Alexander Kukush

Editorial: Total Least Squares and Errors-in-variables Modeling

The total least squares method is a numerical linear algebra tool for finding approximate solutions to overdetermined systems of equation s Ax = b, where both the vectorb as well as the matrixA are assumed to be perturbed. Since its definition by Golub and Van Loan in 1980, the classical total lea st squares method has been extended to solve weighted, structured, and regula ized total least squares problems and was applied in signal processing, system ident ification, computer vision, document retrieval, computer algebra, and other field s.

Consistent least squares fitting of ellipsoids

Summary.A parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered. The ordinary least squares estimator is inconsistent, and due to the nonlinearity of the model, the orthogonal regression estimator is inconsistent as well, i.e., these estimators do not converge to the true value of the parameters, as the sample size tends to infinity. A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator. The correction is explicitly given in terms of the true value of the noise variance.

Consistent estimation in an implicit quadratic measurement error model

An adjusted least squares estimator is derived that yields a consistent estimate of the parameters of an implicit quadratic measurement error model. In addition, a consistent estimator for the measurement error noise variance is proposed. Important assumptions are: (1) all errors are uncorrelated identically distributed and (2) the error distribution is normal. The estimators for the quadratic measurement error model are used to estimate consistently conic sections and ellipsoids. Simulation examples, comparing the adjusted least squares estimator with the ordinary least squares method and the orthogonal regression method, are shown for the ellipsoid fitting problem.

Impact of Uncertainties in Exposure Assessment on Estimates of Thyroid Cancer Risk among Ukrainian Children and Adolescents Exposed from the Chernobyl Accident

The 1986 accident at the Chernobyl nuclear power plant remains the most serious nuclear accident in history, and excess thyroid cancers, particularly among those exposed to releases of iodine-131 remain the best-documented sequelae. Failure to take dose-measurement error into account can lead to bias in assessments of dose-response slope. Although risks in the Ukrainian-US thyroid screening study have been previously evaluated, errors in dose assessments have not been addressed hitherto. Dose-response patterns were examined in a thyroid screening prevalence cohort of 13,127 persons aged <18 at the time of the accident who were resident in the most radioactively contaminated regions of Ukraine. We extended earlier analyses in this cohort by adjusting for dose error in the recently developed TD-10 dosimetry. Three methods of statistical correction, via two types of regression calibration, and Monte Carlo maximum-likelihood, were applied to the doses that can be derived from the ratio of thyroid activity to thyroid mass. The two components that make up this ratio have different types of error, Berkson error for thyroid mass and classical error for thyroid activity. The first regression-calibration method yielded estimates of excess odds ratio of 5.78 Gy−1 (95% CI 1.92, 27.04), about 7% higher than estimates unadjusted for dose error. The second regression-calibration method gave an excess odds ratio of 4.78 Gy−1 (95% CI 1.64, 19.69), about 11% lower than unadjusted analysis. The Monte Carlo maximum-likelihood method produced an excess odds ratio of 4.93 Gy−1 (95% CI 1.67, 19.90), about 8% lower than unadjusted analysis. There are borderline-significant (p = 0.101–0.112) indications of downward curvature in the dose response, allowing for which nearly doubled the low-dose linear coefficient. In conclusion, dose-error adjustment has comparatively modest effects on regression parameters, a consequence of the relatively small errors, of a mixture of Berkson and classical form, associated with thyroid dose assessment.

Asymptotic properties of an estimator in nonlinear functional errors-in-variables models with dependent error terms

Nonlinear functional errors-in-variables models with error terms satisfying mixing conditions are studied. It is pointed out that under certain conditions the least-squares estimator of regression parameters is not consistent. An alternative estimator for regression parameters is proposed. The consistency of the alternative estimator is established.

Consistent fundamental matrix estimation in a quadratic measurement error model arising in motion analysis

Consistent estimators of the rank-deficient fundamental matrix yielding information on the relative orientation of two images in two-view motion analysis are derived. The estimators are derived by minimizing a corrected contrast function in a quadratic measurement error model. In addition, a consistent estimator for the measurement error variance is obtained. Simulation results show the improved accuracy of the newly proposed estimator compared to the ordinary total least-squares estimator.

On the computation of the multivariate structured total least squares estimator

A multivariate structured total least squares problem is considered, in which the extended data matrix is partitioned into blocks and each of the blocks is Toeplitz/Hankel structured, unstructured, or noise free. Two types of numerical solution methods for this problem are proposed: (i) standard local optimization methods in combination with efficient evaluation of the cost function and its first derivative, and (ii) an iterative procedure proposed originally for the element-wise weighted total least squares problem. The computational efficiency of the proposed methods is compared with this of alternative methods. Copyright

Threshold structure of optimal stopping strategies for American type option. I

The paper presents results of theoretical studies of optimal stopping domains of American type options in discrete time. Sufficient conditions on the payoff functions and the price process for the ...

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σu2. The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σu2, both estimators have identical asymptotic covariance matrices up to the order of σu2. We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σu2).

Ordered Random Vectors and Equality in Distribution

Abstract In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors and with equal expected utilities (or distorted expectations) of the sums and for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is . The results in this paper can be considered as generalizations of some recent results on comonotonicity, where necessary conditions related to the distribution of are presented for the random vector to be comonotonic.