Asymptotically Efficient Estimation of a Nonlinear Model of the Heteroscedasticity and the Calibration of Measurement Systems

The measurement system calibration includes the estimation of the sensor error models in order to get an optimal estimation of the measured parameters. The paper is devoted to the estimation of a nonlinear parametric model of the heteroscedasticity and its application to the calibration of measurement systems. The heteroscedasticity occurs in regression when the measurement noise variance is nonconstant. The maximum likelihood (ML) estimation 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 which is intractable in practice. To overcome this difficulty, another linear quasi-ML estimator is proposed. It is strongly consistent, asymptotically Gaussian, and only slightly less efficient than the Cramér-Rao lower bound. By using this estimator as an initial condition, an asymptotically efficient estimation is obtained by using one-step noniterative Newton method. The theoretical findings have been applied to the calibration of the EGNOS/GPS positioning algorithm in the (sub-)urban environments.