Asymptotically Efficient Estimation of a Nonlinear Model of the Heteroscedasticity*

The paper is devoted to the estimation of a nonlinear parametric model of the heteroscedasticity. The heteroscedasticity occurs in regression when the measurement noise variance is non-constant. Sometimes, the noise variance can be represented as a parameterized function of independent variables, so-called variance function. The maximum likelihood estimation (MLE) of variance function parameters leads to a system of nonlinear equations. The iterative solution of these nonlinear equations is based entirely on a successful choice of initial conditions. Hence, in the practice, the nonlinear MLE is intractable. To overcome this difficulty, another linear quasi-MLE estimator is proposed. It is strongly consistent, asymptotically Gaussian and only slightly less efficient than the Cramer-Rao lower bound. By using this estimator as an initial condition, an asymptotically efficient estimation is obtained by using one-step non-iterative Newton method. This approach has been applied to the GPS navigation in the constrained (urban) environment.