Statistical Inference with Fractional Brownian Motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.     

On an Adaptive Estimator of the Least Contrast in a Model with Nonlinear Functional Relations

We consider an implicit nonlinear functional model with errors in variables. On the basis of the concept of deconvolution, we propose a new adaptive estimator of the least contrast of the regression parameter. We formulate sufficient conditions for the consistency of this estimator. We consider several examples within the framework of the L ₁- and L ₂-approaches.     

Infill Asymptotics Inside Increasing Domains for the Least Squares Estimator in Linear Models

A linear model observed in a spatial domain is considered. Consistency and asymptotic normality of the least squares estimator is proved when the observations become dense in a sequence of increasing domains and the error terms are weakly dependent. Similar statements are obtained for the linear errors-in-variables model. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the rosenthal inequality for mixing fields

A proof of the Rosenthal inequality for α-mixing random fields is given. The statements and proofs are modifications of the corresponding results obtained by Doukhan and Utev.     

Asymptotic normality of a projective estimator of an infinite-dimensional parameter of nonlinear regression

A model of nonlinear regression is studied in infinite-dimensional space. Observation errors are equally distributed and have the identity correlation operator. A projective estimator of a parameter is constructed, and the conditions under which it is true are established. For a parameter that belongs to an ellipsoid in a Hilbert space, we prove that the estimators are asymptotically normal; for this purpose, the representation of the estimator in terms of the Lagrange factor is used and the asymptotics of this factor are studied. An example of the nonparametric estimator of a signal is examined for iterated observations under an additive noise.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1205–1214, September, 1993.     

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.Mathematics Subject Classification (2000): 65D15, 65D10, 15A63Revised version received August 15, 2003     

The efficiency of adjusted least squares in the linear functional relationship

A linear functional errors-in-variables model with unknown slope parameter and Gaussian errors is considered. The measurement error variance is supposed to be known, while the variance of errors in the equation is unknown. In this model a risk bound of asymptotic minimax type for arbitrary estimators is established. The bound lies above that one which was found previously in the case of both variances known. The bound is attained by an adjusted least-squares estimators.