Matrix differential calculus with applications in statistics and econometrics

A brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics This exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how...

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Detalles Bibliográficos
Otros Autores:	Neudecker, Heinz, author (author)
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	Hoboken, N.J.: Wiley 2019. Hoboken, NJ : 2019.
Edición:	Third edition
Colección:	Wiley series in probability and statistics.
Materias:	Matrices. Differential calculus. Statistics. Econometrics.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009630609706719

Tabla de Contenidos:

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 dλ
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 λ1 and convexity of λn
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 (Σapi)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′) ∩M(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 σ2
3 The best quadratic and positive unbiased estimator of σ2
4 The best quadratic unbiased estimator of σ2
5 Best quadratic invariant estimation of σ2
6 The best quadratic and positive invariant estimator of σ2
7 The best quadratic invariant estimator of σ2
8 Best quadratic unbiased estimation: multivariate normal case
9 Bounds for the bias of the least-squares estimator of σ2, I
10 Bounds for the bias of the least-squares estimator of σ2, II.
11 The prediction of disturbances.

Matrix differential calculus with applications in statistics and econometrics

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