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Advanced Calculus With Applications In Statistics (Second Edition) by André I. Khuri



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Advanced Calculus With Applications In Statistics (Second Edition) by André I. Khuri , University of Florida, Gainesville, Florida. "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.

Advanced Calculus With Applications In Statistics (Second Edition) by André I. Khuri cover the following topics.



  • 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
    Further Reading and Annotated Bibliography
    Exercises

  • 2. Basic Concepts in Linear Algebra
    2.1. Vector Spaces and Subspaces
    2.2. Linear Transformations
    2.3. Matrices and Determinants
    2.3.1. Basic Operations on Matrices
    2.3.2. The Rank of a Matrix
    2.3.3. The Inverse of a Matrix
    2.3.4. Generalized Inverse of a Matrix
    2.3.5. Eigenvalues and Eigenvectors of a Matrix
    2.3.6. Some Special Matrices
    2.3.7. The Diagonalization of a Matrix
    2.3.8. Quadratic Forms
    2.3.9. The Simultaneous Diagonalization of Matrices
    2.3.10. Bounds on Eigenvalues
    2.4. Applications of Matrices in Statistics
    2.4.1. The Analysis of the Balanced Mixed Model
    2.4.2. The Singular-Value Decomposition
    2.4.3. Extrema of Quadratic Forms
    2.4.4. The Parameterization of Orthogonal Matrices
    Further Reading and Annotated Bibliography
    Exercises

  • 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
    3.4.1. Some Properties of Continuous Functions
    3.4.2. Lipschitz Continuous Functions
    3.5. Inverse Functions
    3.6. Convex Functions
    3.7. Continuous and Convex Functions in Statistics
    Further Reading and Annotated Bibliography
    Exercises

  • 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
    4.4.1. A Sufficient Condition for a Local Optimum
    4.5. Applications in Statistics
    4.5.1. Functions of Random Variables
    4.5.2. Approximating Response Functions
    4.5.3. The Poisson Process
    4.5.4. Minimizing the Sum of Absolute Deviations
    Further Reading and Annotated Bibliography
    Exercises

  • 5. Infinite Sequences and Series
    5.1. Infinite Sequences
    5.1.1. The Cauchy Criterion
    5.2. Infinite Series
    5.2.1. Tests of Convergence for Series of Positive Terms
    5.2.2. Series of Positive and Negative Terms
    5.2.3. Rearrangement of Series
    5.2.4. Multiplication of Series
    5.3. Sequences and Series of Functions
    5.3.1. Properties of Uniformly Convergent Sequences and Series
    5.4. Power Series
    5.5. Sequences and Series of Matrices
    5.6. Applications in Statistics
    5.6.1. Moments of a Discrete Distribution
    5.6.2. Moment and Probability Generating Functions
    5.6.3. Some Limit Theorems
    5.6.3.1. The Weak Law of Large Numbers ŽKhinchine’s Theorem.,
    5.6.3.2. The Strong Law of Large Numbers ŽKolmogorov’s Theorem.
    5.6.3.3. The Continuity Theorem for Probability Generating Functions
    5.6.4. Power Series and Logarithmic Series Distributions
    5.6.5. Poisson Approximation to Power Series Distributions
    5.6.6. A Ridge Regression Application
    Further Reading and Annotated Bibliography
    Exercises

  • 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
    6.3.1. Functions of Bounded Variation
    6.4. Properties of the Riemann Integral
    6.4.1. Change of Variables in Riemann Integration
    6.5. Improper Riemann Integrals
    6.5.1. Improper Riemann Integrals of the Second Kind
    6.6. Convergence of a Sequence of Riemann Integrals
    6.7. Some Fundamental Inequalities
    6.7.1. The CauchySchwarz Inequality
    6.7.2. H¨older’s Inequality
    6.7.3. Minkowski’s Inequality
    6.7.4. Jensen’s Inequality
    6.8. RiemannStieltjes Integral
    6.9. Applications in Statistics
    6.9.1. The Existence of the First Negative Moment of a Continuous Distribution
    6.9.2. Transformation of Continuous Random Variables
    6.9.3. The RiemannStieltjes Representation of the Expected Value
    6.9.4. Chebyshev’s Inequality
    Further Reading and Annotated Bibliography
    Exercises

  • 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
    7.4.1. The Total Derivative
    7.4.2. Directional Derivatives
    7.4.3. Differentiation of Composite Functions
    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
    7.9.1. The Riemann Integral on Cells
    7.9.2. Iterated Riemann Integrals on Cells
    7.9.3. Integration over General Sets
    7.9.4. Change of Variables in n-Tuple Riemann Integrals
    7.10. Differentiation under the Integral Sign
    7.11. Applications in Statistics
    7.11.1. Transformations of Random Vectors
    7.11.2. Maximum Likelihood Estimation
    7.11.3. Comparison of Two Unbiased Estimators
    7.11.4. Best Linear Unbiased Estimation
    7.11.5. Optimal Choice of Sample Sizes in Stratified Sampling
    Further Reading and Annotated Bibliography
    Exercises

  • 8. Optimization in Statistics
    8.1. The Gradient Methods
    8.1.1. The Method of Steepest Descent
    8.1.2. The NewtonRaphson Method
    8.1.3. The DavidonFletcherPowell Method
    8.2. The Direct Search Methods
    8.2.1. The NelderMead Simplex Method
    8.2.2. Price’s Controlled Random Search Procedure
    8.2.3. The Generalized Simulated Annealing Method
    8.3. Optimization Techniques in Response Surface Methodology
    8.3.1. The Method of Steepest Ascent
    8.3.2. The Method of Ridge Analysis
    8.3.3. Modified Ridge Analysis
    8.4. Response Surface Designs
    8.4.1. First-Order Designs
    8.4.2. Second-Order Designs
    8.4.3. Variance and Bias Design Criteria
    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
    8.8.1. The EM Algorithm
    8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components
    8.10. Scheff´e’s Confidence Intervals
    8.10.1. The Relation of Scheff´e’s Confidence Intervals to the F-Test
    Further Reading and Annotated Bibliography
    Exercises

  • 9. Approximation of Functions
    9.1. Weierstrass Approximation
    9.2. Approximation by Polynomial Interpolation
    9.2.1. The Accuracy of Lagrange Interpolation
    9.2.2. A Combination of Interpolation and Approximation
    9.3.1. Properties of Spline Functions
    9.3.2. Error Bounds for Spline Approximation
    9.4. Applications in Statistics
    9.4.1. Approximate Linearization of Nonlinear Models by Lagrange Interpolation
    9.4.2. Splines in Statistics
    9.4.2.1. The Use of Cubic Splines in Regression
    9.4.2.2. Designs for Fitting Spline Models
    9.4.2.3. Other Applications of Splines in Statistics
    Further Reading and Annotated Bibliography
    Exercises

  • 10. Orthogonal Polynomials
    10.1. Introduction
    10.2. Legendre Polynomials
    10.2.1. Expansion of a Function Using Legendre Polynomials
    10.3. Jacobi Polynomials
    10.4. Chebyshev Polynomials
    10.4.1. Chebyshev Polynomials of the First Kind
    10.4.2. Chebyshev Polynomials of the Second Kind
    10.5. Hermite Polynomials
    10.6. Laguerre Polynomials
    10.7. Least-Squares Approximation with Orthogonal Polynomials
    9.3. Approximation by Spline Functions
    10.8. Orthogonal Polynomials Defined on a Finite Set
    10.9. Applications in Statistics
    10.9.1. Applications of Hermite Polynomials
    10.9.1.1. Approximation of Density Functions and Quantiles of Distributions
    10.9.1.2. Approximation of a Normal Integral
    10.9.1.3. Estimation of Unknown Densities
    10.9.2. Applications of Jacobi and Laguerre Polynomials
    10.9.3. Calculation of Hypergeometric Probabilities Using Discrete Chebyshev Polynomials
    Further Reading and Annotated Bibliography
    Exercises

  • 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
    11.5.1. Parseval’s Theorem
    11.6. The Fourier Transform
    11.6.1. Fourier Transform of a Convolution
    11.7. Applications in Statistics
    11.7.1.Applications in Time Series
    11.7.2. Representation of Probability Distributions
    11.7.3. Regression Modeling
    11.7.4. The Characteristic Function
    11.7.4.1. Some Properties of Characteristic Functions
    Further Reading and Annotated Bibliography
    Exercises

  • 12. Approximation of Integrals
    12.1. The Trapezoidal Method
    12.1.1. Accuracy of the Approximation
    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
    12.8.1. Variation Reduction
    12.8.2. Integrals in Higher Dimensions
    12.9. Applications in Statistics
    12.9.1. The GaussHermite Quadrature
    12.9.2. Minimum Mean Squared Error Quadrature
    12.9.3. Moments of a Ratio of Quadratic Forms
    12.9.4. Laplace’s Approximation in Bayesian Statistics
    12.9.5. Other Methods of Approximating Integrals in Statistics
    Further Reading and Annotated Bibliography
    Exercises,

  • Appendix. Solutions to Selected Exercises
    General Bibliography
    Index
    Preface

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