This 2002 book presents the reader with mathematical tools taken from matrix calculus and zero-one matrices and demonstrates how these tools greatly facilitate the application of classical statistical procedures to econometric models. The matrix calculus results are derived from a few basic rules that are generalizations of the rules of ordinary calculus. These results are summarized in a useful table. Well-known zero-one matrices, together with some newer ones, are defined, their mathematical roles explained, and their useful properties presented. The basic building blocks of classical statistics, namely the score vector, the information matrix, and the Cramer-Rao lower bound, are obtained for a sequence of linear econometric models of increasing statistical complexity. From these are obtained interactive interpretations of maximum likelihood estimators, linking them with efficient econometric estimators. Classical test statistics are also derived and compared for hypotheses of interest.
The book aims to present a wide range of the newest results on multivariate statistical models, distribution theory and applications of multivariate statistical methods. A paper on Pearson–Kotz–Dirichlet distributions by Professor N Balakrishnan contains main results of the Samuel Kotz Memorial Lecture. Extensions of linear models to multivariate exponential dispersion models and Growth Curve models are presented, and several papers on classification methods are included. Applications range from insurance mathematics to medical and industrial statistics and sampling algorithms. Contents:Variable Selection and Post-Estimation of Regression Parameters Using Quasi-Likelihood Approach (S Fallahpour and S E Ahmed)Maximum Likelihood Estimates for Markov-Additive Processes of Arrivals by Aggregated Data (A M Andronov)A Simple and Efficient Method of Estimation of the Parameters of a Bivariate Birnbaum-Saunders Distribution Based on Type-II Censored Samples (N Balakrishnan and X Zhu)Analysis of Contingent Valuation Data with Self-Selected Rounded WTP-Intervals Collected by Two-Steps Sampling Plans (Yu K Belyaev and B Kriström)Optimal Classification of Multivariate GRF Observations (K Dučinskas and L Dreižienė)Multivariate Exponential Dispersion Models (B Jørgensen and J R Martínez)Statistical Inference with the Limited Expected Value Function (M Käärik and H Kadarik)Shrinkage Estimation via Penalized Least Squares in Linear Regression with an Application to Hip Fracture Treatment Costs (A Liski, E P Liski and U Häkkinen)K-Nearest Neighbors as Pricing Tool in Insurance: A Comparative Study (K Pärna, R Kangro, A Kaasik and M Möls)Statistical Study of Factors Affecting Knee Joint Space and Osteophytes in the Population with Early Knee Osteoarthritis (T von Rosen, A E Tamm, A O Tamm and I Traat)Simultaneous Confidence Region for ρ and σ2 in a Multivariate Linear Model with Uniform Correlation Structure (I Žežula and D Klein) Readership: Graduated students and Professional researchers in mathematics. Keywords:Multivariate Distributions;Multivariate Statistical Models;Applications of Multivariate Statistical MethodsKey Features:Among the authors several prominent ones appear: N Balakrishnan, E Ahmed, Y Belyaev, B JorgensenOnly few books are published which are dedicated to the problems of multivariate statistics only thus it valuable for people who work in multivariate statisticsApplications in different areas demonstrate the usefulness of the theory in practice
This book provides a unified treatment of matrix differential calculus, specifically written for econometricians and statisticians. Divided into six parts, the book begins with a treatment of matrix algebra, discussing the Schur, Jordan, and singular-value decompositions, the Hadamard and Kronecker products, and more. The second section is the theoretical core of the book and presents a thorough development of the theory of differentials. Practically-oriented, part three contains the rules for working with differentials and lists the differentials of important scalar, vector, and matrix functions. The fourth deals with inequalities, such as Cauchy-Schwarz's and Minkowski's, while the fifth section is devoted to applications of matrix differential calculus to the linear regression model. The book closes by detailing maximum likelihood estimation, an ideal source for demonstrating the power of the propagated techniques. Features numerous exercises.
Designed to demonstrate the essential mathematical concepts—comprehensively and economically—without re-teaching basic material or laboring over superfluous ideas, this text locates the necessary information in a practical economics context. Utilizing clear exposition and dynamic pedagogical features, Mathematical Tools for Economics provides students with the analytical skills they need to better grasp their field of study. A short introduction to mathematics for students of economics Demonstrates essential mathematical concepts necessary for economic analysis, such as matrix algebra and calculus, simultaneous linear equations, and concrete and discrete time Incorporates applications to econometrics and statistics, and includes computational exercises illustrating the methods and concepts discussed in the text Clear explanations and dynamic pedagogical features provide students with the analytical skills they need to better grasp their field of study. Mathematical Tools for Economics is supported by an instructor's manual featuring solutions, available at www.blackwellpublishing.com/turkington