Moskvychova K. Properties of correlogram estimator of random noise covariance function in nonlinear regression model

The thesis is devoted to the study of asymptotic properties of residual correlogram as an estimator of random stationary Gaussian noise covariance function in continuous time nonlinear regression model. A theorem on the probabilities of large deviations for the least squares estimator of the vector parameter of nonlinear regression function and theorem on the probabilities of large deviations in uniform metric of the correlogram of stationary Gaussian noise are obtained. Using these results a theorem was proved on exponential convergence to zero of the probabilities of large deviations in uniform metric of the normed difference of residual correlogram and random noise covariance function. As simple corollaries of this fact, the enhanced properties of the residual correlogram weak consistency are obtained. A functional central limit theorem is proved in the space of continuous functions for the normed residual correlogram in nonlinear regression model in consideration. The result obtained shows that the limiting almost surely continuous Gaussian process coincides with the limiting process in the central limit theorem for standard correlogram of random noise in our regression model. A stochastic asymptotic expansion of the normed residual correlogram is received and the first three terms of the expansion are written in explicit form. Based on this stochastic asymptotic expansion, in the case when all partial derivatives by parameters of the regression function exist and are continuous up to the orders 4, 3, and 2 inclusive, asymptotic expansions of bias, mean square deviation, and variance of the residual correlogram are obtained.