Advanced Calculus With Applications in Statistics

Abstract

Preface. 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. Scheffe'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. Index.