Preface xi,1 Index Notation 1,1 1 Introduction 1,1 2 The summation convention 2. 1 3 Tensors 3,1 4 Examples 5,1 5 Elementary matrix theory 8. 1 5 1 Projections and projection matrices 8,1 5 2 Determinantal factorization 9. 1 5 3 Geometric orthogonality 12,1 5 4 Generalized inverse matrices 13. 1 5 5 Spectral decomposition 14,1 6 Invariants 16,1 7 Direct product spaces 18. 1 7 1 Kronecker product 18,1 7 2 Factorial design and Yates s algorithm 19. 1 8 Bibliographic notes 21,1 9 Further results and exercises 1 22. 2 Elementary Theory of Cumulants 28,2 1 Introduction 28. 2 2 Generating functions 29,2 2 1 Definitions 29,2 2 2 Some examples 31. 2 3 Cumulants and moments 32,2 3 1 Index notation for generating functions 32. 2 3 2 Compositions integer partitions and set partitions 34. 2 3 3 Composition of series 35,2 3 4 Moments and cumulants 38. 2 4 Linear and affine transformations 39,2 5 Univariate cumulants and power notation 40. vi CONTENTS,2 6 Interpretation of cumulants 41,2 7 The central limit theorem 43. 2 7 1 Sums of independent random variables 43,2 7 2 Standardized sums 44. 2 8 Derived scalars 46,2 9 Conditional cumulants 48. 2 10 Bibliographic notes 50,2 11 Further results and exercises 2 51. 3 Generalized Cumulants 61,3 1 Introduction and definitions 61. 3 2 The fundamental identity for generalized cumulants 62. 3 3 Cumulants of homogeneous polynomials 64,3 4 Polynomial transformations 65. 3 5 Complementary set partitions 68,3 5 1 Equivalence classes 68. 3 5 2 Symbolic computation 69,3 6 Elementary lattice theory 70. 3 6 1 Generalities 70, 3 6 2 Mo bius function for the partition lattice 72. 3 6 3 Inclusion exclusion and the binary lattice 74. 3 6 4 Cumulants and the partition lattice 75,3 6 5 Further relationships among cumulants 77. 3 7 Some examples involving linear models 80,3 8 Cumulant spaces 82. 3 9 Gaussian moments 85,3 9 1 Isserlis formulae 85. 3 9 2 Complex valued random vectors 86,3 10 Laplace approximation 88. 3 10 1 Bi partition expansions 88,3 10 2 Formal Laplace expansion 89. 3 11 Bibliographic notes 90,3 12 Further results and exercises 3 92. 4 Sample Cumulants 98,4 1 Introduction 98,4 2 k statistics 99. 4 2 1 Definitions and notation 99,4 2 2 Some general formulae for k statistics 100. 4 2 3 Joint cumulants of ordinary k statistics 103. 4 2 4 Pattern matrices and pattern functions 105,CONTENTS vii. 4 3 Related symmetric functions 108,4 3 1 Generalized k statistics 108. 4 3 2 Symmetric means and polykays 111,4 4 Derived scalars 115. 4 5 Computation 118,4 5 1 Practical issues 118,4 5 2 From power sums to symmetric means 119. 4 5 3 From symmetric means to polykays 120,4 6 Application to sampling 122. 4 6 1 Simple random sampling 122,4 6 2 Joint cumulants of k statistics 124. 4 7 k statistics based on least squares residuals 129. 4 7 1 Multivariate linear regression 129,4 7 2 Univariate linear regression 134. 4 8 Free samples 136,4 8 1 Randomization by unitary conjugation 136. 4 8 2 Spectral k statistics 137,4 9 Bibliographic notes 140. 4 10 Further results and exercises 4 141,5 Edgeworth Series 151. 5 1 Introduction 151,5 2 A formal series expansion 152. 5 2 1 Density 152,5 2 2 Log density 154,5 2 3 Cumulative distribution function 155. 5 3 Expansions based on the normal density 157,5 3 1 Multivariate case 157. 5 3 2 Univariate case 158,5 3 3 Regularity conditions 159. 5 4 Some properties of Hermite tensors 161,5 4 1 Tensor properties 161. 5 4 2 Orthogonality 162,5 4 3 Generalized Hermite tensors 163. 5 4 4 Factorization of Hermite tensors 164, 5 5 Linear regression and conditional cumulants 165. 5 5 1 Covariant representation of cumulants 165,5 5 2 Orthogonalized variables 166. 5 5 3 Linear regression 168,5 6 Conditional cumulants 168. 5 6 1 Formal expansions 168,5 6 2 Asymptotic expansions 170. 5 7 Normalizing transformation 171,viii CONTENTS,5 8 Bibliographic notes 172. 5 9 Further results and exercises 5 173,6 Saddlepoint Approximation 177. 6 1 Introduction 177,6 2 Legendre transformation of K 178. 6 2 1 Definition 178,6 2 2 Applications 181,6 2 3 Some examples 185. 6 2 4 Transformation properties 186, 6 2 5 Expansion of the Legendre transformation 188. 6 2 6 Tail probabilities in the univariate case 189. 6 3 Derivation of the saddlepoint approximation 191. 6 4 Approximation to conditional distributions 193. 6 4 1 Conditional density 193,6 4 2 Conditional tail probability 194. 6 5 Bibliographic notes 195,6 6 Further results and exercises 6 195. 7 Likelihood Functions 200,7 1 Introduction 200,7 2 Log likelihood derivatives 202. 7 2 1 Null cumulants 202,7 2 2 Non null cumulants 204. 7 2 3 Tensor derivatives 205,7 3 Large sample approximation 208. 7 3 1 Log likelihood derivatives 208,7 3 2 Maximum likelihood estimation 209. 7 4 Maximum likelihood ratio statistic 210,7 4 1 Invariance properties 210. 7 4 2 Expansion in arbitrary coordinates 211,7 4 3 Invariant expansion 212. 7 4 4 Bartlett factor 212,7 4 5 Tensor decomposition of W 214. 7 5 Some examples 215,7 5 1 Exponential regression model 215. 7 5 2 Poisson regression model 217,7 5 3 Inverse Gaussian regression model 218. 7 6 Bibliographic notes 220,7 7 Further results and exercises 7 221. 8 Ancillary Statistics 227,8 1 Introduction 227,CONTENTS ix. 8 2 Joint cumulants 229,8 2 1 Joint cumulants of A and U 229. 8 2 2 Conditional cumulants given A 230,8 3 Local ancillarity 231. 8 4 Stable combinations of cumulants 234,8 5 Orthogonal statistic 236. 8 6 Conditional distribution given A 238,8 7 Bibliographic notes 243. 8 8 Further results and exercises 8 244,Appendix Complementary Set Partitions 247. References 261,Author Index 273,Subject Index 277, In matters of aesthetics and mathematical notation no one loves an. index According to one school of thought indices are the pawns. of an arcane and archaic notation the front line troops the cannon. fodder first to perish in the confrontation of an inner product Only. their shadows persist Like the putti of a Renaissance painting or the. cherubim of an earlier era or like mediaeval gargoyles indices are mere. embellishments unnecessary appendages that adorn an otherwise bare. mathematical symbol Tensor analysis it is claimed despite all evidence. to the contrary has nothing whatever to do with indices Coordinate. free methods and operator calculus are but two of the rallying slogans. for mathematicians of this persuasion Computation on the other. hand is a reactionary and subversive word Stripped of its appendages. freed from its coordinate shackles a plain unadorned letter is the very. model of a modern mathematical operator, Yet this extreme scorn and derision for indices is not universal It can. be argued for example that a plain unadorned letter conceals more. than it reveals A more enlightened opinion shared by the author is. that for many purposes it is the symbol itself that plain unadorned. letter not the indices that is the superfluous appendage Like the. grin on Alice s Cat the indices can remain long after the symbol has. gone Just as the grin rather than the Cat is the visible display of the. Cat s disposition so too it is the indices not the symbol that is the. visible display of the nature of the mathematical object Surprising as. it may seem indices are capable of supporting themselves without the. aid of crutches, In matters of index notation and tensor analysis there are few neutral. observers Earlier workers preferred index notation partly perhaps out. of an understandable concern for computation Many modern workers. prefer coordinate free notation in order to foster geometrical insight The. above parodies convey part of the flavour and passion of the arguments. that rage for and against the use of indices, Index notation is the favoured mode of exposition used in this book. although flexibility and tolerance are advocated To avoid unfamiliar. objects indices are usually supported by symbols but many of the most. important formulae such as the fundamental identity 3 3 could easily. xii PREFACE, be given in terms of indices alone without the assistance of supporting. symbols The fundamental operation of summing over connecting. partitions has everything to do with the indices and nothing to do with. the supporting symbol Also in Section 4 3 2 where supporting symbols. tend to intrude the Cheshire notation is used, Chapter 1 introduces the reader to a number of aspects of index. notation groups invariants and tensor calculus Examples are drawn. from linear algebra physics and statistics Chapters 2 and 3 dealing. with moments cumulants and invariants form the core of the book and. are required reading for all subsequent chapters, Chapter 4 covers the topics of sample cumulants symmetric functions. polykays simple random sampling and cumulants of k statistics This. material is not used in subsequent chapters Unless the reader has a. particular interest in these fascinating topics I recommend that this. chapter be omitted on first reading, Chapters 5 and 6 dealing with Edgeworth and saddlepoint approx. imations are presented as a complementary pair though they can be. read independently Chapter 6 refers backwards to Chapter 5 at only. The final two chapters are again presented as a pair Chapter 7 deal. ing with likelihood functions log likelihood derivatives likelihood ratio. statistics and Bartlett factors makes use only of material in Chapters. 2 and 3 Chapter 8 dealing with ancillary statistics makes consider. able use of saddlepoint approximations and Legendre transforms from. Chapter 6 and other expansions given in Chapter 7 The book concludes. with a derivation of Barndorff Nielsen s formula for the conditional dis. tribution of the maximum likelihood estimator given a suitable ancillary. Exercises are an integral part of the book though results derived as. exercises are rarely used in the remainder of the book One exception is. the Taylor expansion for the log determinant derived in Exercise 1 16. subsequently used in Section 8 6 Exercises vary considerably in length. and I suppose difficulty I have decided against difficulty ratings on. the grounds that the composer of the exercise is least suited to judge. difficulty Results derived in one exercise are commonly used in the. exercises immediately following, The book is intended mainly for graduate students in statistics and. as a reference work for professional statisticians Readers should have. some familiarity with linear algebra eigenvalue decompositions linear. models and for later chapters with likelihood functions likelihood ratio. statistics and so on As soon as students become comfortable using. index notation most of the first four chapters will be within their grasp. Final year undergraduate mathematics students at U K universities can. master most of Chapter 2 and parts of Chapters 3 and 4. PREFACE xiii, In the three years that it has taken me to write this book I have. benefitted greatly from the help and advice of colleagues at Imperial. College AT T Bell Labs and University of Chicago I am especially in. debted to Colin Mallows for numerous discussions concerning invariants. for debunking hastily conceived conjectures and for calculations leading. to a number of exercises Allan Wilks is responsible for relieving me. of the arduous task of compiling by hand the Tables in the Appendix. Others who have read and commented on parts of the manuscript or who. have made useful recommendations short of abandoning the enterprise. include R Bahadur J M Chambers D R Cox M Handcock C Inclan. V Johnson P Lang D Pregibon N Reid L A Shepp I M Skovgaard. T P Speed J M Steele S M Stigler J W Tukey P Vos D L Wallace. Daming Xu and S L Zabell I am grateful also to H E Daniels and to. I M Skovgaard for providing copies of unpublished papers on saddlepoint. approximation and to O E Barndorff Nielsen P Blaesild for providing. unpublished papers on strings and related topics, In a book of this kind it is unreasonable to expect all formulae and all. exercises to be free from error however careful the checking All formulae. have been checked but undoubtedly errors remain I would be grateful. therefore if diligent eagle eyed readers could notify me of any further. The book was typeset partly using the TEX system Knuth 1986. and partly using the TROFF system The figures were typeset using. the PICTEX system Wichura 1986 I am grateful to M Wichura for. assistance with PICTEX and to Diana Wilson for typesetting advice. Finally it is a great pleasure to thank the typist whose unfailing. courtesy inexhaustible patience and careful attention to detail have. made this book possible,Chicago P McCullagh,December 1986. The reprinted edition has been lightly edited and updated particularly. in sections 1 5 2 3 3 6 3 9 3 10 and 4 8,Chicago P McCullagh. August 2017,Index Notation,1 1 Introduction, It is a fact not widely acknowledged that with appropriate choice of. notation many multivariate statistical calculations can be made simpler. and more transparent than the corresponding univariate calculations. This simplicity is achieved through the systematic use of index notation. and special arrays called tensors For reasons that are given in the fol. lowing sections matrix notation a reliable workhorse for many second. order calculations is totally unsuitable for more complicated calculations. involving either non linear functions or higher order moments The aim. of this book is to explain how index notation simplifies certain statistical. calculations particularly those involving moments or cumulants of non. linear functions Other applications where index notation greatly simpli. fies matters include k statistics Edgeworth and conditional Edgeworth. approximations saddlepoint and Laplace approximations calculations. involving conditional cumulants moments of maximum likelihood es. timators likelihood ratio statistics and the construction of ancillary. statistics These topics are the subject matter of later chapters. In some ways the most obvious and at least initially one of the. most disconcerting aspects of index notation is that the components. of the vector of primary interest usually a parameter or a random. variable X are indexed using superscripts Thus 2 is the second. component of the vector which is not to be confused with the square. of any component Likewise X 3 is the third component of X and so on. For that reason powers are best avoided unless the context leaves no. room for ambiguity In principle 2 3 is the product of two components. which implies that the square of 2 is written as 2 2 In view of the. considerable advantages achieved this is a modest premium to pay. 2 INDEX NOTATION,1 2 The summation convention, Index notation is a convention for the manipulation of multi dimen. sional arrays The values inserted in these arrays are real or complex. numbers called either components or coefficients depending on the. context Technically speaking each vector in the original space has. components with respect to the given basis each vector in the dual. space of linear functionals has coefficients with respect to the dual basis. In the setting of parametric inference and in manipulations associ. ated with likelihood functions it is appropriate to take the unknown. parameter as the vector of interest see the first example in Section 2 4. Here however we take as our vector of interest the p dimensional ran. dom variable X with components X 1 X p An array P of constants. a1 ap used in the formation of a linear combination ai X i is called. the coefficient vector This terminology is merely a matter of convention. but it appears to be useful and the notation does emphasize it Thus. for example i E X i is a one dimensional array whose components. are the means of the components of X and ij E X i X j is a two. dimensional array whose components are functions of the joint distribu. tions of pairs of variables, Probably the most convenient aspect of index notation is the implied. summation over any index repeated once as a superscript and once as a. subscript The range of summation is not stated explicitly but is implied. by the positions of the repeated index and by conventions regarding the. range of the index Thus,ai X ai X i 1 1, specifies a linear combination of the Xs with coefficients a1 ap. Quadratic and cubic forms in X with coefficients aij and aijk are written. in the form,aij X i X j and aijk X i X j X k 1 2, and the extension to homogeneous polynomials of arbitrary degree is. For the sake of simplicity and with no loss of generality we take all. multiply indexed arrays to be symmetric under index permutation but. of course subscripts may not be interchanged with superscripts The. value of this convention is clearly apparent when we deal with scalars. such as aij akl ijkl which by convention only is the same as aik ajl ijkl. and ail ajk ijkl For instance if p 2 and aij ij 1 if i j and 0. otherwise then without the convention,aij akl ijkl aik ajl ijkl 1122 2211 1212 2121. and this is not zero unless ijkl is symmetric under index permutation. Expressions 1 1 and 1 2 produce one dimensional or scalar quan. tities in this case scalar random variables Suppose instead we wish to. construct a vector random variable Y with components Y 1 Y q each. of which is linear in X i e,Y r ari X i 1 3, and r 1 q is known as a free index Similarly if the components. of Y are homogeneous quadratic forms in X we may write. Y r arij X i X j 1 4, Non homogeneous quadratic polynomials in X may be written in the. Y r ar ari X i arij X i X j, Where two sets of indices are required as in 1 3 and 1 4 one referring. to the components of X and the other to the components of Y we use the. sets of indices i j k and r s t Occasionally it will be necessary. to introduce a third set but this usage will be kept to a. All of the above expressions could with varying degrees of difficulty. be written using matrix notation For example 1 1 is typically written. as aT X where a and X are column vectors the quadratic expression. in 1 2 is written XT AX where A is symmetric and 1 3 becomes. Y A X where A is of order q p From these examples it is evident. that there is a relationship of sorts between column vectors and the. use of superscripts but the notation XT AX for aij X i X j violates the. relationship The most useful distinction is not in fact between rows. and columns but between coefficients and components and it is for that. reason that index notation is preferred here,1 3 Tensors. The term tensor is used in this book in a well defined sense similar in. spirit to its meaning in differential geometry but with minor differences. in detail It is not used as a synonym for array index notation or the. summation convention A cumulant tensor for example is a symmetric. array whose components are functions of the joint distribution of the. random variable of interest X say The values of these components in. any one coordinate system are real numbers but when we describe the. array as a tensor we mean that the values in one coordinate system. Y say can be obtained from those in any other system X say by.

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