This book presents a method of two-stage maximization of a likelihood function, which helps to solve a series of non-solved before well-posed and ill-posed problems of pseudosolution computing systems of linear algebraic equations (or, in statistical terminology, parameter estimators of functional relationships) and linear integral equations in the presence of deterministic and random errors in the initial data. A solution of the problem of reciprocal influence of passive errors of regressors and of active errors of predictors is given by computing point estimators of functional relationships.
In Chapter 1, the basic problem of confluent, confluent-variance and confluent-regression analysis of passive experiments, a problem of estimation of unknown parameters, is solved algebraically. The problem of robust estimation of normal parameters of incomplete-rank confluent and confluent-regression models is solved also. In Chapter 2, models of passive-active-regression experiments are constructed. A picture of exposition of experimental researches in the framework of confluent-influent-regression models is finished. This allows to understand better a picture of researches and to carry out correctly parameter estimation. A method of effective correction of rounding errors is constructed also for the procedure of numerical solution of systems of linear algebraic equations and of numerical computation of parameter estimates. Regularized estimation methods for the case of incomplete-rank matrices are developed.
In Chapter 3, it is allowed for deterministic and random errors in variational methods that construct pseudosolutions of linear integral equations of the second kind and regularized pseudosolutions and quasisolutions of linear integral equations of the first kind. Both passive errors (i.e., errors during observation or measurement) in the right-hand side and passive or active errors (i.e., errors during specification) in the core are considered. Representation methods of a priori information on sought pseudosolutions using mixed models and statistical regularization methods are considered. Numerical realizations of these methods are constructed.
This book is intended for students, postgraduate students, scientists, and other researchers handling economical and technical data. It is especially intended for those who constantly use regression analysis in their own research and for those who create mathematical software for computers.