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Matrix Differential Calculus with Applications in Statistics and Econometrics


Matrix Differential Calculus with Applications in Statistics and Econometrics


Wiley Series in Probability and Statistics 3. Aufl.

von: Jan R. Magnus, Heinz Neudecker

97,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 15.03.2019
ISBN/EAN: 9781119541165
Sprache: englisch
Anzahl Seiten: 504

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Beschreibungen

<p><b>A brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics</b></p> <p>This exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how easy it is to use this theory once you have mastered the technique. Jan Magnus, who, along with the late Heinz Neudecker, pioneered the theory, develops it further in this new edition and provides many examples along the way to support it.</p> <p>Matrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioral sciences to econometrics. It is still relevant and used today in a wide range of subjects such as the biosciences and psychology. <i>Matrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition </i>contains all of the essentials of multivariable calculus with an emphasis on the use of differentials. It starts by presenting a concise, yet thorough overview of matrix algebra, then goes on to develop the theory of differentials. The rest of the text combines the theory and application of matrix differential calculus, providing the practitioner and researcher with both a quick review and a detailed reference.</p> <ul> <li>Fulfills the need for an updated and unified treatment of matrix differential calculus</li> <li>Contains many new examples and exercises based on questions asked of the author over the years</li> <li>Covers new developments in field and features new applications</li> <li>Written by a leading expert and pioneer of the theory</li> <li>Part of the Wiley Series in Probability and Statistics </li> </ul> <p><i>Matrix Differential Calculus With Applications in Statistics and Econometrics Third Edition </i>is an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.</p>
<p>Preface xiii</p> <p><b>Part One — Matrices</b></p> <p><b>1 Basic properties of vectors and matrices 3</b></p> <p>1 Introduction 3</p> <p>2 Sets 3</p> <p>3 Matrices: addition and multiplication 4</p> <p>4 The transpose of a matrix 6</p> <p>5 Square matrices 6</p> <p>6 Linear forms and quadratic forms 7</p> <p>7 The rank of a matrix 9</p> <p>8 The inverse 10</p> <p>9 The determinant 10</p> <p>10 The trace 11</p> <p>11 Partitioned matrices 12</p> <p>12 Complex matrices 14</p> <p>13 Eigenvalues and eigenvectors 14</p> <p>14 Schur’s decomposition theorem 17</p> <p>15 The Jordan decomposition 18</p> <p>16 The singular-value decomposition 20</p> <p>17 Further results concerning eigenvalues 20</p> <p>18 Positive (semi)definite matrices 23</p> <p>19 Three further results for positive definite matrices 25</p> <p>20 A useful result 26</p> <p>21 Symmetric matrix functions 27</p> <p>Miscellaneous exercises<i> </i>28</p> <p>Bibliographical notes<i> </i>30</p> <p><b>2 Kronecker products, vec operator, and Moore-Penrose inverse 31</b></p> <p>1 Introduction 31</p> <p>2 The Kronecker product 31</p> <p>3 Eigenvalues of a Kronecker product 33</p> <p>4 The vec operator 34</p> <p>5 The Moore-Penrose (MP) inverse 36</p> <p>6 Existence and uniqueness of the MP inverse 37</p> <p>7 Some properties of the MP inverse 38</p> <p>8 Further properties 39</p> <p>9 The solution of linear equation systems 41</p> <p>Miscellaneous exercises<i> </i>43</p> <p>Bibliographical notes<i> </i>45</p> <p><b>3 Miscellaneous matrix results 47</b></p> <p>1 Introduction 47</p> <p>2 The adjoint matrix 47</p> <p>3 Proof of Theorem 3.1 49</p> <p>4 Bordered determinants 51</p> <p>5 The matrix equation<i> AX</i> = 0 51</p> <p>6 The Hadamard product 52</p> <p>7 The commutation matrix<i> K<sub>mn</sub> </i>54</p> <p>8 The duplication matrix<i> D<sub>n</sub></i> 56</p> <p>9 Relationship between <i>D<sub>n+1 </sub></i>and <i>D<sub>n</sub></i>, I 58</p> <p>10 Relationship between <i>D<sub>n+1 </sub></i>and <i>D<sub>n</sub></i>, II 59</p> <p>11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60</p> <p>12 Necessary and sufficient conditions for<i> r(A : B) = r(A) + r(B) </i>63</p> <p>13 The bordered Gramian matrix 65</p> <p>14 The equations<i> X<sub>1</sub>A + X<sub>2</sub>B′ = G<sub>1</sub>,X<sub>1</sub>B = G<sub>2</sub> </i>67</p> <p>Miscellaneous exercises<i> </i>69</p> <p>Bibliographical notes<i> </i>70</p> <p><b>Part Two — Differentials: the theory</b></p> <p><b>4 Mathematical preliminaries 73</b></p> <p>1 Introduction 73</p> <p>2 Interior points and accumulation points 73</p> <p>3 Open and closed sets 75</p> <p>4 The Bolzano-Weierstrass theorem 77</p> <p>5 Functions 78</p> <p>6 The limit of a function 79</p> <p>7 Continuous functions and compactness 80</p> <p>8 Convex sets 81</p> <p>9 Convex and concave functions 83</p> <p>Bibliographical notes<i> </i>86</p> <p><b>5 Differentials and differentiability 87</b></p> <p>1 Introduction 87</p> <p>2 Continuity 88</p> <p>3 Differentiability and linear approximation 90</p> <p>4 The differential of a vector function 91</p> <p>5 Uniqueness of the differential 93</p> <p>6 Continuity of differentiable functions 94</p> <p>7 Partial derivatives 95</p> <p>8 The first identification theorem 96</p> <p>9 Existence of the differential, I 97</p> <p>10 Existence of the differential, II 99</p> <p>11 Continuous differentiability 100</p> <p>12 The chain rule 100</p> <p>13 Cauchy invariance 102</p> <p>14 The mean-value theorem for real-valued functions 103</p> <p>15 Differentiable matrix functions 104</p> <p>16 Some remarks on notation 106</p> <p>17 Complex differentiation 108</p> <p>Miscellaneous exercises<i> </i>110</p> <p>Bibliographical notes<i> </i>110</p> <p><b>6 The second differential 111</b></p> <p>1 Introduction 111</p> <p>2 Second-order partial derivatives 111</p> <p>3 The Hessian matrix 112</p> <p>4 Twice differentiability and second-order approximation, I 113</p> <p>5 Definition of twice differentiability 114</p> <p>6 The second differential 115</p> <p>7 Symmetry of the Hessian matrix 117</p> <p>8 The second identification theorem 119</p> <p>9 Twice differentiability and second-order approximation, II 119</p> <p>10 Chain rule for Hessian matrices 121</p> <p>11 The analog for second differentials 123</p> <p>12 Taylor’s theorem for real-valued functions 124</p> <p>13 Higher-order differentials 125</p> <p>14 Real analytic functions 125</p> <p>15 Twice differentiable matrix functions 126</p> <p>Bibliographical notes<i> </i>127</p> <p><b>7 Static optimization 129</b></p> <p>1 Introduction 129</p> <p>2 Unconstrained optimization 130</p> <p>3 The existence of absolute extrema 131</p> <p>4 Necessary conditions for a local minimum 132</p> <p>5 Sufficient conditions for a local minimum: first-derivative test 134</p> <p>6 Sufficient conditions for a local minimum: second-derivative test 136</p> <p>7 Characterization of differentiable convex functions 138</p> <p>8 Characterization of twice differentiable convex functions 141</p> <p>9 Sufficient conditions for an absolute minimum 142</p> <p>10 Monotonic transformations 143</p> <p>11 Optimization subject to constraints 144</p> <p>12 Necessary conditions for a local minimum under constraints 145</p> <p>13 Sufficient conditions for a local minimum under constraints 149</p> <p>14 Sufficient conditions for an absolute minimum under constraints 154</p> <p>15 A note on constraints in matrix form 155</p> <p>16 Economic interpretation of Lagrange multipliers 155</p> <p>Appendix: the implicit function theorem 157</p> <p>Bibliographical notes<i> </i>159</p> <p><b>Part Three — Differentials: the practice</b></p> <p><b>8 Some important differentials 163</b></p> <p>1 Introduction 163</p> <p>2 Fundamental rules of differential calculus 163</p> <p>3 The differential of a determinant 165</p> <p>4 The differential of an inverse 168</p> <p>5 Differential of the Moore-Penrose inverse 169</p> <p>6 The differential of the adjoint matrix 172</p> <p>7 On differentiating eigenvalues and eigenvectors 174</p> <p>8 The continuity of eigenprojections 176</p> <p>9 The differential of eigenvalues and eigenvectors: symmetric case 180</p> <p>10 Two alternative expressions for dλ 183</p> <p>11 Second differential of the eigenvalue function 185</p> <p>Miscellaneous exercises<i> </i>186</p> <p>Bibliographical notes<i> </i>189</p> <p><b>9 First-order differentials and Jacobian matrices 191</b></p> <p>1 Introduction 191</p> <p>2 Classification 192</p> <p>3 Derisatives 192</p> <p>4 Derivatives 194</p> <p>5 Identification of Jacobian matrices 196</p> <p>6 The first identification table 197</p> <p>7 Partitioning of the derivative 197</p> <p>8 Scalar functions of a scalar 198</p> <p>9 Scalar functions of a vector 198</p> <p>10 Scalar functions of a matrix, I: trace 199</p> <p>11 Scalar functions of a matrix, II: determinant 201</p> <p>12 Scalar functions of a matrix, III: eigenvalue 202</p> <p>13 Two examples of vector functions 203</p> <p>14 Matrix functions 204</p> <p>15 Kronecker products 206</p> <p>16 Some other problems 208</p> <p>17 Jacobians of transformations 209</p> <p>Bibliographical notes<i> </i>210</p> <p><b>10 Second-order differentials and Hessian matrices 211</b></p> <p>1 Introduction 211</p> <p>2 The second identification table 211</p> <p>3 Linear and quadratic forms 212</p> <p>4 A useful theorem 213</p> <p>5 The determinant function 214</p> <p>6 The eigenvalue function 215</p> <p>7 Other examples 215</p> <p>8 Composite functions 217</p> <p>9 The eigenvector function 218</p> <p>10 Hessian of matrix functions, I 219</p> <p>11 Hessian of matrix functions, II 219</p> <p>Miscellaneous exercises<i> </i>220</p> <p><b>Part Four — Inequalities</b></p> <p><b>11 Inequalities 225</b></p> <p>1 Introduction 225</p> <p>2 The Cauchy-Schwarz inequality 226</p> <p>3 Matrix analogs of the Cauchy-Schwarz inequality 227</p> <p>4 The theorem of the arithmetic and geometric means 228</p> <p>5 The Rayleigh quotient 230</p> <p>6 Concavity of λ<i><sub>1 </sub></i>and convexity of λ<i><sub>n </sub></i>232</p> <p>7 Variational description of eigenvalues 232</p> <p>8 Fischer’s min-max theorem 234</p> <p>9 Monotonicity of the eigenvalues 236</p> <p>10 The Poincar´e separation theorem 236</p> <p>11 Two corollaries of Poincar´e’s theorem 237</p> <p>12 Further consequences of the Poincar´e theorem 238</p> <p>13 Multiplicative version 239</p> <p>14 The maximum of a bilinear form 241</p> <p>15 Hadamard’s inequality 242</p> <p>16 An interlude: Karamata’s inequality 242</p> <p>17 Karamata’s inequality and eigenvalues 244</p> <p>18 An inequality concerning positive semidefinite matrices 245</p> <p>19 A representation theorem for ( ∑a<sup>p</sup><sub>i</sub> )<sup>1/p</sup> 246</p> <p>20 A representation theorem for (tr<i>A</i><sup>p</sup>)<sup>1/p </sup>247</p> <p>21 Hölder’s inequality 248</p> <p>22 Concavity of log|A| 250</p> <p>23 Minkowski’s inequality 251</p> <p>24 Quasilinear representation of |A|<sup>1/n</sup> 253</p> <p>25 Minkowski’s determinant theorem 255</p> <p>26 Weighted means of order<i> p</i> 256</p> <p>27 Schlömilch’s inequality 258</p> <p>28 Curvature properties of<i> M<sub>p</sub></i>(<i>x</i>,<i> a</i>) 259</p> <p>29 Least squares 260</p> <p>30 Generalized least squares 261</p> <p>31 Restricted least squares 262</p> <p>32 Restricted least squares: matrix version 264</p> <p>Miscellaneous exercises<i> </i>265</p> <p>Bibliographical notes<i> </i>269</p> <p><b>Part Five — The linear model</b></p> <p><b>12 Statistical preliminaries 273</b></p> <p>1 Introduction 273</p> <p>2 The cumulative distribution function 273</p> <p>3 The joint density function 274</p> <p>4 Expectations 274</p> <p>5 Variance and covariance 275</p> <p>6 Independence of two random variables 277</p> <p>7 Independence of n random variables 279</p> <p>8 Sampling 279</p> <p>9 The one-dimensional normal distribution 279</p> <p>10 The multivariate normal distribution 280</p> <p>11 Estimation 282</p> <p>Miscellaneous exercises<i> </i>282</p> <p>Bibliographical notes<i> </i>283</p> <p><b>13 The linear regression model 285</b></p> <p>1 Introduction 285</p> <p>2 Affine minimum-trace unbiased estimation 286</p> <p>3 The Gauss-Markov theorem 287</p> <p>4 The method of least squares 290</p> <p>5 Aitken’s theorem 291</p> <p>6 Multicollinearity 293</p> <p>7 Estimable functions 295</p> <p>8 Linear constraints: the case<i> M(R′) </i><i>⊂M(X</i><i>′) </i>296</p> <p>9 Linear constraints: the general case 300</p> <p>10 Linear constraints: the case<i> M(R′) ∩M(X′) </i>= {0} 302</p> <p>11 A singular variance matrix: the case<i> M(X) </i><i>⊂M(V ) </i>304</p> <p>12 A singular variance matrix: the case<i> r(X′V +X) = r(X) </i>305</p> <p>13 A singular variance matrix: the general case, I 307</p> <p>14 Explicit and implicit linear constraints 307</p> <p>15 The general linear model, I 310</p> <p>16 A singular variance matrix: the general case, II 311</p> <p>17 The general linear model, II 314</p> <p>18 Generalized least squares 315</p> <p>19 Restricted least squares 316</p> <p>Miscellaneous exercises<i> </i>318</p> <p>Bibliographical notes<i> </i>319</p> <p><b>14 Further topics in the linear model 321</b></p> <p>1 Introduction 321</p> <p>2 Best quadratic unbiased estimation of σ2 322</p> <p>3 The best quadratic and positive unbiased estimator of σ2 322</p> <p>4 The best quadratic unbiased estimator of σ2 324</p> <p>5 Best quadratic invariant estimation of σ2 326</p> <p>6 The best quadratic and positive invariant estimator of σ2 327</p> <p>7 The best quadratic invariant estimator of σ2 329</p> <p>8 Best quadratic unbiased estimation: multivariate normal case 330</p> <p>9 Bounds for the bias of the least-squares estimator of σ<sup>2</sup>, I 332</p> <p>10 Bounds for the bias of the least-squares estimator of σ<sup>2</sup>, II 333</p> <p>11 The prediction of disturbances 335</p> <p>12 Best linear unbiased predictors with scalar variance matrix 336</p> <p>13 Best linear unbiased predictors with fixed variance matrix, I 338</p> <p>14 Best linear unbiased predictors with fixed variance matrix, II 340</p> <p>15 Local sensitivity of the posterior mean 341</p> <p>16 Local sensitivity of the posterior precision 342</p> <p>Bibliographical notes<i> </i>344</p> <p><b>Part Six — Applications to maximum likelihood estimation</b></p> <p><b>15 Maximum likelihood estimation 347</b></p> <p>1 Introduction 347</p> <p>2 The method of maximum likelihood (ML) 347</p> <p>3 ML estimation of the multivariate normal distribution 348</p> <p>4 Symmetry: implicit versus explicit treatment 350</p> <p>5 The treatment of positive definiteness 351</p> <p>6 The information matrix 352</p> <p>7 ML estimation of the multivariate normal distribution: distinct means 354</p> <p>8 The multivariate linear regression model 354</p> <p>9 The errors-in-variables model 357</p> <p>10 The nonlinear regression model with normal errors 359</p> <p>11 Special case: functional independence of mean and variance parameters 361</p> <p>12 Generalization of Theorem 15.6 362</p> <p>Miscellaneous exercises<i> </i>364</p> <p>Bibliographical notes<i> </i>365</p> <p><b>16 Simultaneous equations 367</b></p> <p>1 Introduction 367</p> <p>2 The simultaneous equations model 367</p> <p>3 The identification problem 369</p> <p>4 Identification with linear constraints on <i>B </i>and Γ only 371</p> <p>5 Identification with linear constraints on <i>B</i>, Γ, and ∑ 371</p> <p>6 Nonlinear constraints 373</p> <p>7 FIML: the information matrix (general case) 374</p> <p>8 FIML: asymptotic variance matrix (special case) 376</p> <p>9 LIML: first-order conditions 378</p> <p>10 LIML: information matrix 381</p> <p>11 LIML: asymptotic variance matrix 383</p> <p>Bibliographical notes<i> </i>388</p> <p><b>17 Topics in psychometrics 389</b></p> <p>1 Introduction 389</p> <p>2 Population principal components 390</p> <p>3 Optimality of principal components 391</p> <p>4 A related result 392</p> <p>5 Sample principal components 393</p> <p>6 Optimality of sample principal components 395</p> <p>7 One-mode component analysis 395</p> <p>8 One-mode component analysis and sample principal components 398</p> <p>9 Two-mode component analysis 399</p> <p>10 Multimode component analysis 400</p> <p>11 Factor analysis 404</p> <p>12 A zigzag routine 407</p> <p>13 A Newton-Raphson routine 408</p> <p>14 Kaiser’s varimax method 412</p> <p>15 Canonical correlations and variates in the population 414</p> <p>16 Correspondence analysis 417</p> <p>17 Linear discriminant analysis 418</p> <p>Bibliographical notes<i> </i>419</p> <p><b>Part Seven — Summary</b></p> <p><b>18 Matrix calculus: the essentials 423</b></p> <p>1 Introduction 423</p> <p>2 Differentials 424</p> <p>3 Vector calculus 426</p> <p>4 Optimization 429</p> <p>5 Least squares 431</p> <p>6 Matrix calculus 432</p> <p>7 Interlude on linear and quadratic forms 434</p> <p>8 The second differential 434</p> <p>9 Chain rule for second differentials 436</p> <p>10 Four examples 438</p> <p>11 The Kronecker product and vec operator 439</p> <p>12 Identification 441</p> <p>13 The commutation matrix 442</p> <p>14 From second differential to Hessian 443</p> <p>15 Symmetry and the duplication matrix 444</p> <p>16 Maximum likelihood 445</p> <p>Further reading 448</p> <p>Bibliography 449</p> <p>Index of symbols 467</p> <p>Subject index 471</p>
<p><b>JAN R. MAGNUS</b> is Emeritus Professor at the Department of Econometrics & Operations Research, Tilburg University, and Extraordinary Professor at the Department of Econometrics & Operations Research, Vrije University, Amsterdam. He is research fellow of CentER and the Tinbergen Institute. He has co-authored nine books and is the author of over 100 scientific papers. <p><b>HEINZ NEUDECKER</b> (1933-2017) was Professor of Econometrics at the University of Amsterdam from 1972 until his retirement in 1998.
<p><b>A BRAND NEW, FULLY UPDATED EDITION OF A POPULAR CLASSIC ON MATRIX DIFFERENTIAL CALCULUS WITH APPLICATIONS IN STATISTICS AND ECONOMETRICS</b> <p>This exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how easy it is to use this theory once you have mastered the technique. Jan Magnus, who, along with the late Heinz Neudecker, pioneered the theory, develops it further in this new edition and provides many examples along the way to support it. <p>Matrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioral sciences to econometrics. It is still relevant and used today in a wide range of subjects such as the biosciences and psychology. <i>Matrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition</i> contains all of the essentials of multivariable calculus with an emphasis on the use of differentials. It starts by presenting a concise, yet thorough overview of matrix algebra, then goes on to develop the theory of differentials. The rest of the text combines the theory and application of matrix differential calculus, providing the practitioner and researcher with both a quick review and a detailed reference. <ul> <li>Fulfills the need for an updated and unified treatment of matrix differential calculus</li> <li>Contains many new examples and exercises based on questions asked of the author over the years</li> <li>Covers new developments in field and features new applications</li> <li>Written by a leading expert and pioneer of the theory</li> <li>Part of the Wiley Series in Probability and Statistics</li> </ul> <p><i>Matrix Differential Calculus With Applications in Statistics and Econometrics, Third Edition</i> is an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.

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