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Advanced Calculus with Applications in Statistics, 2nd Edition

ISBN: 978-0-471-39104-3
704 pages
November 2002, ©2003
Advanced Calculus with Applications in Statistics, 2nd Edition (0471391042) cover image
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. The Second Edition provides substantial new coverage of the material, including three new chapters and a large appendix that contains solutions to almost all of the exercises in the book. Applications of some of these methods in statistics are discusses. 
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Preface to the First Edition.

1. An Introduction to Set Theory.

1.1. The Concept of a Set.

1.2. Set Operations.

1.3. Relations and Functions.

1.4. Finite, Countable, and Uncountable Sets.

1.5. Bounded Sets.

1.6. Some Basic Topological Concepts.

1.7. Examples in Probability and Statistics.

2. Basic Concepts in Linear Algebra.

2.1. Vector Spaces and Subspaces.

2.2. Linear Transformations.

2.3. Matrices and Determinants.

2.4. Applications of Matrices in Statistics.

3. Limits and Continuity of Functions.

3.1. Limits of a Function.

3.2. Some Properties Associated with Limits of Functions.

3.3. The o, O Notation.

3.4. Continuous Functions.

3.5. Inverse Functions.

3.6. Convex Functions.

3.7. Continuous and Convex Functions in Statistics.

4. Differentiation.

4.1. The Derivative of a Function.

4.2. The Mean Value Theorem.

4.3. Taylor's Theorem.

4.4. Maxima and Minima of a Function.

4.5. Applications in Statistics.

5. Infinite Sequences and Series.

5.1. Infinite Sequences.

5.2. Infinite Series.

5.3. Sequences and Series of Functions.

5.4. Power Series.

5.5. Sequences and Series of Matrices.

5.6. Applications in Statistics.

6. Integration.

6.1. Some Basic Definitions.

6.2. The Existence of the Riemann Integral.

6.3. Some Classes of Functions That Are Riemann Integrable.

6.4. Properties of the Riemann Integral.

6.5. Improper Riemann Integrals.

6.6. Convergence of a Sequence of Riemann Integrals.

6.7. Some Fundamental Inequalities.

6.8. RiemannStieltjes Integral.

6.9. Applications in Statistics.

7. Multidimensional Calculus.

7.1. Some Basic Definitions.

7.2. Limits of a Multivariable Function.

7.3. Continuity of a Multivariable Function.

7.4. Derivatives of a Multivariable Function.

7.5. Taylor's Theorem for a Multivariable Function.

7.6. Inverse and Implicit Function Theorems.

7.7. Optima of a Multivariable Function.

7.8. The Method of Lagrange Multipliers.

7.9. The Riemann Integral of a Multivariable Function.

7.10. Differentiation under the Integral Sign.

7.11. Applications in Statistics.

8. Optimization in Statistics.

8.1. The Gradient Methods.

8.2. The Direct Search Methods.

8.3. Optimization Techniques in Response Surface Methodology.

8.4. Response Surface Designs.

8.5. Alphabetic Optimality of Designs.

8.6. Designs for Nonlinear Models.

8.7. Multiresponse Optimization.

8.8. Maximum Likelihood Estimation and the EM Algorithm.

8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components.

8.10. Scheffé's Confidence Intervals.

9. Approximation of Functions.

9.1. Weierstrass Approximation.

9.2. Approximation by Polynomial Interpolation.

9.3. Approximation by Spline Functions.

9.4. Applications in Statistics.

10. Orthogonal Polynomials.

10.1. Introduction.

10.2. Legendre Polynomials.

10.3. Jacobi Polynomials.

10.4. Chebyshev Polynomials.

10.5. Hermite Polynomials.

10.6. Laguerre Polynomials.

10.7. Least-Squares Approximation with Orthogonal Polynomials.

10.8. Orthogonal Polynomials Defined on a Finite Set.

10.9. Applications in Statistics.

11. Fourier Series.

11.1. Introduction.

11.2. Convergence of Fourier Series.

11.3. Differentiation and Integration of Fourier Series.

11.4. The Fourier Integral.

11.5. Approximation of Functions by Trigonometric Polynomials.

11.6. The Fourier Transform.

11.7. Applications in Statistics.

12. Approximation of Integrals.

12.1. The Trapezoidal Method.

12.2. Simpson’s Method.

12.3. NewtonCotes Methods.

12.4. Gaussian Quadrature.

12.5. Approximation over an Infinite Interval.

12.6. The Method of Laplace.

12.7. Multiple Integrals.

12.8. The Monte Carlo Method.

12.9. Applications in Statistics.

Appendix. Solutions to Selected Exercises.

General Bibliography.


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ANDRE I. KHURI, PhD, is a Professor in the Department of Statistics at the University of Florida, Gainesville.
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  • An application-oriented and appropriately rigorous introduction to the central themes of advanced calculus for statistics students, with enough theoretical explanation to be suitable for mathematics students as well.
  • End-of-chapter applications for flexibility in use by either audience.
  • New sections added: topology of the real line, complex numbers, additional properties of eigenvalues, some matrix inequalities, distribution of quadratic forms, tensors, and designs for General Linear Models (GLMs), among others.
  • Solutions to selected exercises now included to encourage independent study and content reinforcement.
  • Even more examples and exercises.
  • Unique blend of mathematics and statistics with no known competitor
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"This is an exceptional book, which I would recommend for anyone beginning a career in statistical research." (Journal of the American Statistical Association, September 2004)
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