Advanced Calculus with Applications in Statistics

Informationen zum Autor ANDRE I. KHURI , PhD, is a Professor in the Department of Statistics at the University of Florida, Gainesville. Klappentext Praise for the First Edition "An enticing approach to the subject. . . . Students contemplating a career in statistics will acquire a valuable understanding of the underlying structure of statistical theory. . . statisticians should consider purchasing it as an additional reference on advanced calculus." - Journal of the American Statistical Association "This book is indeed a pleasure to read. It is simple to understand what the author is attempting to accomplish, and to follow him as he proceeds. . . . I would highly recommend the book for one's personal collection or suggest your librarian purchase a copy." - Journal of the Operational Research Society Knowledge of advanced calculus has become imperative to the understanding of the recent advances in statistical methodology. The First Edition of Advanced Calculus with Applications in Statistics has served as a reliable resource for both practicing statisticians and students alike. In light of the tremendous growth of the field of statistics since the book's publication, André Khuri has reexamined his popular work and substantially expanded it to provide the most up-to-date and comprehensive coverage of the subject. Retaining the original's much-appreciated application-oriented approach, Advanced Calculus with Applications in Statistics , Second Edition supplies a rigorous introduction to the central themes of advanced calculus suitable for both statisticians and mathematicians alike. The Second Edition adds significant new material on: Basic topological concepts Orthogonal polynomials Fourier series Approximation of integrals Solutions to selected exercises The volume's user-friendly text is notable for its end-of-chapter applications, designed to be flexible enough for both statisticians and mathematicians. Its well thought-out solutions to exercises encourage independent study and reinforce mastery of the content. Any statistician, mathematician, or student wishing to master advanced calculus and its applications in statistics will find this new edition a welcome resource. Zusammenfassung Designed to help motivate the learning of advanced calculus by demonstrating its relevance in the field of statistics! this successful text features detailed coverage of optimization techniques and their applications in statistics while introducing the reader to approximation theory. Inhaltsverzeichnis Preface xv Preface to the First Edition xvii 1. An Introduction to Set Theory 1 1.1. The Concept of a Set 1 1.2. Set Operations 2 1.3. Relations and Functions 4 1.4. Finite Countable and Uncountable Sets 6 1.5. Bounded Sets 9 1.6. Some Basic Topological Concepts 10 1.7. Examples in Probability and Statistics 13 Further Reading and Annotated Bibliography 15 Exercises 17 2. Basic Concepts in Linear Algebra 21 2.1. Vector Spaces and Subspaces 21 2.2. Linear Transformations 25 2.3. Matrices and Determinants 27 2.3.1. Basic Operations on Matrices 28 2.3.2. The Rank of a Matrix 33 2.3.3. The Inverse of a Matrix 34 2.3.4. Generalized Inverse of a Matrix 36 2.3.5. Eigenvalues and Eigenvectors of a Matrix 36 2.3.6. Some Special Matrices 38 2.3.7. The Diagonalization of a Matrix 38 2.3.8. Quadratic Forms 39 2.3.9. The Simultaneous Diagonalization of Matrices 40 2.3.10. Bounds on Eigenvalues 41 2.4. Applications of Matrices in Statistics 43 2.4.1. The Analysis of the Balanced Mixed Model 43 2.4.2. The Singular-Value Decomposition 45 2.4.3. Extrema of Quadratic Forms 48 2.4.4. The Paramete...

Autorentext
ANDRE I. KHURI, PhD, is a Professor in the Department of Statistics at the University of Florida, Gainesville.

Klappentext
Praise for the First Edition "An enticing approach to the subject. . . . Students contemplating a career in statistics will acquire a valuable understanding of the underlying structure of statistical theory. . . statisticians should consider purchasing it as an additional reference on advanced calculus."
-Journal of the American Statistical Association "This book is indeed a pleasure to read. It is simple to understand what the author is attempting to accomplish, and to follow him as he proceeds. . . . I would highly recommend the book for one's personal collection or suggest your librarian purchase a copy."
-Journal of the Operational Research Society Knowledge of advanced calculus has become imperative to the understanding of the recent advances in statistical methodology. The First Edition of Advanced Calculus with Applications in Statistics has served as a reliable resource for both practicing statisticians and students alike. In light of the tremendous growth of the field of statistics since the book's publication, André Khuri has reexamined his popular work and substantially expanded it to provide the most up-to-date and comprehensive coverage of the subject. Retaining the original's much-appreciated application-oriented approach, Advanced Calculus with Applications in Statistics, Second Edition supplies a rigorous introduction to the central themes of advanced calculus suitable for both statisticians and mathematicians alike. The Second Edition adds significant new material on: Basic topological concepts Orthogonal polynomials Fourier series Approximation of integrals Solutions to selected exercises The volume's user-friendly text is notable for its end-of-chapter applications, designed to be flexible enough for both statisticians and mathematicians. Its well thought-out solutions to exercises encourage independent study and reinforce mastery of the content. Any statistician, mathematician, or student wishing to master advanced calculus and its applications in statistics will find this new edition a welcome resource.

Zusammenfassung
Designed to help motivate the learning of advanced calculus by demonstrating its relevance in the field of statistics, this successful text features detailed coverage of optimization techniques and their applications in statistics while introducing the reader to approximation theory.

Inhalt
Preface xv Preface to the First Edition xvii 1. An Introduction to Set Theory 1 1.1. The Concept of a Set 1 1.2. Set Operations 2 1.3. Relations and Functions 4 1.4. Finite Countable and Uncountable Sets 6 1.5. Bounded Sets 9 1.6. Some Basic Topological Concepts 10 1.7. Examples in Probability and Statistics 13 Further Reading and Annotated Bibliography 15 Exercises 17 2. Basic Concepts in Linear Algebra 21 2.1. Vector Spaces and Subspaces 21 2.2. Linear Transformations 25 2.3. Matrices and Determinants 27 2.3.1. Basic Operations on Matrices 28 2.3.2. The Rank of a Matrix 33 2.3.3. The Inverse of a Matrix 34 2.3.4. Generalized Inverse of a Matrix 36 2.3.5. Eigenvalues and Eigenvectors of a Matrix 36 2.3.6. Some Special Matrices 38 2.3.7. The Diagonalization of a Matrix 38 2.3.8. Quadratic Forms 39 2.3.9. The Simultaneous Diagonalization of Matrices 40 2.3.10. Bounds on Eigenvalues 41 2.4. Applications of Matrices in Statistics 43 2.4.1. The Analysis of the Balanced Mixed Model 43 2.4.2. The Singular-Value Decomposition 45 2.4.3. Extrema of Quadratic Forms 48 2.4.4. The Parameterization of Orthogonal Matrices 49 Further Reading and Annotated Bibliography 50 Exercises 53 3. Limits and Continuity of Functions 57 3.1. Limits of a Function 57 3.2. Some Properties Associated with Limits of Functions 63 3.3. The o O Not…

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