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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 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
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
2.3.9. 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
2.4.4. The Parameterization of Orthogonal Matrices
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
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
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
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 CauchySchwarz Inequality
6.7.2. H¨older’s Inequality
6.7.3. Minkowski’s Inequality
6.7.4. Jensen’s Inequality
6.8. RiemannStieltjes 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 RiemannStieltjes Representation of the Expected Value
6.9.4. Chebyshev’s Inequality
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
Exercises

• 8. Optimization in Statistics
8.1.1. The Method of Steepest Descent
8.1.2. The NewtonRaphson Method
8.1.3. The DavidonFletcherPowell Method
8.2. The Direct Search Methods
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
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
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
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
Exercises

• 12. Approximation of Integrals
12.1. The Trapezoidal Method
12.1.1. Accuracy of the Approximation
12.2. Simpson’s Method
12.3. NewtonCotes Methods
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
12.9.2. 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
Exercises,

• Appendix. Solutions to Selected Exercises
General Bibliography
Index
Preface

• Open Now

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