Savych I. Asymptotic properties of Koenker-Bassett estimator of parameter of nonlinear regression with strongly dependent random noise.

The thesis is devoted to the study of time continuous nonlinear regression models parameters Koenker-Bassett estimator asymptotic properties when the random noise is a local functional of Gaussian stationary process with singular spectrum. In the modern theory of statistical inference the estimator of quantile regression, or the Koencker-Bassett estimator, occupies an important place and is an estimator of observations unknown quantiles being defined using asymmetric loss function. In the thesis a simplifying assumption on the equality to zero of observations errors is made. This narrows the idea of quantile regression but provides an opportunity to consider ordinary regression models with asymmetric errors of observations, and receive robust Koenker-Bassett etimators of regression parameters. Some sufficient conditions on sharpened weak consistency of the Koenker-Bassett estimator are obtained. Two different cases of unbounded and bounded parametric sets are considered as well as different types of estimator normalization. To prove Koenker-Bassett estimator asymptotic normality besides consistency the following important results are obtained and utilized. Firstly, taking into account the phenomenon of the noise strong dependence sufficient conditions on indicated process spectral density ??admissibility are formulated, where ? is a spectral measure of nonlinear regression function. These sufficient conditions are fulfilled for spectral density of our regression model if the sets of measure ? atoms and spectrum singularity points do not intersect. The last fact gives an opportunity to write down in explicit form the covariance matrix of Koenker-Bassett estimator limiting normal distribution. Secondly, the reduction theorem is proved that reduces the problem of Koenker-Bassett estimator asymptotic distribution obtaining to the problem of asymptotic distribution obtaining for integral of an indicator process generated by random noise and weighted by regression function gradient. In the proof of reduction theorem the Huber method of parametric set partition and expansions in Fourier series by Hermite polynomials have been used. Thirdly, a central limit theorem for weighted integrals of nonlinear transformations of Gaussian stationary random process with singular spectrum is obtained. In the proof of the last result the Nualart-Peccati-Tudor multivariate central limit theorem for a family of vectors of Wiener-Ito multiple stochastic integrals over isonormal process generated by the random noise and diagram machinery were used. In the thesis the least moduli estimator asymptotic properties are investigated as a partial case of Koenker-Bassett estimator. In particular a new formula for the least moduli estimator limiting normal distribution covariance matrix is derived for Gaussian stationary random process with singular spectrum taken in the capacity of the model noise. Besides it is proved that under conditions imposed on the model the Hermite rank of local nonlinear transformation of Gaussian process must be equal to one. Correspondent examples are given in the text.