Regression Analysis Under A Priori Parameter Restrictions [electronic resource] /
Knopov, Pavel S.author.
Korkhin, Arnold S.author.
Construction of various models of objects under uncertainty is one of the most important problems in modern decision making theory. Regression models are some of the most prevalent tools for modeling under uncertainty and are widely applied in different branches of science such as in industrial research, agriculture, medicine, and business and economics. Regression Analysis Under A Priori Parameter Restrictions will be of interest to a broad spectrum of readers in applied mathematics, mathematical statistics, identification theory, systems analysis, econometrics, finance, optimization, and other scientific disciplines. Requiring a background in algebra, probability theory, mathematical statistics, and mathematical programming, this work may also be a useful supplement for advanced graduate courses in estimation theory, regression analysis, mathematical statistics, econometrics, mathematical programming and optimal control, and stochastic optimization.  The material contained in this monograph successfully combines interesting theoretical results with methods and algorithms for solving practical problems. It focuses on the construction of regression models with linear and non-linear constraint inequalities and is the first book in which the theoretical results lying in the background of construction and studying regression models with inequality constraints on parameters are presented systematically and solidly.   Problems are described and studied in a clear, precise, and rigorous method and include: calculation of estimates for regression parameters, determination of their asymptotic properties and accuracy of estimation, point and interval prediction by the regression, parameters of which are estimated under inequality constraints. The authors’ approach lends itself to numerous applications in various practical problems, several of which are discussed in detail.
Description based upon print version of record.
Regression Analysis Under A Priori Parameter Restrictions; Preface; Acknowledgments; Contents; Notation; Chapter 1: Estimation of Regression Model Parameterswith Specific Constraints; 1.1 Estimation of the Parameters of a Linear Regression with Inequality Constraints; 1.1.1 Method of Estimating the Solution to (1.7); 1.1.2 Algorithm of Finding the Solution to (1.9); 1.1.3 Special Case of the Problem (1.7); 1.2 Estimation of Parameters of Nonlinear Regression with Nonlinear Inequality Constraints; 1.2.1 Statement of the Problem and a Method of Its Solution; 1.2.1.1 Estimation with Constraints
1.2.1.2 Estimation Without Constraints1.2.2 Solution to the Auxiliary Problem; 1.2.3 Compatibility of Constraints in the Auxiliary Problem; 1.2.4 Calculation of the Constants  and; 1.3 Estimation of Multivariate Linear Regression Parameters with Nonlinear Equality Constraints; Chapter 2: Asymptotic Properties of Parameters in Nonlinear Regression Models; 2.1 Consistency of Estimates in Nonlinear Regression Models; 2.2 Asymptotic Properties of Nonlinear Regression Parameters Estimates Obtained by the Least Squares Method Under a Priory Inequality Constraints (Convex Case); 2.2.1 Introduction
2.2.2 Auxiliary Results2.2.3 Fundamental Results; 2.3 Asymptotic Properties of Nonlinear Regression Parameters Estimates by the Least Squares Method Under a Priory Inequality Constraints (Non-Convex Case); 2.3.1 Assumptions and Auxiliary Results; 2.3.2 Fundamental Result; 2.4 Limit Distribution of the Estimate of Regression Parameters Which Are Subject to Equality Constraints; 2.5 Asymptotic Properties of the Least Squares Estimates of Parameters of a Linear Regression with Non-Stationary Variables Under Convex Restrictions on Parameters; 2.5.1 Settings; 2.5.2 Consistency of Estimator
2.5.3 Limit Distribution of the Parameter EstimateChapter 3: Method of Empirical Means in Nonlinear Regression and Stochastic Optimization Models; 3.1 Consistency of Estimates Obtained by the Method of Empirical Means with Independent Or Weakly Dependent Observations; 3.2 Regression Models for Long Memory Systems; 3.3 Statistical Methods in Stochastic Optimization and Estimation Problems; 3.4 Empirical Mean Estimates Asymptotic Distribution; 3.4.1 Asymptotic Distribution of Empirical Estimates for Models with Independent and Weakly Dependent Observations
3.4.2 Asymptotic Distribution of Estimates for Long Memory Stochastic Systems3.4.3 Asymptotic Distribution of the Least Squares Estimates for Long Memory Stochastic Systems; 3.5 Large Deviations of Empirical Means in Estimation and Optimization Problems; 3.5.1 Large Deviations of the Empirical Means Method for Dependent Observations; 3.5.2 Large Deviations of Empiric Estimates for Non-Stationary Observations; 3.5.3 Large Deviations in Nonlinear Regression Problems; Chapter 4: Determination of Accuracy of Estimation of Regression Parameters Under Inequality Constraints
4.1 Preliminary Analysis of the Problem
Includes bibliographical references and index.
Construction of various models of objects under uncertainty is one of the most important problems in modern decision making theory. Regression models are some of the most prevalent tools for modeling under uncertainty and are widely applied in different branches of science such as in industrial research, agriculture, medicine, and business and economics. Regression Analysis Under A Priori Parameter Restrictions will be of interest to a broad spectrum of readers in applied mathematics, mathematical statistics, identification theory, systems analysis, econometrics, finance, optimization, and other scientific disciplines. Requiring a background in algebra, probability theory, mathematical statistics, and mathematical programming, this work may also be a useful supplement for advanced graduate courses in estimation theory, regression analysis, mathematical statistics, econometrics, mathematical programming and optimal control, and stochastic optimization.  The material contained in this monograph successfully combines interesting theoretical results with methods and algorithms for solving practical problems. It focuses on the construction of regression models with linear and non-linear constraint inequalities and is the first book in which the theoretical results lying in the background of construction and studying regression models with inequality constraints on parameters are presented systematically and solidly.   Problems are described and studied in a clear, precise, and rigorous method and include: calculation of estimates for regression parameters, determination of their asymptotic properties and accuracy of estimation, point and interval prediction by the regression, parameters of which are estimated under inequality constraints. The authors’ approach lends itself to numerous applications in various practical problems, several of which are discussed in detail.
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