目录：Matrix Differential Calculus with Applications in Statistics and Econometrics,3rd_[Magnus2019]
2024-09-02 05:09:25
目录：Matrix Differential Calculus with Applications in Statistics and Econometrics,3rd_[Magnus2019]
Title -16
Contents -14
Preface -6
Part One — Matrices 1
1 Basic properties of vectors and matrices 3
1.1 Introduction 3
1.2 Sets 3
1.3 Matrices: addition and multiplication 4
1.4 The transpose of a matrix 6
1.5 Square matrices 6
1.6 Linear forms and quadratic forms 7
1.7 The rank of a matrix 9
1.8 The inverse 10
1.9 The determinant 10
1.10 The trace 11
1.11 Partitioned matrices 12
1.12 Complex matrices 14
1.13 Eigenvalues and eigenvectors 14
1.14 Schur’s decomposition theorem 17
1.15 The Jordan decomposition 18
1.16 The singular-value decomposition 20
1.17 Further results concerning eigenvalues 20
1.18 Positive (semi)definite matrices 23
1.19 Three further results for positive definite matrices 25
1.20 A useful result 26
1.21 Symmetric matrix functions 27
Miscellaneous exercises 28
Bibliographical notes 30
2 Kronecker products, vec operator, and Moore-Penrose inverse 31
2.1 Introduction 31
2.2 The Kronecker product 31
2.3 Eigenvalues of a Kronecker product 33
2.4 The vec operator 34
2.5 The Moore-Penrose (MP) inverse 36
2.6 Existence and uniqueness of the MP inverse 37
2.7 Some properties of the MP inverse 38
2.8 Further properties 39
2.9 The solution of linear equation systems 41
Miscellaneous exercises 43
Bibliographical notes 45
3 Miscellaneous matrix results 47
3.1 Introduction 47
3.2 The adjoint matrix 47
3.3 Proof of Theorem 3.1 49
3.4 Bordered determinants 51
3.5 The matrix equation AX = 0 51
3.6 The Hadamard product 52
3.7 The commutation matrix K mn 54
3.8 The duplication matrix D n 56
3.9 Relationship between D n+1 and D n , I 58
3.10 Relationship between D n+1 and D n , II 59
3.11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60
12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B)63
13 The bordered Gramian matrix 65
14 The equations X 1 A + X 2 B ′ = G 1 ,X 1 B = G 2 67
Miscellaneous exercises 69
Bibliographical notes 70
Part Two — Differentials: the theory
4 Mathematical preliminaries 73
4.1 Introduction 73
4.2 Interior points and accumulation points 73
4.3 Open and closed sets 75
4.4 The Bolzano-Weierstrass theorem 77
4.5 Functions 78
4.6 The limit of a function 79
4.7 Continuous functions and compactness 80
4.8 Convex sets 81
4.9 Convex and concave functions 83
Bibliographical notes 86
5 Differentials and differentiability 87
5.1 Introduction 87
5.2 Continuity 88
5.3 Differentiability and linear approximation 90
5.4 The differential of a vector function 91
5.5 Uniqueness of the differential 93
5.6 Continuity of differentiable functions 94
5.7 Partial derivatives 95
5.8 The first identification theorem 96
5.9 Existence of the differential, I 97
5.10 Existence of the differential, II 99
5.11 Continuous differentiability 100
5.12 The chain rule 100
5.13 Cauchy invariance 102
5.14 The mean-value theorem for real-valued functions 103
5.15 Differentiable matrix functions 104
5.16 Some remarks on notation 106
5.17 Complex differentiation 108
Miscellaneous exercises 110
Bibliographical notes 110
6 The second differential 111
6.1 Introduction 111
6.2 Second-order partial derivatives 111
6.3 The Hessian matrix 112
6.4 Twice differentiability and second-order approximation, I 113
6.5 Definition of twice differentiability 114
6.6 The second differential 115
6.7 Symmetry of the Hessian matrix 117
6.8 The second identification theorem 119
6.9 Twice differentiability and second-order approximation, II 119
6.10 Chain rule for Hessian matrices 121
6.11 The analog for second differentials 123
6.12 Taylor’s theorem for real-valued functions 124
6.13 Higher-order differentials 125
6.14 Real analytic functions 125
6.15 Twice differentiable matrix functions 126
Bibliographical notes 127
7 Static optimization 129
7.1 Introduction 129
7.2 Unconstrained optimization 130
7.3 The existence of absolute extrema 131
7.4 Necessary conditions for a local minimum 132
7.5 Sufficient conditions for a local minimum: first-derivative test 134
7.6 Sufficient conditions for a local minimum: second-derivative test 136
7.7 Characterization of differentiable convex functions 138
7.8 Characterization of twice differentiable convex functions 141
7.9 Sufficient conditions for an absolute minimum 142
7.10 Monotonic transformations 143
7.11 Optimization subject to constraints 144
7.12 Necessary conditions for a local minimum under constraints 145
7.13 Sufficient conditions for a local minimum under constraints 149
7.14 Sufficient conditions for an absolute minimum under constraints 154
7.15 A note on constraints in matrix form 155
7.16 Economic interpretation of Lagrange multipliers 155
Appendix: the implicit function theorem 157
Bibliographical notes 159
Part Three — Differentials: the practice 161
8 Some important differentials 163
8.1 Introduction 163
8.2 Fundamental rules of differential calculus 163
8.3 The differential of a determinant 165
8.4 The differential of an inverse 168
8.5 Differential of the Moore-Penrose inverse 169
8.6 The differential of the adjoint matrix 172
8.7 On differentiating eigenvalues and eigenvectors 174
8.8 The continuity of eigenprojections 176
8.9 The differential of eigenvalues and eigenvectors: symmetric case 180
8.10 Two alternative expressions for dλ 183
8.11 Second differential of the eigenvalue function 185
Miscellaneous exercises 186
Bibliographical notes 189
9 First-order differentials and Jacobian matrices 191
9.1 Introduction 191
9.2 Classification 192
9.3 Derisatives 192
9.4 Derivatives 194
9.5 Identification of Jacobian matrices 196
9.6 The first identification table 197
9.7 Partitioning of the derivative 197
9.8 Scalar functions of a scalar 198
9.9 Scalar functions of a vector 198
9.10 Scalar functions of a matrix, I: trace 199
9.11 Scalar functions of a matrix, II: determinant 201
9.12 Scalar functions of a matrix, III: eigenvalue 202
9.13 Two examples of vector functions 203
9.14 Matrix functions 204
9.15 Kronecker products 206
9.16 Some other problems 208
9.17 Jacobians of transformations 209
Bibliographical notes 210
10 Second-order differentials and Hessian matrices 211
10.1 Introduction 211
10.2 The second identification table 211
10.3 Linear and quadratic forms 212
10.4 A useful theorem 213
10.5 The determinant function 214
10.6 The eigenvalue function 215
10.7 Other examples 215
10.8 Composite functions 217
10.9 The eigenvector function 218
10.10 Hessian of matrix functions, I 219
10.11 Hessian of matrix functions, II 219
Miscellaneous exercises 220
Part Four — Inequalities 223
11 Inequalities 225
11.1 Introduction 225
11.2 The Cauchy-Schwarz inequality 226
11.3 Matrix analogs of the Cauchy-Schwarz inequality 227
11.4 The theorem of the arithmetic and geometric means 228
11.5 The Rayleigh quotient 230
11.6 Concavity of λ 1 and convexity of λ n 232
11.7 Variational description of eigenvalues 232
11.8 Fischer’s min-max theorem 234
11.9 Monotonicity of the eigenvalues 236
11.10 The Poincar´ e separation theorem 236
11.11 Two corollaries of Poincar´ e’s theorem 237
11.12 Further consequences of the Poincar´ e theorem 238
11.13 Multiplicative version 239
11.14 The maximum of a bilinear form 241
11.15 Hadamard’s inequality 242
11.16 An interlude: Karamata’s inequality 242
11.17 Karamata’s inequality and eigenvalues 244
11.18 An inequality concerning positive semidefinite matrices 245
11.19 A representation theorem for (Papi) 1/p 246
11.20 A representation theorem for (trA p ) 1/p 247
11.21 Hölder’s inequality 248
11.22 Concavity of log|A| 250
11.23 Minkowski’s inequality 251
11.24 Quasilinear representation of |A| 1/n 253
11.25 Minkowski’s determinant theorem 255
11.26 Weighted means of order p 256
11.27 Schl¨ omilch’s inequality 258
11.28 Curvature properties of M p (x,a) 259
11.29 Least squares 260
11.30 Generalized least squares 261
11.31 Restricted least squares 262
11.32 Restricted least squares: matrix version 264
Miscellaneous exercises 265
Bibliographical notes 269
Part Five — The linear model 271
12 Statistical preliminaries 273
12.1 Introduction 273
12.2 The cumulative distribution function 273
12.3 The joint density function 274
12.4 Expectations 274
12.5 Variance and covariance 275
12.6 Independence of two random variables 277
12.7 Independence of n random variables 279
12.8 Sampling 279
12.9 The one-dimensional normal distribution 279
12.10 The multivariate normal distribution 280
12.11 Estimation 282
Miscellaneous exercises 282
Bibliographical notes 283
13 The linear regression model 285
13.1 Introduction 285
13.2 Affine minimum-trace unbiased estimation 286
13.3 The Gauss-Markov theorem 287
13.4 The method of least squares 290
13.5 Aitken’s theorem 291
13.6 Multicollinearity 293
13.7 Estimable functions 295
13.8 Linear constraints: the case M(R ′ ) ⊂ M(X ′ ) 296
13.9 Linear constraints: the general case 300
13.10 Linear constraints: the case M(R ′ ) ∩ M(X ′ ) = {0} 302
13.11 A singular variance matrix: the case M(X) ⊂ M(V ) 304
13.12 A singular variance matrix: the case r(X'V+ X) = r(X) 305
13.13 A singular variance matrix: the general case, I 307
13.14 Explicit and implicit linear constraints 307
13.15 The general linear model, I 310
13.16 A singular variance matrix: the general case, II 311
13.17 The general linear model, II 314
13.18 Generalized least squares 315
13.19 Restricted least squares 316
Miscellaneous exercises 318
Bibliographical notes 319
14 Further topics in the linear model 321
14.1 Introduction 321
14.2 Best quadratic unbiased estimation of σ 2 322
14.3 The best quadratic and positive unbiased estimator of σ 2 322
14.4 The best quadratic unbiased estimator of σ 2 324
14.5 Best quadratic invariant estimation of σ 2 326
14.6 The best quadratic and positive invariant estimator of σ 2 327
14.7 The best quadratic invariant estimator of σ 2 329
14.8 Best quadratic unbiased estimation: multivariate normal case 330
14.9 Bounds for the bias of the least-squares estimator of σ 2 , I 332
14.10 Bounds for the bias of the least-squares estimator of σ 2 , II 333
14.11 The prediction of disturbances 335
14.12 Best linear unbiased predictors with scalar variance matrix 336
14.13 Best linear unbiased predictors with fixed variance matrix, I 338
14.14 Best linear unbiased predictors with fixed variance matrix, II 340
14.15 Local sensitivity of the posterior mean 341
14.16 Local sensitivity of the posterior precision 342
Bibliographical notes 344
Part Six — Applications to maximum likelihood estimation 345
15 Maximum likelihood estimation 347
15.1 Introduction 347
15.2 The method of maximum likelihood (ML) 347
15.3 ML estimation of the multivariate normal distribution 348
15.4 Symmetry: implicit versus explicit treatment 350
15.5 The treatment of positive definiteness 351
15.6 The information matrix 352
15.7 ML estimation of the multivariate normal distribution:distinct means 354
15.8 The multivariate linear regression model 354
15.9 The errors-in-variables model 357
15.10 The nonlinear regression model with normal errors 359
15.11 Special case: functional independence of mean and variance parameters 361
15.12 Generalization of Theorem 15.6 362
Miscellaneous exercises 364
Bibliographical notes 365
16 Simultaneous equations 367
16.1 Introduction 367
16.2 The simultaneous equations model 367
16.3 The identification problem 369
16.4 Identification with linear constraints on B and Γ only 371
16.5 Identification with linear constraints on B, Γ, and Σ 371
16.6 Nonlinear constraints 373
16.7 FIML: the information matrix (general case) 374
16.8 FIML: asymptotic variance matrix (special case) 376
16.9 LIML: first-order conditions 378
16.10 LIML: information matrix 381
16.11 LIML: asymptotic variance matrix 383
Bibliographical notes 388
17 Topics in psychometrics 389
17.1 Introduction 389
17.2 Population principal components 390
17.3 Optimality of principal components 391
17.4 A related result 392
17.5 Sample principal components 393
17.6 Optimality of sample principal components 395
17.7 One-mode component analysis 395
17.8 One-mode component analysis and sample principal components 398
17.9 Two-mode component analysis 399
17.10 Multimode component analysis 400
17.11 Factor analysis 404
17.12 A zigzag routine 407
17.13 A Newton-Raphson routine 408
17.14 Kaiser’s varimax method 412
17.15 Canonical correlations and variates in the population 414
17.16 Correspondence analysis 417
17.17 Linear discriminant analysis 418
Bibliographical notes 419
Part Seven — Summary 421
18 Matrix calculus: the essentials 423
18.1 Introduction 423
18.2 Differentials 424
18.3 Vector calculus 426
18.4 Optimization 429
18.5 Least squares 431
18.6 Matrix calculus 432
18.7 Interlude on linear and quadratic forms 434
18.8 The second differential 434
18.9 Chain rule for second differentials 436
18.10 Four examples 438
18.11 The Kronecker product and vec operator 439
18.12 Identification 441
18.13 The commutation matrix 442
18.14 From second differential to Hessian 443
18.15 Symmetry and the duplication matrix 444
18.16 Maximum likelihood 445
Further reading 448
Bibliography 449
Index of symbols 467
Subject index 471
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