Real valued $M$-estimators ${\widehat{\theta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(\theta \right))$ in a statistical model with observations $Y_i\sim F_{{\theta }_0}$ are replaced by ${ℝ}^p$-valued $M$-estimators ${\widehat{\beta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(u\left(z_i^T\beta \right)\right))$ in a new model with observations $Y_i\sim F_{u\left(z_i^t{\beta }_0\right)}$, where $z_i\in {ℝ}^p$ are regressors, ${\beta }_0\in {ℝ}^p$ is a structural parameter and $u:ℝ\to ℝ$ a structural function of the new model. Sufficient conditions for the consistency of ${\widehat{\beta }}_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” ${\widehat{\theta }}_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If...

The paper deals with modified minimax quadratic estimation of variance and covariance components under full ellipsoidal restrictions. Based on the, so called, linear approach to estimation variance components, i. e. considering useful local transformation of the original model, we can directly adopt the results from the linear theory. Under normality assumption we can can derive the explicit form of the estimator which is formally find to be the Kuks–Olman type estimator.

A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary...