In the case of the nonlinear regression model, methods and procedures have been developed to obtain estimates of the parameters. These methods are much more complicated than the procedures used if the model considered is linear. Moreover, unlike the linear case, the properties of the resulting estimators are unknown and usually depend on the true values of the estimated parameters. It is sometimes possible to approximate the nonlinear model by a linear one and use the much more developed linear...

A construction of confidence regions in nonlinear regression models is difficult mainly in the case that the dimension of an estimated vector parameter is large. A singularity is also a problem. Therefore some simple approximation of an exact confidence region is welcome. The aim of the paper is to give a small modification of a confidence ellipsoid constructed in a linearized model which is sufficient under some conditions for an approximation of the exact confidence region.

If an observation vector in a nonlinear regression model is normally distributed, then an algorithm for a determination of the exact $(1-\alpha )$-confidence region for the parameter of the mean value of the observation vector is well known. However its numerical realization is tedious and therefore it is of some interest to find some condition which enables us to construct this region in a simpler way.

In nonlinear regression models with constraints a linearization of the model leads to a bias in estimators of parameters of the mean value of the observation vector. Some criteria how to recognize whether a linearization is possible is developed. In the case that they are not satisfied, it is necessary to decide whether some quadratic corrections can make the estimator better. The aim of the paper is to contribute to the solution of the problem.

A linearization of the nonlinear regression model causes a bias in estimators of model parameters. It can be eliminated, e.g., either by a proper choice of the point where the model is developed into the Taylor series or by quadratic corrections of linear estimators. The aim of the paper is to obtain formulae for biases and variances of estimators in linearized models and also for corrected estimators.

The paper deals with the linear model with uncorrelated observations. The dispersions of the values observed are linear-quadratic functions of the unknown parameters of the mean (measurements by devices of a given class of precision). Investigated are the locally best linear-quadratic unbiased estimators as improvements of locally best linear unbiased estimators in the case that the design matrix has none, one or two linearly dependent rows.

The aim of the paper is to estimate a function $\gamma =tr\left(D\beta {\beta }^{\text{'}}\right)+tr(C\sum )$ (with $d,C$ known matrices) in a regression model $(Y,X\beta ,\sum )$ with an unknown parameter $\beta$ and covariance matrix $\sum$. Stochastically independent replications $Y_1,...,Y_m$ of the stochastic vector $Y$ are considered, where the estimators of $X\beta$ and $\sum$ are $\overline{Y}=\frac{1}{m}{\sum }_{i=1}^m{Y}_i$ and $\widehat{\sum }={(m-1)}^{-1}{\sum }_{i=1}^m(Y_i-\overline{Y}){(Y_i-\overline{Y})}^{\text{'}}$, respectively. Locally and uniformly best inbiased estimators of the function $\gamma$, based on $\overline{Y}$ and $\widehat{\sum }$, are given.

This paper presents a semi-global mathematical model for an analysis of a signal of amperometric biosensors. Artificial neural networks were applied to an analysis of the biosensor response to multi-component mixtures. A large amount of the learning and test data was synthesized using computer simulation of the biosensor response. The biosensor signal was analyzed with respect to the concentration of each component of the mixture. The paradigm of locally weighted linear regression was used for retraining...