ESTIMATION OF PARAMETER WITH SEVERAL VALUES

Abstract

Context. The problem of estimating a parameter with several values on different parts of the data interval is considered. The object of this research is the estimation of several values of an unknown parameter. Objective. The approach to the estimation of several values of an unknown parameter for a given data model is to be developed. Method. The approach to solve the estimation problem of the unknown parameter with several values is based on the constructing a function of the residual between the data and their model and on the subsequent applying the minimum-extent criterion to it. The minimum-extent criterion allows detecting the values of unknown parameter in the form of local minima for the quasi-extent functional of residual function. In the discrete case, the proposed approach is to search for the main local minima of the multiextremal objective function. To solve this problem in the one-dimensional case a simple method is proposed. The performance of this method is illustrated by the examples of the problems both with one unknown linear parameter of the model and with one unknown non-linear parameter of the model. Results. Unlike the traditional approaches based on the criterion of least squares or criterion of mean-absolute deviation which provide the possibility of estimating just one value of unknown parameter, the proposed approach provides estimating the several values of unknown parameter. Numerical simulation of the one-dimensional approximation problem with models containing the one unknown linear parameter and the one unknown non-linear parameter confirmed the feasibility of the proposed approach and its performance when the necessary smoothing does not lead to the loss of weak local minima. Conclusions. To estimate the several values of unknown parameter it is advisable to use the approach which consists in solving the minimization problem of the quasi-extent functional for the residual function of data. This approach provides an individualization of the values of unknown parameter by forming the corresponding local minima of the objective function. The results of numerical simulation of the one-dimensional problem for both the linear and non-linear parameter confirmed the performance of the proposed approach.