calculate the function a in the Gaussian martingale representation ∫0ta(s)dWs that minimizes this distance. So, we have the minimax problem that is solved by the methods of convex analysis. Since the minimizing function a can not be either presented analytically or calculated explicitly, we perform discretization of the problem and evaluate the discretized version of the function a numerically

Estimation in Linear-Rate Simple Survival Models with Measurement Errors and Censoring

A simple exponential regression model is considered where the rate parameter of the response variable linearly depends on the explanatory variable. We consider complications of the model: censoring of the response variable (either upper censoring or interval observations), the additive classical error or multiplicative Berkson error in the explanatory variable, or a combination of censoring with Berkson errors. We construct or use already-known estimators in the models, and verify their performance in simulations

Consistency of the total least squares estimator in the linear errors-in-variables regression

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid

Maximum likelihood estimation for Gaussian process with nonlinear drift

We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained generalize the paper [Yu. Mishura, K. Ralchenko, S. Shklyar, Maximum likelihood drift estimation for Gaussian process with stationary increments, Austrian J. Stat., 46(3–4): 67–78, 2017] in two directions: the drift may be nonlinear, and the noise may have nonstationary increments. As an example, the model with subfractional Brownian motion is considered