# Kukush A. G.

### Asymptotically independent estimators in a structural linear model with measurement errors

Kukush A. G., Shklyar S. V., Tsaregorodtsev Ya. V.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1505-1517

We consider a structural linear regression model with measurement errors. A new parameterization is proposed, in which the expectation of the response variable plays the role of a new parameter instead of the intercept. This enables us to form three groups of asymptotically independent estimators in the case where the ratio of variances of the errors is known and two groups of this kind if the variance of the measurement error in the covariate is known. In this case, it is not assumed that the errors and the latent variable are normally distributed.

### Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1026–1033

We consider a linear multivariate errors-in-variables model *AX* ? *B*, where the matrices *A* and *B* are observed with errors and the matrix parameter *X* is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to *X* as the number of rows in *A* tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found.

### Correction of nonlinear orthogonal regression estimator

Fazekas L., Kukush A. G., Zwanzig S.

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1101–1118

For any nonlinear regression function, it is shown that the orthogonal regression procedure delivers an inconsistent estimator. A new technical approach to the proof of inconsistency based on the implicit-function theorem is presented. For small measurement errors, the leading term of the asymptotic expansion of the estimator is derived. We construct a corrected estimator, which has a smaller asymptotic deviation for small measurement errors.

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

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1204-1209

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.

### On the rosenthal inequality for mixing fields

Fazekas L., Kukush A. G., Tómács T.

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 266-276

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

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1205–1214

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.

### Asymptotic behavior of the solution to the cauchy problem for stochastic parabolic equation

Dorogovtsev A. Ya., Kukush A. G.

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 162-169

### Asymptotic behavior of solutions of the heat-conduction equation with white noise in the right side

Dorogovtsev A. Ya., Ivasyshen S. D., Kukush A. G.

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 8 – 20