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E-raamat: Multivariate Statistical Methods: Going Beyond the Linear

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Multivariate Statistical Methods: Going Beyond the LinearSuurem pilt
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This book presents a general method for deriving higher-order statistics of multivariate distributions with simple algorithms that allow for actual calculations. Multivariate nonlinear statistical models require the study of higher-order moments and cumulants. The main tool used for the definitions is the tensor derivative, leading to several useful expressions concerning Hermite polynomials, moments, cumulants, skewness, and kurtosis. A general test of multivariate skewness and kurtosis is obtained from this treatment. Exercises are provided for each chapter to help the readers understand the methods. Lastly, the book includes a comprehensive list of references, equipping readers to explore further on their own.


Arvustused

The book under review is a very well-written monograph, which gives an up-to-date, self-contained, and thorough analysis of the cumulants and related statistical measures like the skewness and kurtosis for non-Gaussian multivariate distributions. From my point of view, the author has written an interesting book, which could be a reference book for researchers interested in multivariate analysis as well as text for advanced graduate-level courses. (Apostolos Batsidis, zbMATH 1512.62005, 2023)

1 Some Introductory Algebra
1(60)
1.1 Permutations
1(4)
1.2 Tensor Product, vec Operator, and Commutation
5(8)
1.2.1 Tensor Product
5(1)
1.2.2 The vec Operator
6(1)
1.2.3 Commutation Matrices
7(3)
1.2.4 Commuting T-Products of Vectors
10(3)
1.3 Symmetrization and Multilinear Algebra
13(13)
1.3.1 Symmetrization
13(3)
1.3.2 Multi-Indexing, Elimination, and Duplication
16(10)
1.4 Partitions and Diagrams
26(23)
1.4.1 Generating all Partitions
29(1)
1.4.2 The Number of All Partitions
30(4)
1.4.3 Canonical Partitions
34(1)
1.4.4 Partitions and Permutations
35(2)
1.4.5 Partitions with Lattice Structure
37(1)
1.4.6 Indecomposable Partitions
38(2)
1.4.7 Alternative Ways of Checking Indecomposability
40(3)
1.4.8 Diagrams
43(6)
1.5 Appendix
49(3)
1.5.1 Proof of Lemma 1.1
49(1)
1.5.2 Proof of Lemma 1.3
50(1)
1.5.3 Proof of Lemma 1.5
50(1)
1.5.4 Star Product
51(1)
1.6 Exercises
52(7)
1.7 Bibliographic Notes
59(2)
2 The Tensor Derivative of Vector Functions
61(46)
2.1 Derivatives of Composite Functions
61(9)
2.1.1 Faa di Bruno's Formula
63(4)
2.1.2 Mixed Higher-Order Derivatives
67(3)
2.2 T-derivative
70(20)
2.2.1 Differentials and Derivatives
70(2)
2.2.2 The Operator of T-derivative
72(2)
2.2.3 Basic Rules
74(3)
2.2.4 T-derivative of T-products
77(11)
2.2.5 Taylor Series Expansion
88(2)
2.3 Multi-Variable Faa di Bruno's Formula
90(8)
2.4 Appendix
98(3)
2.4.1 Proof of Faa di Bruno's Lemma
98(1)
2.4.2 Proof of Faa di Bruno's T-formula
99(1)
2.4.3 Moment Commutators
100(1)
2.5 Exercises
101(5)
2.6 Bibliographic Notes
106(1)
3 T-Moments and T-Cumulants
107(76)
3.1 Multiple Moments
107(3)
3.2 Tensor Moments
110(8)
3.3 Cumulants for Multiple Variables
118(8)
3.3.1 Definition of Cumulants
118(2)
3.3.2 Definition of T-cumulants
120(4)
3.3.3 Basic Properties
124(2)
3.4 Expressions between Moments and Cumulants
126(23)
3.4.1 Expression for Cumulants via Moments
126(12)
3.4.2 Expressions for Moments via Cumulants
138(5)
3.4.3 Expression of the Cumulant of Products via Products of Cumulants
143(6)
3.5 Additional Matters
149(18)
3.5.1 Expressions of Moments and Cumulants via Preceding Moments and Cumulants
149(2)
3.5.2 Cumulants and Fourier Transform
151(3)
3.5.3 Conditional Cumulants
154(7)
3.5.4 Cumulants of the Log-likelihood Function
161(6)
3.6 Appendix
167(7)
3.6.1 Proof of Lemma 3.6 and Theorem 3.7
167(3)
3.6.2 A Hint for Proof of Lemma 3.8
170(2)
3.6.3 Proof of Lemma 3.2
172(1)
3.6.4 Proof of Lemma 3.5
173(1)
3.7 Exercises
174(6)
3.8 Bibliographic Notes
180(3)
4 Gaussian Systems, T-Hermite Polynomials, Moments, and Cumulants
183(58)
4.1 Hermite Polynomials in One Variable
183(2)
4.2 Hermite Polynomials of Several Variables
185(5)
4.3 Moments and Cumulants for Gaussian Systems
190(7)
4.3.1 Moments of Gaussian Systems and Hermite Polynomials
190(3)
4.3.2 Cumulants for Product of Gaussian Variates and Hermite Polynomials
193(4)
4.4 Products of Hermite Polynomials, Linearization
197(5)
4.5 T-Hermite Polynomials
202(14)
4.6 Moments, Cumulants, and Linearization
216(10)
4.6.1 Cumulants for T-Hermite Polynomials
220(3)
4.6.2 Products for T-Hermite Polynomials
223(3)
4.7 Gram-Charlier Expansion
226(6)
4.8 Appendix
232(2)
4.8.1 Proof of Theorem 4.2
232(1)
4.8.2 Proof of (4.79)
233(1)
4.9 Exercises
234(4)
4.10 Bibliographic Notes
238(3)
5 Multivariate Skew Distributions
241(72)
5.1 The Multivariate Skew-Normal Distribution
241(11)
5.1.1 The Inverse Mill's Ratio and the Central Folded Normal Distribution
242(2)
5.1.2 Skew-Normal Random Variates
244(3)
5.1.3 Canonical Fundamental Skew-Normal (CFUSN) Distribution
247(5)
5.2 Elliptically Symmetric and Skew-Spherical Distributions
252(23)
5.2.1 Elliptically Contoured Distributions
253(8)
5.2.2 Multivariate Moments and Cumulants
261(5)
5.2.3 Canonical Fundamental Skew-Spherical Distribution
266(9)
5.3 Multivariate Skew-t Distribution
275(10)
5.3.1 Multivariate t-Distribution
275(2)
5.3.2 Skew-t Distribution
277(1)
5.3.3 Higher-Order Cumulants of Skew-t Distributions
278(7)
5.4 Scale Mixtures of Skew-Normal Distribution
285(2)
5.5 Multivariate Skew-Normal-Cauchy Distribution
287(7)
5.5.1 Moments of h(|Z|)
292(2)
5.6 Multivariate Laplace
294(3)
5.7 Appendix
297(10)
5.7.1 Spherically Symmetric Distribution
297(6)
5.7.2 T-Derivative of an Inner Product
303(1)
5.7.3 Proof of (5.44)
304(2)
5.7.4 Proof of Lemma 5.6
306(1)
5.8 Exercises
307(3)
5.9 Bibliographic Notes
310(3)
6 Multivariate Skewness and Kurtosis
313(38)
6.1 Multivariate Skewness of Random Vectors
313(8)
6.2 Multivariate Kurtosis of Random Vectors
321(6)
6.3 Indices Based on Distinct Elements of Cumulant Vectors
327(1)
6.4 Testing Multivariate Skewness
328(9)
6.4.1 Estimation of Skewness
329(4)
6.4.2 Testing Zero Skewness
333(4)
6.5 Testing Multivariate Kurtosis
337(6)
6.5.1 Estimation of Kurtosis
338(3)
6.5.2 Testing Zero Kurtosis
341(2)
6.6 A Simulation Study
343(3)
6.7 Appendix
346(1)
6.7.1 Estimated Hermite Polynomials
346(1)
6.8 Exercises
347(1)
6.9 Bibliographic Notes
348(3)
A Formulae
351(30)
A.1 Bell Polynomials
351(2)
A.1.1 Incomplete (Partial) Bell Polynomials
351(1)
A.1.2 Bell Polynomials
352(1)
A.2 Commutators
353(6)
A.2.1 Moment Commutators
353(3)
A.2.2 Commutators Connected to T-Hermite Polynomials
356(3)
A.3 Derivatives of Composite Functions
359(2)
A.4 Moments, Cumulants
361(2)
A.4.1 T-Moments, T-Cumulants
361(2)
A.5 Hermite Polynomials
363(5)
A.5.1 Product of Hermite Polynomials
363(2)
A.5.2 T-Hermite Polynomials
365(3)
A.6 Function G
368(8)
A.6.1 Moments, Cumulants for Skew-t Generator R
370(4)
A.6.2 Moments of Beta Powers
374(2)
A.7 Complementary Error Function
376(2)
A.8 Derivatives of i-Mill's Ratio
378(3)
Notations 381(4)
Solutions 385(24)
References 409(8)
Index 417
György Terdik received his PhD in 1982 at the Department of Probability Theory, State University of Leningrad, USSR. He has been a full-time professor at the Faculty of Informatics, University of Debrecen, Hungary since 2008. He has spent 10 semesters visiting different universities in the US including UC Berkeley and UC Santa Barbara, and the Case Western Reserve University, among others.





His research interests include multivariate nonlinear statistics, time series analysis, modelling high speed communication networks, bilinear and multi-fractal models, directional statistics, and spherical processes, spatial dependence and interaction between space and time.
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