Matrix Differential Calculus with Applications in Statistics and Econometrics

Name: Matrix Differential Calculus with Applications in Statistics and Econometrics
Brand: Wiley-ISTE
Price: 97.99 EUR
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Jan R. Magnus Heinz Neudecker(Autor*in)

Wiley-ISTE (Verlag)

3. Auflage

Erschienen am 15. März 2019

504 Seiten

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978-1-119-54119-6 (ISBN)

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Inhalt

Cover
Title Page
Copyright
Contents
Preface
Part One - Matrices
Chapter 1 Basic properties of vectors and matrices
1 Introduction
2 Sets
3 Matrices: addition and multiplication
4 The transpose of a matrix
5 Square matrices
6 Linear forms and quadratic forms
7 The rank of a matrix
8 The inverse
9 The determinant
10 The trace
11 Partitioned matrices
12 Complex matrices
13 Eigenvalues and eigenvectors
14 Schur's decomposition theorem
15 The Jordan decomposition
16 The singular-value decomposition
17 Further results concerning eigenvalues
18 Positive (semi)definite matrices
19 Three further results for positive definite matrices
20 A useful result
21 Symmetric matrix functions
Miscellaneous exercises
Bibliographical notes
Chapter 2 Kronecker products, vec operator, and Moore-Penrose inverse
1 Introduction
2 The Kronecker product
3 Eigenvalues of a Kronecker product
4 The vec operator
5 The Moore-Penrose (MP) inverse
6 Existence and uniqueness of the MP inverse
7 Some properties of the MP inverse
8 Further properties
9 The solution of linear equation systems
Miscellaneous exercises
Bibliographical notes
Chapter 3 Miscellaneous matrix results
1 Introduction
2 The adjoint matrix
3 Proof of Theorem 3.1
4 Bordered determinants
5 The matrix equation AX = 0
6 The Hadamard product
7 The commutation matrix Kmn
8 The duplication matrix Dn
9 Relationship between Dn+1 and Dn, I
10 Relationship between Dn+1 and Dn, II
11 Conditions for a quadratic form to be positive (negative) subject to linear constraints
12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B)
13 The bordered Gramian matrix
14 The equations X1A + X2B´ = G1,X1B = G2
Miscellaneous exercises
Bibliographical notes
Part Two - Differentials: the theory
Chapter 4 Mathematical preliminaries
1 Introduction
2 Interior points and accumulation points
3 Open and closed sets
4 The Bolzano-Weierstrass theorem
5 Functions
6 The limit of a function
7 Continuous functions and compactness
8 Convex sets
9 Convex and concave functions
Bibliographical notes
Chapter 5 Differentials and differentiability
1 Introduction
2 Continuity
3 Differentiability and linear approximation
4 The differential of a vector function
5 Uniqueness of the differential
6 Continuity of differentiable functions
7 Partial derivatives
8 The first identification theorem
9 Existence of the differential, I
10 Existence of the differential, II
11 Continuous differentiability
12 The chain rule
13 Cauchy invariance
14 The mean-value theorem for real-valued functions
15 Differentiable matrix functions
16 Some remarks on notation
17 Complex differentiation
Miscellaneous exercises
Bibliographical notes
Chapter 6 The second differential
1 Introduction
2 Second-order partial derivatives
3 The Hessian matrix
4 Twice differentiability and second-order approximation, I
5 Definition of twice differentiability
6 The second differential
7 Symmetry of the Hessian matrix
8 The second identification theorem
9 Twice differentiability and second-order approximation, II
10 Chain rule for Hessian matrices
11 The analog for second differentials
12 Taylor's theorem for real-valued functions
13 Higher-order differentials
14 Real analytic functions
15 Twice differentiable matrix functions
Bibliographical notes
Chapter 7 Static optimization
1 Introduction
2 Unconstrained optimization
3 The existence of absolute extrema
4 Necessary conditions for a local minimum
5 Sufficient conditions for a local minimum: first-derivative test
6 Sufficient conditions for a local minimum: second-derivative test
7 Characterization of differentiable convex functions
8 Characterization of twice differentiable convex functions
9 Sufficient conditions for an absolute minimum
10 Monotonic transformations
11 Optimization subject to constraints
12 Necessary conditions for a local minimum under constraints
13 Sufficient conditions for a local minimum under constraints
14 Sufficient conditions for an absolute minimum under constraints
15 A note on constraints in matrix form
16 Economic interpretation of Lagrange multipliers
Appendix: the implicit function theorem
Bibliographical notes
Part Three - Differentials: the practice
Chapter 8 Some important differentials
1 Introduction
2 Fundamental rules of differential calculus
3 The differential of a determinant
4 The differential of an inverse
5 Differential of the Moore-Penrose inverse
6 The differential of the adjoint matrix
7 On differentiating eigenvalues and eigenvectors
8 The continuity of eigenprojections
9 The differential of eigenvalues and eigenvectors: symmetric case
10 Two alternative expressions for d?
11 Second differential of the eigenvalue function
Miscellaneous exercises
Bibliographical notes
Chapter 9 First-order differentials and Jacobian matrices
1 Introduction
2 Classification
3 Derisatives
4 Derivatives
5 Identification of Jacobian matrices
6 The first identification table
7 Partitioning of the derivative
8 Scalar functions of a scalar
9 Scalar functions of a vector
10 Scalar functions of a matrix, I: trace
11 Scalar functions of a matrix, II: determinant
12 Scalar functions of a matrix, III: eigenvalue
13 Two examples of vector functions
14 Matrix functions
15 Kronecker products
16 Some other problems
17 Jacobians of transformations
Bibliographical notes
Chapter 10 Second-order differentials and Hessian matrices
1 Introduction
2 The second identification table
3 Linear and quadratic forms
4 A useful theorem
5 The determinant function
6 The eigenvalue function
7 Other examples
8 Composite functions
9 The eigenvector function
10 Hessian of matrix functions, I
11 Hessian of matrix functions, II
Miscellaneous exercises
Part Four - Inequalities
Chapter 11 Inequalities
1 Introduction
2 The Cauchy-Schwarz inequality
3 Matrix analogs of the Cauchy-Schwarz inequality
4 The theorem of the arithmetic and geometric means
5 The Rayleigh quotient
6 Concavity of ?1 and convexity of ?n
7 Variational description of eigenvalues
8 Fischer's min-max theorem
9 Monotonicity of the eigenvalues
10 The Poincar´e separation theorem
11 Two corollaries of Poincar´e's theorem
12 Further consequences of the Poincar´e theorem
13 Multiplicative version
14 The maximum of a bilinear form
15 Hadamard's inequality
16 An interlude: Karamata's inequality
17 Karamata's inequality and eigenvalues
18 An inequality concerning positive semidefinite matrices
19 A representation theorem for (Sapi)1/p
20 A representation theorem for (trAp)1/p
21 H¨older's inequality
22 Concavity of log |A|
23 Minkowski's inequality
24 Quasilinear representation of |A|1/n
25 Minkowski's determinant theorem
26 Weighted means of order p
27 Schl¨omilch's inequality
28 Curvature properties of Mp(x, a)
29 Least squares
30 Generalized least squares
31 Restricted least squares
32 Restricted least squares: matrix version
Miscellaneous exercises
Bibliographical notes
Part Five - The linear model
Chapter 12 Statistical preliminaries
1 Introduction
2 The cumulative distribution function
3 The joint density function
4 Expectations
5 Variance and covariance
6 Independence of two random variables
7 Independence of n random variables
8 Sampling
9 The one-dimensional normal distribution
10 The multivariate normal distribution
11 Estimation
Miscellaneous exercises
Bibliographical notes
Chapter 13 The linear regression model
1 Introduction
2 Affine minimum-trace unbiased estimation
3 The Gauss-Markov theorem
4 The method of least squares
5 Aitken's theorem
6 Multicollinearity
7 Estimable functions
8 Linear constraints: the case M(R´) ?M(X´)
9 Linear constraints: the general case
10 Linear constraints: the case M(R´) nM(X´) = {0}
11 A singular variance matrix: the case M(X) ?M(V )
12 A singular variance matrix: the case r(X´V +X) = r(X)
13 A singular variance matrix: the general case, I
14 Explicit and implicit linear constraints
15 The general linear model, I
16 A singular variance matrix: the general case, II
17 The general linear model, II
18 Generalized least squares
19 Restricted least squares
Miscellaneous exercises
Bibliographical notes
Chapter 14 Further topics in the linear model
1 Introduction
2 Best quadratic unbiased estimation of s2
3 The best quadratic and positive unbiased estimator of s2
4 The best quadratic unbiased estimator of s2
5 Best quadratic invariant estimation of s2
6 The best quadratic and positive invariant estimator of s2
7 The best quadratic invariant estimator of s2
8 Best quadratic unbiased estimation: multivariate normal case
9 Bounds for the bias of the least-squares estimator of s2, I
10 Bounds for the bias of the least-squares estimator of s2, II
11 The prediction of disturbances
12 Best linear unbiased predictors with scalar variance matrix
13 Best linear unbiased predictors with fixed variance matrix, I
14 Best linear unbiased predictors with fixed variance matrix, II
15 Local sensitivity of the posterior mean
16 Local sensitivity of the posterior precision
Bibliographical notes
Part Six - Applications to maximum likelihood estimation
Chapter 15 Maximum likelihood estimation
1 Introduction
2 The method of maximum likelihood (ML)
3 ML estimation of the multivariate normal distribution
4 Symmetry: implicit versus explicit treatment
5 The treatment of positive definiteness
6 The information matrix
7 ML estimation of the multivariate normal distribution: distinct means
8 The multivariate linear regression model
9 The errors-in-variables model
10 The nonlinear regression model with normal errors
11 Special case: functional independence of mean and variance parameters
12 Generalization of Theorem 15.6
Miscellaneous exercises
Bibliographical notes
Chapter 16 Simultaneous equations
1 Introduction
2 The simultaneous equations model
3 The identification problem
4 Identification with linear constraints on B and G only
5 Identification with linear constraints on B, G, and S
6 Nonlinear constraints
7 FIML: the information matrix (general case)
8 FIML: asymptotic variance matrix (special case)
9 LIML: first-order conditions
10 LIML: information matrix
11 LIML: asymptotic variance matrix
Bibliographical notes
Chapter 17 Topics in psychometrics
1 Introduction
2 Population principal components
3 Optimality of principal components
4 A related result
5 Sample principal components
6 Optimality of sample principal components
7 One-mode component analysis
8 One-mode component analysis and sample principal components
9 Two-mode component analysis
10 Multimode component analysis
11 Factor analysis
12 A zigzag routine
13 A Newton-Raphson routine
14 Kaiser's varimax method
15 Canonical correlations and variates in the population
16 Correspondence analysis
17 Linear discriminant analysis
Bibliographical notes
Part Seven - Summary
Chapter 18 Matrix calculus: the essentials
1 Introduction
2 Differentials
3 Vector calculus
4 Optimization
5 Least squares
6 Matrix calculus
7 Interlude on linear and quadratic forms
8 The second differential
9 Chain rule for second differentials
10 Four examples
11 The Kronecker product and vec operator
12 Identification
13 The commutation matrix
14 From second differential to Hessian
15 Symmetry and the duplication matrix
16 Maximum likelihood
Further reading
Bibliography
Index of symbols
Subject index
EULA

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