Tensor methods in statistics

302 Pages·1995·16.96 MB·English

by  McCullagh

MONOGRAPHS ON
STATISTICS AND APPLIED PROBABILITY
General Editors
D.R. Cox, D.V. Hinkley, N. Reid, D.B. Rubin
and B.W. Silverman
1 Stochastic Population Models in Ecology and Epidemiology
M.S. Bartlett (1960)
2 Queues D.R. Cox and W.L. Smith (1961)
3 Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964)
4 The Statistical Analysis of Series of Events D.R. Cox and
P.A.W. Lewis (1966)
5 Population Genetics W.J. Ewens (1969)
6 Probability, Statistics and Time M.S. Bartlett (1975)
7 Statistical Inference S.D. Silvey (1975)
8 The Analysis of Contingency Tables B.S. Everitt (1977)
9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977)
10 Stochastic Abundance Models S. Engen (1978)
11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1978)
12 Point Processes D.R. Cox and V. Isham (1980)
13 Identification of Outliers D.M. Hawkins (1980)
14 Optimal Design S.D. Silvey (1980)
15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981)
16 Classification A.D. Gordon (1981)
17 Distribution-free Statistical Methods J.S. Maritz (1981)
18 Residuals and Influence in Regression R.D. Cook
and S. Weisberg (1982)
19 Applications of Queueing Theory G.F. Newell (1982)
20 Risk Theory, 3rd edition R.E. Beard, T. Pentikainen and
E. Pesonen (1984)
21 Analysis of Survival Data D.R. Cox and D. Oakes (1984)
22 An Introduction to Latent Variable Models B.S. Everitt (1984)
23 Bandit Problems D A. Berry and B. Fristedt (1985)
24 Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985)
25 The Statistical Analysis of Compositional Data J. Aitchison (1986)
26 Density Estimation for Statistical and Data Analysis
B.W. Silverman (1986)
27 Regression Analysis with Applications G.B. Wetherill (1986)
28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill (1986)
29 Tensor Methods in Statistics P. McCullagh (1987)
30 Transformation and Weighting in Regression R.J. Carroll and
D. Ruppert (1988)
31 Asymptotic Techniques for use in Statistics O.E. Bamdorff-Nielsen
and D.R. Cox (1989)
32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989)
33 Analysis of Infectious Disease Data N.G. Becker (1989)
34 Design and Analysis of Cross-Over Trials B. Jones and
M.G. Kenward (1989)
35 Empirical Bayes Method, 2nd edition J.S. Maritz and T. Lwin (1989)
36 Symmetric Multivariate and Related Distributions K.-TFang,
S. Kotz and K. Ng (1989)
37 Generalized Linear Models, 2nd edition P. McCullagh and
J.A. Nelder (1989)
38 Cyclic Designs J.A. John (1987)
39 Analog Estimation Methods in Econometrics C.F. Manski (1988)
40 Subset Selection in Regression A.J. Miller (1990)
41 Analysis of Repeated Measures M. Crowder and D.J. Hand (1990)
42 Statistical Reasoning with Imprecise Probabilities P. Walley (1990)
43 Generalized Additive Models T.J. Hastie and R.J. Tibshirani (1990)
44 Inspection Errors for Attributes in Quality Control
N.L. Johnson, S. Kotz and X. Wu (1991)
45 The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992)
46 The Analysis of Quantal Response Data B.J.T. Morgan (1992)
47 Longitudinal Data with Serial Correlation: A State-space Approach
R.H. Jones (1993)
48 Differential Geometry and Statistics Michael K. Murray
and John W. Rice (1993)
49 Markov Models and Optimization M.H.A. Davis (1993)
50 Networks and Chaos - Statistical and Probalistic Aspects
O.E. Bamdorff-Nielson,
J.H. Jensen and W.S. Kendall (1993)
51 Number Theoretic Methods in Statistics K.-T. Fang
and Y. Wang (1993)
52 Inference and Asymptotics D.R. Cox
and O.E. Bamdorff-Nielson (1994)
53 Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikdinen
and M. Pesonen (1993)
54 Statistical Concepts and Applications in Medicine J. Aitchison
and I.J. Launder (1994)
55 Predictive Inference S. Geisser (1993)
56 Model Free Curve Estimation M. Tartar and M.Lock (1994)
57 An Introduction to the Bootstrap B. Efron
and R.Tibshirani (1993)
58 Nonparametric Regression and Generalized Linear Models
P.J. Green and B. W. Silverman (1994)
59 Multidimensional Scaling
T. Cox and M. Cox (1994)
(Full details concerning this series are available from the publisher)
Tensor Methods
in Statistics
PETER MCCULLAGH
The University of Chicago
Department of Statistics
Chicago, USA
m
CHAPMAN & HALL
Boca Raton London New York
London ■GlasgowC R• CW Preeisns his eainm im •p rNinet owf tYheork • Tokyo • Melbourne • Madras
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First published 1987 by CRC Press
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Reissued 2018 by CRC Press
© 1987 by P. McCullagh
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Library of Congress Cataloging-in-Publication Data
McCullagh. P. (Peter), 1952-
Tensor methods in statistics.
(Monographs on statistics and applied probability)
Bibliography : p
Includes indexes.
1. Calculus of tensors. 2. Mathematical statistics.
I. Title. II. Series.
QA433.M325 1987 519.5 87-13826
ISBN 0-412-27480-9
A Library of Congress record exists under LC control number: 87013826
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Contents
Preface page xii
1 Index notation 1
1.1 Introduction 1
1.2 The summation convention 2
1.3 Tensors 4
1.4 Examples 6
1.5 Elementary matrix theory 8
1.5.1 Generalized inverse matrices 8
1.5.2 Spectral decomposition 10
1.6 Invariants 12
1.7 Direct product spaces 14
1.7.1 Kronecker product 14
1.7.2 Factorial design and Yates’s algorithm 15
1.8 Bibliographic notes 17
1.9 Further results and exercises 1 18
2 Elementary theory of cumulants 24
2.1 Introduction 24
2.2 Generating functions 25
2.2.1 Definitions 25
2.2.2 Some examples 27
2.3 Cumulants and moments 29
2.4 Linear and affine transformations 32
2.5 Univariate cumulants and power notation 34
2.6 Interpretation of cumulants 35
2.7 The central limit theorem 37
2.7.1 Sums of independent random variables 37
2.7.2 Standardized sums 38
2.8 Derived scalars 40
vii
viii CONTENTS
2.9 Conditional cumulants 43
2.10 Bibliographic notes 45
2.11 Further results and exercises 2 46
3 Generalized cumulants 57
3.1 Introduction and definitions 57
3.2 The fundamental identity for generalized cumulants 58
3.3 Cumulants of homogeneous polynomials 61
3.4 Polynomial transformations 62
3.5 Classifying complementary partitions 65
3.6 Elementary lattice theory 66
3.6.1 Generalities 66
3.6.2 Cumulants and the partition lattice 69
3.7 Some examples involving linear models 74
3.8 Cumulant spaces 77
3.9 Bibliographic notes 80
3.10 Further results and exercises 3 81
4 Sample cumulants 88
4.1 Introduction 88
4.2 ^-statistics 89
4.2.1 Definitions and notation 89
4.2.2 Some general formulae for ^-statistics 91
4.2.3 Joint cumulants of ordinary ^-statistics 93
4.2.4 Pattern matrices and pattern functions 96
4.3 Related symmetric functions 97
4.3.1 Generalized fc-statistics 97
4.3.2 Symmetric means and polykays 102
4.4 Derived scalars 107
4.5 Computation 110
4.5.1 Practical issues 110
4.5.2 From power sums to symmetric means 112
4.5.3 From symmetric means to polykays 113
4.6 Application to sampling 115
4.6.1 Simple random sampling 115
4.6.2 Joint cumulants of ^-statistics 117
4.7 ^-statistics based on least squares residuals 123
4.7.1 Multivariate linear regression 123
4.7.2 Univariate linear regression 130
CONTENTS ix
4.8 Bibliographic notes 132
4.9 Further results and exercises 4 133
5 Edgeworth series 142
5.1 Introduction 142
5.2 A formal series expansion 143
5.2.1 Approximation for the density 143
5.2.2 Approximation for the log density 146
5.2.3 Approximation for the cumulative
distribution function 147
5.3 Expansions based on the normal density 149
5.3.1 Multivariate case 149
5.3.2 Univariate case 150
5.3.3 Regularity conditions 152
5.4 Some properties of Hermite tensors 154
5.4.1 Tensor properties 154
5.4.2 Orthogonality 155
5.4.3 Generalized Hermite tensors 156
5.4.5 Factorization of Hermite tensors 157
5.5 Linear regression and conditional cumulants 159
5.5.1 Covariant representation of cumulants 159
5.5.2 Orthogonalized variables 159
5.5.3 Linear regression 161
5.6 Conditional cumulants 162
5.6.1 Formal expansions 162
5.6.2 Asymptotic expansions 165
5.7 Normalizing transformation 165
5.8 Bibliographic notes 167
5.9 Further results and exercises 5 168
6 Saddlepoint approximation 172
6.1 Introduction 172
6.2 Legendre transformation of K(£) 174
6.2.1 Definition 174
6.2.2 Applications 177
6.2.3 Some examples 181
6.2.4 Transformation properties 182
6.2.5 Expansion of the Legendre transformation 184
6.2.6 Tail probabilities in the univariate case 186
6.3 Derivation of the saddlepoint approximation 188