Advanced Calculus with Applications in Statistics

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

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 Notation 65

3.4. Continuous Functions 66

3.4.1. Some Properties of Continuous Functions 71

3.4.2. Lipschitz Continuous Functions 75

3.5. Inverse Functions 76

3.6. Convex Functions 79

3.7. Continuous and Convex Functions in Statistics 82

Further Reading and Annotated Bibliography 87

Exercises 88

4. Differentiation 93

4.1. The Derivative of a Function 93

4.2. The Mean Value Theorem 99

4.3. Taylor’s Theorem 108

4.4. Maxima and Minima of a Function 112

4.4.1. A Sufficient Condition for a Local Optimum 114

4.5. Applications in Statistics 115

4.5.1. Functions of Random Variables 116

4.5.2. Approximating Response Functions 121

4.5.3. The Poisson Process 122

4.5.4. Minimizing the Sum of Absolute Deviations 124

Further Reading and Annotated Bibliography 125

Exercises 127

5. Infinite Sequences and Series 132

5.1. Infinite Sequences 132

5.1.1. The Cauchy Criterion 137

5.2. Infinite Series 140

5.2.1. Tests of Convergence for Series of Positive Terms 144

5.2.2. Series of Positive and Negative Terms 158

5.2.3. Rearrangement of Series 159

5.2.4. Multiplication of Series 162

5.3. Sequences and Series of Functions 165

5.3.1. Properties of Uniformly Convergent Sequences and Series 169

5.4. Power Series 174

5.5. Sequences and Series of Matrices 178

5.6. Applications in Statistics 182

5.6.1. Moments of a Discrete Distribution 182

5.6.2. Moment and Probability Generating Functions 186

5.6.3. Some Limit Theorems 191

5.6.3.1. The Weak Law of Large Numbers ŽKhinchine’s Theorem. 192

5.6.3.2. The Strong Law of Large Numbers ŽKolmogorov’s Theorem. 192

5.6.3.3. The Continuity Theorem for Probability Generating Functions 192

5.6.4. Power Series and Logarithmic Series Distributions 193

5.6.5. Poisson Approximation to Power Series Distributions 194

5.6.6. A Ridge Regression Application 195

Further Reading and Annotated Bibliography 197

Exercises 199

6. Integration 205

6.1. Some Basic Definitions 205

6.2. The Existence of the Riemann Integral 206

6.3. Some Classes of Functions That Are Riemann Integrable 210

6.3.1. Functions of Bounded Variation 212

6.4. Properties of the Riemann Integral 215

6.4.1. Change of Variables in Riemann Integration 219

6.5. Improper Riemann Integrals 220

6.5.1. Improper Riemann Integrals of the Second Kind 225

6.6. Convergence of a Sequence of Riemann Integrals 227

6.7. Some Fundamental Inequalities 229

6.7.1. The Cauchy_Schwarz Inequality 229

6.7.2. H¨older’s Inequality 230

6.7.3. Minkowski’s Inequality 232

6.7.4. Jensen’s Inequality 233

6.8. Riemann_Stieltjes Integral 234

6.9. Applications in Statistics 239

6.9.1. The Existence of the First Negative Moment of a Continuous Distribution 242

6.9.2. Transformation of Continuous Random Variables 246

6.9.3. The Riemann_Stieltjes Representation of the Expected Value 249

6.9.4. Chebyshev’s Inequality 251

Further Reading and Annotated Bibliography 252

Exercises 253

7. Multidimensional Calculus 261

7.1. Some Basic Definitions 261

7.2. Limits of a Multivariable Function 262

7.3. Continuity of a Multivariable Function 264

7.4. Derivatives of a Multivariable Function 267

7.4.1. The Total Derivative 270

7.4.2. Directional Derivatives 273

7.4.3. Differentiation of Composite Functions 276

7.5. Taylor’s Theorem for a Multivariable Function 277

7.6. Inverse and Implicit Function Theorems 280

7.7. Optima of a Multivariable Function 283

7.8. The Method of Lagrange Multipliers 288

7.9. The Riemann Integral of a Multivariable Function 293

7.9.1. The Riemann Integral on Cells 294

7.9.2. Iterated Riemann Integrals on Cells 295

7.9.3. Integration over General Sets 297

7.9.4. Change of Variables in n-Tuple Riemann Integrals 299

7.10. Differentiation under the Integral Sign 301

7.11. Applications in Statistics 304

7.11.1. Transformations of Random Vectors 305

7.11.2. Maximum Likelihood Estimation 308

7.11.3. Comparison of Two Unbiased Estimators 310

7.11.4. Best Linear Unbiased Estimation 311

7.11.5. Optimal Choice of Sample Sizes in Stratified Sampling 313

Further Reading and Annotated Bibliography 315

Exercises 316

8. Optimization in Statistics 327

8.1. The Gradient Methods 329

8.1.1. The Method of Steepest Descent 329

8.1.2. The Newton_Raphson Method 331

8.1.3. The Davidon_Fletcher_Powell Method 331

8.2. The Direct Search Methods 332

8.2.1. The Nelder_Mead Simplex Method 332

8.2.2. Price’s Controlled Random Search Procedure 336

8.2.3. The Generalized Simulated Annealing Method 338

8.3. Optimization Techniques in Response Surface Methodology 339

8.3.1. The Method of Steepest Ascent 340

8.3.2. The Method of Ridge Analysis 343

8.3.3. Modified Ridge Analysis 350

8.4. Response Surface Designs 355

8.4.1. First-Order Designs 356

8.4.2. Second-Order Designs 358

8.4.3. Variance and Bias Design Criteria 359

8.5. Alphabetic Optimality of Designs 362

8.6. Designs for Nonlinear Models 367

8.7. Multiresponse Optimization 370

8.8. Maximum Likelihood Estimation and the EM Algorithm 372

8.8.1. The EM Algorithm 375

8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components 378

8.10. Scheff´e’s Confidence Intervals 382

8.10.1. The Relation of Scheff´e’s Confidence Intervals to the F-Test 385

Further Reading and Annotated Bibliography 391

Exercises 395

9. Approximation of Functions 403

9.1. Weierstrass Approximation 403

9.2. Approximation by Polynomial Interpolation 410

9.2.1. The Accuracy of Lagrange Interpolation 413

9.2.2. A Combination of Interpolation and Approximation 417

9.3. Approximation by Spline Functions 418

9.3.1. Properties of Spline Functions 418

9.3.2. Error Bounds for Spline Approximation 421

9.4. Applications in Statistics 422

9.4.1. Approximate Linearization of Nonlinear Models by Lagrange Interpolation 422

9.4.2. Splines in Statistics 428

9.4.2.1. The Use of Cubic Splines in Regression 428

9.4.2.2. Designs for Fitting Spline Models 430

9.4.2.3. Other Applications of Splines in Statistics 431

Further Reading and Annotated Bibliography 432

Exercises 434

10. Orthogonal Polynomials 437

10.1. Introduction 437

10.2. Legendre Polynomials 440

10.2.1. Expansion of a Function Using Legendre Polynomials 442

10.3. Jacobi Polynomials 443

10.4. Chebyshev Polynomials 444

10.4.1. Chebyshev Polynomials of the First Kind 444

10.4.2. Chebyshev Polynomials of the Second Kind 445

10.5. Hermite Polynomials 447

10.6. Laguerre Polynomials 451

10.7. Least-Squares Approximation with Orthogonal Polynomials 453

10.8. Orthogonal Polynomials Defined on a Finite Set 455

10.9. Applications in Statistics 456

10.9.1. Applications of Hermite Polynomials 456

10.9.1.1. Approximation of Density Functions and Quantiles of Distributions 456

10.9.1.2. Approximation of a Normal Integral 460

10.9.1.3. Estimation of Unknown Densities 461

10.9.2. Applications of Jacobi and Laguerre Polynomials 462

10.9.3. Calculation of Hypergeometric Probabilities Using Discrete Chebyshev Polynomials 462

Further Reading and Annotated Bibliography 464

Exercises 466

11. Fourier Series 471

11.1. Introduction 471

11.2. Convergence of Fourier Series 475

11.3. Differentiation and Integration of Fourier Series 483

11.4. The Fourier Integral 488

11.5. Approximation of Functions by Trigonometric Polynomials 495

11.5.1. Parseval’s Theorem 496

11.6. The Fourier Transform 497

11.6.1. Fourier Transform of a Convolution 499

11.7. Applications in Statistics 500

11.7.1. Applications in Time Series 500

11.7.2. Representation of Probability Distributions 501

11.7.3. Regression Modeling 504

11.7.4. The Characteristic Function 505

11.7.4.1. Some Properties of Characteristic Functions 510

Further Reading and Annotated Bibliography 510

Exercises 512

12. Approximation of Integrals 517

12.1. The Trapezoidal Method 517

12.1.1. Accuracy of the Approximation 518

12.2. Simpson’s Method 521

12.3. Newton_Cotes Methods 523

12.4. Gaussian Quadrature 524

12.5. Approximation over an Infinite Interval 528

12.6. The Method of Laplace 531

12.7. Multiple Integrals 533

12.8. The Monte Carlo Method 535

12.8.1. Variation Reduction 537

12.8.2. Integrals in Higher Dimensions 540

12.9. Applications in Statistics 541

12.9.1. The Gauss_Hermite Quadrature 542

12.9.2. Minimum Mean Squared Error Quadrature 543

12.9.3. Moments of a Ratio of Quadratic Forms 546

12.9.4. Laplace’s Approximation in Bayesian Statistics 548

12.9.5. Other Methods of Approximating Integrals in Statistics 549

Further Reading and Annotated Bibliography 550

Exercises 552

Appendix. Solutions to Selected Exercises 557

General Bibliography 652

Index 665