Confidence interval estimation of nonlinear single-parameter paired regression

Getmanskaya I. V.

Moscow Technical University of Communications and Informatics

Abstract: A mathematical model of a phenomenon is usually a functional dependence, possibly nonlinear, of its quantitative characteristics, some of which are dependent and independent variables, and others are parameters. Due to unaccounted-for factors or errors in measuring characteristics, random components are introduced into the model, and it is built in a probabilistic scheme of problem formulation, and is used and refined by statistical methods. An adequate model allows us to solve the problem of estimating its parameters based on the measured values of dependent and independent variables. The solution of such problems by traditional methods is based on the laws of the distribution of random components of the model. In this paper, it is assumed that only their basic numerical characteristics are known, which are known, for example, as errors of the measuring device or the result of long-term observations. Estimating the parameters of a general population or regression models using statistical methods is, in fact, an approximation, the main characteristic of which is accuracy. In statistics, the accuracy of an approximation is estimated using a confidence interval. In this paper, a point estimate is considered as the center of the confidence interval, as the average of a certain subset of implementations of the estimated parameter. To determine the reliability of the confidence interval, information is needed on the variance of the center of the interval (point estimate), which is expressed in terms of a previously found approximation of a certain accuracy of the variance of the components of the point estimate of the estimated parameter. The construction of the confidence interval is based on the classical laws of large numbers: Chebyshev's inequality in centered form and one of the forms of the central limit theorem. A numerical interval estimation experiment is performed for 2 regressions with different values of the volume of the initial data, the estimated parameter and the standard deviation of the dependent argument, as well as in different intervals of change of the independent argument. The experimental results are consistent with the theoretical grounds given in the work.

Keywords: regression, regression coefficient, nonlinear, point, interval, consistent estimation, mathematical expectation, variance, standard deviation, average numerical characteristics of a random variable.

UDC: 519.23

MSC: 93A30