Strings: Mathematical Theory and Statistical Examples

Strings: Mathematical Theory and Statistical ExamplesAuthor(s): O. E. Barndorff-Nielsen and P. BlaesildSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 411, No. 1840 (May 8, 1987), pp. 155-176Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2398180 .

Accessed: 12/06/2014 17:08

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 185.44.77.89 on Thu, 12 Jun 2014 17:08:01 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/2398180?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Proc. R. Soc. Lond. A 411, 155-176 (1987) Printed in GAreat Britain

Strings: mathematical theory and statistical examples

BY 0. E. BARNDORFF-NIELSEN AND P. BLAESILD Department of Theoretical Statistics, Institute of Mathematics, University of

Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

(Communicated by Sir David Cox, F.R.S. - Received 27 August 1986)

Strings, in the sense of the present paper, are sequences of multiarrays with two types of indices; tensorial and structural, and they are characterized by a transformation law that generalizes those for tensors, affine connections and derivatives of scalars. The original definition Barndorff- Nielsen (Proc. R. Soc. Lond. A 406, 127-137 (I986)) is here extended and a systematic study of the mathematics of strings is undertaken. In particular, a convolutive multiplication of strings is introduced and is used in the discussion of intertwining, a type of operation that produces tensors from strings and strings from tensors and connection strings, a special kind of string. It appears that tensors and connection strings have a role, respectively, as 'coordinates' and 'coordinate frames' in the calculus of strings. A definition of 'covariant differentiation of strings' is proposed and is related to convolutive multiplication and to intertwining. The general theory is illustrated by various examples from the context of statistical inference. Finally, a brief comparison is made between strings and the somewhat related concept of extensors.

1. INTRODUCTION

In Barndorff-Nielsen (i986b) a differential-geometric concept termed strings was introduced and studied to some extent, and various statistical examples and applications were given. A string is a sequence of multiarrays that satisfies a certain transformation law (see (2. 1) and (2.5)), and the concept encompasses those of tensors, (affine) connections, and successive derivatives of scalars.

The present paper extends the definition of strings by abandoning a symmetry condition and it discusses further properties of and operations on strings; associated statistical illustrations are also presented. In particular, a definition of covariant differentiation of elements of strings is proposed and this leads to a concept of canonical strings, which are obtained by successive covariant differentiation.

The general definition of strings is given in ? 2, and ? 3 is concerned with elemental algebraic properties of strings. In particular, we introduce three types of products of strings, the first leading to double strings whereas the other two do result in single strings. Covariant differentiation and strings form the subject of ?4, and ?5 discusses intertwining of ordinary strings as well as p-double strings. Intertwining establishes a relation between strings and sequences of tensors, and it provides a method for constructing tensors, having desirable properties, from strings. Somewhat more specifically, every string has a representation as the intertwining of any given connection string and a uniquely determined sequence

[ 155 ]



156 0. E. Barndorff-Nielsen and P. Bloesild of tensors, so that tensors have a role as 'coordinates' in the calculus of strings. A relation between covariant differentiation and intertwining is also noted. Section 6 contains a number of illustrations in the context of statistical inference.

The concept of strings is formally reminiscent of the concept of extensors, introduced in Craig (I937) and studied primarily by Kawaguchi and by Craig, see in particular Kawaguchi (I939, 1940), Craig (I943), Morey & Craig (I972) and the references given there. We are grateful to Professor Shun-Ichi Amari for referring us to those works. A brief discussion of extensors and their relation to strings are given in the concluding ?7.

2. THE GENERAL DEFINITION OF STRINGS

In this section the concept of strings is introduced and illustrated by examples of general interest. Examples of strings in statistics are discussed in ?6. At the end of the present section two generalizations of strings, double strings and strings with weight, are introduced.

We shall be concerned with multiarrays on a differentiable manifold X#, and with the behaviour of such arrays under reparametrizations, i.e. changes of coordinates.

Consider a differentiable manifold X/ of dimension d, and let w = (w1, ..., wd)

denote (local) coordinates on S. We shall refer to w as a parametrization of X#, and we denote generic coordinates of w by , (X k, etc. Let Vr be an alternative parametrization of X/, whose generic coordinates we shall denote by a, .b, c, etc. We may think of Vr as a function of w, and vice versa, and we let aa = aa at = a/a, 0ia = da (i, V.( a = at. 1a, ?i = a etc. Furthermore, if A denotes i ~ /a a / t= r lab = aa b t.i a set of indices al, ..., ar we use the notation

IA= /a, ... ar a, ... aar

A multiarray M on Xx is a class of families of multidimensional arrays, each family corresponding to a particular parametrization of XA and each element of a family corresponding to a point in X/. We say that M is a (r, s) array if its generic element under the parametrization w is of the form

M jr = Mjl - ir((O))

all r + s indices running from 1 to d, and we refer to il, ir as contravariant indices and to ji, ..., j. as covariant indices. Under the parametrization Vi the generic element of M is =

As indicated, we shall often drop the arguments w or Vf from our formulas because the indices i, j, k, etc., respectively a, b, c, etc., distinguish between the parametrizations.

For the type of multiarrays with which we shall be mainly occupied the covariant indices fall, in fact, in two groups and we will indicate this by writing

aa wihs k *... kt

and by speaking of a (r, s; t) array with kl, ..., kt as the structural indices.



Strings 157

We often consider a partition of a set of indices C = {cl, ..., ct} into r subsets or blocks C1, ..., C. Throughout the paper we assume that the ordering of the elements in the partition is in accordance with the ordering of the blocks of C, that is within each block the order of the elements is the same as their mutual order within C and, moreover,

ci c-Cl

and cnm ECm,

where nm = min {n: cn 6 C\(Cl U ... U Cm_i)}, m = 2, ..., r. Let T denote a non-negative integer or oo and let T = {1, ..., T) if T is finite

and T= {1, 2, ...} if T is infinite. A string M on X of contravariant degree r (r = 0, 1, 2, ...) and covariant degree s (s = 0, 1, ...), or more briefly a (r, s) string, of length T is now defined as a collection of (r, s; t) arrays

ji.. Js k, ...kt ((), t c T, which satisfy the transformation rule

Ma i.ar2 ... { Er k k}

X!fa/,... Vfa/r WIbl6 . lbs (2.1)

for all t E T and every pair of parametrizations w and Vf . In (2.1) C/Tr indicates summation over all partitions of C= {cl, ..., ct} into r

subsets Cl, ..., C. Furthermore, as we shall also do in the rest of the paper, we have in (2.1) used Einstein's summation convention, according to which, whenever an expression contains the same letter as an upper and a lower index summation over the full range of that index is assumed.

The set of (r, s) strings of length T will be denoted by Y'r(T) and we refer to the indices kl, ..., kt as structural indices and to il, ..., ir and jl, ***, js as contravariant and covariant tensorial indices, respectively. The reason for the latter name is that these indices play the same role in (2.1) as in the well known transformation law for a tensor T of contravariant rank r and covariant rank s - or a (r, s) tensor T - that is

Tal .. a = T 1l .. r */l

a, .. *f ar (O)/b (Oi.Ct/s b -.. h,..is /iij /ir lb1~ /bs

Furthermore, denoting the set of such (r, s) tensors on Xk by g-r one has r(l) = g7 r i.e. every (r, s) string of length 1 is a (r, s+1) tensor, as appears

from (2.1). The concept of a string was introduced in Barndorff-Nielsen (I986b) and that

paper discussed contrastrings, (r, 0) strings, and costrings, (0, s) strings, and gave examples of statistical interest of such strings. A string is said to be symmetric if each of the multiarrays of which it consists is invariant under arbitrary permu- tations of the indices, separately within each of the three types of indices, contravariant tensorial, covariant tensorial and structural. As will be apparent from ?6 most of the basic strings in statistics have this property. However, there exist non-symmetric strings possessing relevant information on a statistical model, cf. example 2.3.



158 0. E. Barndorff-Nielsen and P. Bleasild

Well-known quantities from differential geometry constitute examples of strings as shown below. In accordarwe with this strings in S ? are called scalar strings and strings in Y?I are called connection strings.

Example 2.1. Let f be a CO-function, i.e. f is real-valued and has derivatives of all orders. Let

fkl *** kt = ak* *.* akt f t {1, 2,**} Because, on change of parametrization,

t ... ct (Ok ...lk1)c ()kTX I t = 1, 25,..}' (2.2)

it follows thatf is a scalar string of infinite length, i.e. fe?Y 0(oo). Example 2.2. Let T E g r+. Defining M by

A?P.i .rkk - JO, for t'=1,...,t-1, 1i ... T Js ki.r. kt for t =

one has M E Y r(t), i.e. a (r, s + t) tensor can be considered as a (r, s) string of length t. In example 2.3 and later we use the Kronecker tensor, i.e. the (1, 1) tensor given

by i=j,

0- iij.

Example 2.3. Recall that the components of w are the local coordinates of a d-dimensional manifold S4. Let E?, i = 1, ..., d, denote the corresponding coordinate frames, i.e. the canonical basis for the tangent space Tp( #) at p E X.4 is given by

Eip[f] = (f(p- (@)))l=,

where fe C(O0)(p) and where cp is the mapping which to every point p e .4' assigns the local coordinate Cl = cp(p). An affine connection V on XA is characterized by the array rFik2, the so-called Christoffel symbols, giving the coordinates of VEk Ekl with respect to the coordinate frame, i.e.

VEkEkl = i k k2E.

The Christoffel symbols obey the transformation law

lC2 = rk1k2 /Cj /C2 k C} C a(. Setting Fi1 = 4k, the Kronecker tensor, and comparing (2.1) and (2.3) it follows that {IFi, I J'k2} is a (1, 0) string of length two. This string may be extended to a string of infinite length by repeated covariant differentiation. To be specific, let for t >, 2 Fki be defined by

VEkt VEkt_- ... VEk2 Ek= rik ... kt Ei. (2.4)

It follows from the results in ?4 that F i= {F1 kt t e{1, 2, ... }} is a (1, 0) string of infinite length. In general this string is not symmetric.



Strings 159

In particular, if X is a Riemannian manifold and if R denotes the Riemannian curvature tensor one has

R(Ek3, Ek2, Ek1, Ek4) = (VEk VEk Ek, VEk VEk Ekl, Ek4)

= ((Fr k2 k3 -ki k3 k2) Ei, Ek4)

k(Fr k2 k3- k k3 k2) gik4'

where gi, = (E,, E>) denotes the Riemannian metric. There are several ways to generalize the concept of a string. Here we just present

two. A double string M of degree (r, s) and length (T, U) is a double sequence of multiarrays

i .. Jsrki ... kt 11... lul ET EU

satisfying the transformation law t tl

11 ...

basr c 1... Ct d, .. du ET-1 ki - DVJ ... k.............. i8 IC *Ir V@C ''' /t ID D = C v= Dlv ID..,sT

x l ar r ( ll Ibs (2.5)

Similarly, one can define p-double strings p = 3, 4,. Note that if M is a double string of degree (r, 0) and length (1, U) then M can

also be viewed as an element M of ,r9 of length U. As shown in ?3 (outer) multiplication of two strings has as a result a double

string. Furthermore, we have the following generalization of example 2.1. Example 2.4. Letf be a scalar defined on ?P = Q x ? x ... x 2 (p times), where

2 denotes the domain of variation of w. The values off are denoted byf(wl, ..., up) where w> = ((O, ..., I(t) is an element in ?. Letting a,k = we set

fkll... kit;k2l ...k2t;...;kpi...kptp alkil ... alkltl a2k2l ... a2k2t ... apkpl ... apkpt f (2.6)

and sometimes we abbreviate this expression to

fK,;...;Kp.

From this functionfKl; ;Kp5defined on ?P, we obtain a function Kl; ;Kp defined on ? by evaluating the former on the diagonal of ?P, i.e.

K,; ...; Kp((W) = fKi; ...; Kp(((W, * w). (2.7)

It is easily seen that the collection Jof these functions constitute a p-double string of degree (0, 0).

Setting K2 = ... = Kp= (the empty set) one has

fIkll ... kit ((w) = fKi;q0;... ;0(W, ..., ( ).

The set of such functions f may be shown to be a (0, 0) string of infinite length. Thus f satisfies the transformation law for scalar strings (2.2) without being a scalar string of the type considered in example 2.1.

For applications of double strings in statistical contexts see ?6.



160 0. E. Barndorff-Nielsen and P. Blwesild

We conclude this section by extending the concept of tensors of weight w discussed by Craig (I943), Thomas (I96I). A string M of weight weR and degree (r, s) is a sequence of arrays satisfying the transformation law

Mgl asr c ...ct = 1,)I, {w E E Mitl k t4 kT/ lC CT

X W/al . .. Gyar CO)21 . .. WS (2.8)

Here Iw/RlrI denotes the determinant of the jacobian matrix of the mapping from Vf to w. Note that (2.8) is identical to (2.1) except for the factor Iw/0biw.

A statistical example of a string with weight one is given in ?6.

3. ELEMENTAL OPERATIONS ON STRINGS

In this section we show how elemental operations in tensor calculus, such as addition, subtraction, and multiplication, generalize to strings. Simple multiplication of strings does not yield strings of the same multiplicity, but at the end of the section we discuss two alternative definitions of the product of strings that do have this property.

3 (a) Addition and subtraction Let M and Q be elements of Yr (T) and define M3+ Q by

(Ml?Q)il1...ki = ~ "I i.rkk i?Q '1 r tec-T. (M+ ... is ki... kt MG1.. isk,... kt-?Q3)1 .. s ki... kt5 6

Then M?Qef gr(T). Note that if

M)tl Q rkl kt, for t < t

then (M-Q)il :rk... ktr S +t-

3 (b) Simple multiplication Let MeCf r (T) and Qe Ju(W). The simple product M_s0 Q of M and Q, defined

by M C). is g ml ... mv ki ... kt n, ... nw

A ,... sk,...kt ml ... mvni... w (t,w)ETx W, is easily seen to be a double string of degree (r+u, s+v) and length (T, W), i.e. M C) Q eS9 rs+u(T, W).

Because yr+u(0, 0) = g-r+u and grS(o) = g r, simple (i.e. Kronecker) multiplication of tensors can be considered as a special case. Furthermore, note that if Mi3C"ri(Ti),i= 1,..., p, then Ml (1...0(M is a p-double string of length (T1, ..., T.) and degree (r+ ...+ rp, 81+ +s8).

3 (c) Contraction Let Me Y g4r+u(T) and define Q by identifying u of the covariant and contravari-

ant tensorial indices. Recall that according to Einstein's convention this implies



Strings 161

summation over the full range of values for these indices. For notational convenience, suppose that the last u of the tensorial indices are identified, i.e.

- ~~~~~u tc-T. -Qil ... s ki... kt = 3 ... is 11.. lu ki . kt) t

Then Qe 9 r(T) and Q is said to be obtained from M by contraction.

3 (d) Shifting of tensorial indices

Let M c r+ui(N) and T e 2v and define Q by "I mr m ~ "I Mt r 1;.r. lu Tm'l ..m'vmi. mv, nc-N. ii ...is 11.* . .. ** kn ii ... ism m ...um'v k ...k n I1,..I... , .Iv

From 3 (b) and 3 (c) it follows that Qe E r+v (N). Qis said to be obtained from M and T by lowering u of the contravariant tensorial indices and by lifting v of the covariant tensorial indices, or more briefly by shifting u + v of the tensorial indices. Again, in the equation above, it is only for notational convenience that the u+v indices to be shifted have been chosen to be the last u contravariant and the last v covariant tensorial indices, respectively.

Example 2.3 (continued). Lowering the contravariant index in FeP J (oo) by the riemannian metric tensor one obtains a string in Y 0 (oo) with elements

Fiki...kt = F...ktg, t = 1,g2.i It follows that

Fjk,k2 = (VEk Ekl, Ej)

and that the Riemannian curvature tensor may be expressed in terms of the new string, as

R(Ek3) Ek2) Ek1, EJ) = rk k2 k3- rFk k3 k2.

Example 2.4 (continued). Consider the case p= 2 and assume that for every Cl E Q we have

fk(w, ?) = 0, K = 1,..., d (3.1)

and [fkl;k2(W' o)] is non-singular. (3.2)

Writing 1k1 k2 for the inverse of the matrix fkl; k2 we have that 1k1 k2 E Y 0 and we now define for t = 1, 2, ...

fk 1... kt fj; ki... kt f (3.3)

and f..tA..tJ (3.4) and~~~~~~~~~~~ kik .. . kt =fikl . .. kt; i P ; j

Recalling that S"?(1, oo) is essentially the same as YO?(oo) it follows that the quantities defined in (3.3) and (3.4) each constitute a connection string of infinite length, i.e. an element of Yl(oo). Note, that

fki &k =fkik

Clearly, tensorial indices of p-double strings and of strings with weight may be shifted in a similar way.

6 Vol. 4I i. A



162 0. E. Barndorff-Nielsen and P. Bloesild

3 (e) Convolutive multiplication Let Me f r(T) and Ml C 9'r"(T) and consider the arrays M* M' with entries

0, t= 1,

(M * M')!JIJk, ... kt = j E MJK, M"' t = 2, ..., T. K/2

Here we have used the abbreviations I = i. i I, = *i, J=j *i*j ' = j8 ... and K1 and K2 denote the two blocks that K = {kl, ..., ktl} has been

partitioned into. We now show that 11* i' e "Sir(T). Letting w/ = @/ ... ?J. 8+8, IB ~~~lb, lbs and Vfi = V. a, ... *yar we have / ISi /ir

( MBBc ... ct = BC MBC1 MB2 C/2

(C, IC21

iCl2r, ~ ~~ ~I kl1.. k,MTJ k2, ... 22 VC/27-1=I ClIT1 T2- C2/T2 Jl - llJkl-k

X Ik k. wkl,( k 2 2 J NJ WA1' A (3.6) IC1, /ICl,, IC21,, B B 1 I

A typical term in the sum in (3.6) contains the factor

(Ok ..k.W(.7 IC, ICT' (3.7) where 2 < r < t and where CY, ..., CT denotes a partition of C into r blocks. Let K1, K2 be a partition of K(7-) = {Y1, *.., T} such that

and K2 = {J21, ... I 2} where Ti > 0, i = 1, 2, and where r = rl + -2. For a fixed T and for a fixed partition Cl, .(.. CT the terms in,the sum in (3.6) containing the factor (3.7) are precisely those for which CQ = Q1, U ... U C1T1 and C2 = 021 U ... U 027T2. Consequexitly,

(M * M )BB'cl ... ct

K2MIJK, YIK IC, IC .IBWIB II IF 7-2 C/I K(r)/2

= ; k k

(M* MX )JJ'Al A'.kr@CI / '' @/ BWIBVJII?jIlI' (3.8) T =2 CIT

I ICI B I fI

Because, according to (3.5), (M *M')JIJ'k, = 0 formula (3.8) proves that M*M' is a (r + r', s + s') string of length T. Similarly, one has M' * Me - , t(T) and consequently M** M.r' e r+r (T) where M?** Ml' = M* ?' + ?' * M, i.e.

(M ** M) , )JJ k... kt= (M * M)JJ'k, ... kt + (M * M)J'Jkl ... kt

0, t=l, t

JJKI / MIK2' t2, ..T. (39)



Strings 163

In (3.9) t indicates that we here drop the assumption that the element k1 of K = {k1, ..., ckt} belongs to K1, the first of the two blocks K1 and K2 in the partition of K. Within each block the order of the elements is still assumed to be the same as their mutual order within K.

Obviously, one has

(M **M').7 . .kt = (M ** M)f,J&k.kt (3.10)

Notice that one difference between the products * and ** and the product 0, considered in ? 3(b), is that * and ** have as a result a string whereas (0 has as result a double string.

We term the product operations * and ** convolutive multiplication and symmetric convolutive multiplication, respectively.

Let Me Yr(T), M' e r(T) and M" e Yr(T). Then the following relations for the products * and ** are easily proved

(M* (M * M"))IIJJ'"k -..kt = E MJK, M 'K2 MJ-K3 (3.11) K13

(M*M')*M " = M*(M'**M"), (3.12)

(M*M') **M" = M* (M' **M")+M" * (M*M'), (3.13)

M ** (M' ** M") = (M ** M') ** M". (3.14)

Note that the ** product is associative in contrast to the * product. We conclude this section by defining r(* r), the rth power with respect to the * product of a connection string FeY' c (oo), in the following way

F(*-l 2) = (F*F)l2 kt t = 1, 2, ..., (3.15)

(*lt r il = (F*F(*r l))i2jtr r = 3, 4, ..., t = 1, 2 . (3.16)

Clearly, r(* r) is an (r, 0) string and it follows from (3.11) that

(0, r >t,

r(*r)il...ir = (3.17) k, .. kt lEriK ... FrK, r <, t. Klr K, Kr'

This relation will be of considerable use in ?5.

4. COVARIANT DIFFERENTIATION AND STRINGS

In this section we generalize the concept of covariant differentiation of tensors to covariant differentiation of (r, s; t) arrays. The covariant derivative of such an array is a (r, s; t + 1) array. We discuss how this generalization may, under a certain condition, be used to extend finite strings. In particular, we show that the (1, 0) string of length 2 corresponding to an affine connection V, i.e. {6i I riFl }, may be extended in a canonical way to a connection string of infinite length. Finally, some simple rules of calculation for covariant differentiation of elements and products of strings are given.

6-2




Let V be an affine connection and let {8t, Fi k2} be the corresponding connection string of length 2. According to- Eisenhart (I 926) the (V) covariant derivative of a (r, s) tensor T with respect to wk is the (r, s + 1) tensor Vk T with entries

r s (Vk T)ji = k k(Tj jr)+2 T 1T 1 ir8ir- Frfi (4.1)

Now let M be a- (r, s; t) array. We then define the (V) covariant derivative of k1sr. kt with respect to wk as the array Vk M with entries

r (Vk k) i ..kl ktk ak(Mg 5kl kt)+ M 'j kyr k it

l

? --Jsk ... i... is ki ... kt rjfl k ..(42

Comparing (4.1) and (4.2) one sees that covariant differentiation of a (r, s; t) array may be thought of as ordinary covariant differentiation of tensors but considering the structural indices as fixed.

We shall often use the notation Mtl I..kl r kdk or Vk M for the quantity in (4.2), and we may consider this as a (r, s; t + 1) array. In line with this we often use the notation Til .../ k for the quantity in (4.1).

As noted earlier, an element of fr (1) may be considered as an (r, s + 1) tensor. In the context of covariant differentiation it is, however, important to make a distinction. To illustrate this, consider the Kronecker tensor d. If 8 is considered as a (1, 1) tensor we have

=1' - 11 j ll/k = ak (8)tl) + bj1ri r-4 rjk =1 ?

whereas, assuming 5ef01(1), we obtain

hi,/k - k(kJ1)+ C1F C = k- (4.3)

For Mec 9r(T) it is of course of interest to know under which conditions

{jm1,l IIir t = 1) ...I T, MAl -ijrk..kTIIk}

constitute a (r, s) string of length T+ 1. We now show that this is so if

Ml "IIklkt t = 2, ... , T, (4.4)

where //k2 ... kt symbolizes repeated use of the differentiation rule (4.2), first with respect to w)k2, then with respect to wk3 and so on. In the proof we set

ji1gsk ... is k ... kT k j ... J) s k, ....... kTIlk-

By (2.1) one has

ac(Ma asrcl ... c7,)

T

t-= 1 C/t k( )1 gskl . kt k a is

c t 1 lt .. ki...kt(fIC' *.et.O) /r lbi b

ak0 ---skl..t) t/k .. 1 4r

(oi lb8Wi



Strings 165

t + M,~lir, ( wOki kJT ~ kt al ar~j .o

-T- ) gsk1 ... kt /C * /(T U C) /Ct /i * /ir lb, * lbs

r r ~ Ii Ok kt al . a ol (i + k, ... rk kt ()e ... )Ct lI/il l * ** /iak * ** /ir lb1 lbs Ic

s + E Ml~* k t a/ar) **is (45 + Mi ...risk,. kt /lh1...w(1) i... ?rar /bf.. ic3 lbs> (45 + t=M51 Jskl - kt @/CI @/Ct /il /ir lb, lb6c

...

In (4.5) one has for t = 1, ..., T, according to (4.4), that r

ak(M .. kikr)= M l ... k k EMil .ir k11+ r k

ki~~~~~~~~~~~ EMl.y ...kt i... js ki ... kt k j l k- (4.6 + M~1111 r

F. (4.6)

Applying the transformation law (2.1) to 1 as well as to F one obtains from (4.5) and (4.6) that

r - g Mai... ar Fb C (M ba 1

. barC ..CT) +, ...1:: bs cl ....... .CT -Fa`c

ill, 1 b

Mb 1... b .. bs cl ... CT b6 c

- C { l Wkk kk ..**' C/kt )k

t=lclt il ... is ki ...Tktt k. /el Ia/ c

T11j i .. is k t ... 1 l * (CT U C) Ict X /j hir lb, ...

lbs

T s T~~~~~Il r

+ M I .. js' i ... kt t =1 CIt a = 1

le It/k *k t)C ?/il * /ia 11 /ia+l ?a /ir lb / l bs

Ha Ic ik +~i Ia t/c +i'Mk ?/a ?/c + /ac /i ?*/i L%

T s

t= cl tvB= 1 j .. i.

J*-s k, ..kt

x (O (O ~~Vfr W131 . 0) 3f6-1 W, 3,6+i ..( lC1 ICt */l" /ir lb, lbf6-1 lbf6+1 lbs

x (f'f6 I0) @ k kk + Wj) c /b lbf lc likWb,6 c{F k )Wj6lc+ lbficVf'Wib)

Consequently, it suffices to prove that the last two sums vanish. This is obviously true for the last sum.

To show that the second sum vanishes we note the general formula t

0 = t /cl ... CT I/K1 /KT



166 0. E. Barndorff-Nielsen and P. Blaesild

for t > 1, which follows on repeated differentiation of the equation 8 = w5, Vclj. In particular,

Va,wk 0%c fja.f*c uk /c= /ac /i /i,

and from this we find that the second sum vanishes and the proof is complete. A string M satisfying (4.4) is called a string of derivatives (of the tensor Mfp. l ...) .

Such strings are characterized in ? 5. The string of derivatives r of 8 of infinite length is called the canonical string

corresponding to FP1 k2 (or V). In view of (4.3) this string has elements

Fk k2 k3 ... kt = kllk2 k3 ... kt' t=2,3.

Example 2.3 (continued). It follows from (2.4) that for t = 3, 4, ... one has

k =kktkEi = VEk(rl ...ktEi') = Ei

= r-kl kt//kEi,

which shows that r is the canonical string corresponding to V. We conclude this section by giving some simple calculation rules for covariant

differentation of elements of strings. Let M, Qe ?r(T) and let f, ge CO (Q). Then

If Me5?(T) a,nd QeE/'V(W) then (fM (E Q)jl ktlm;k. mv kk *.. kt lk nlk k- ,/p g sk .. . kt/p m.-vn *--w J,~~~~~~~~ M iSk.... .. kt Q m; .. m l. nwl .. (i)

(MG Q)...~~~~~~~~..du1~ej .. Mtr k4//p

From (4.8) one obtains a rule for covariant differentiation of strings that have been obtained by shifting indices. To illustrate this rule suppose that Me 5?r+1(N) and T E sY 0. If Q iS obtained from M and T by lifting one covariant tensorial index., i.e. QeS?"r+1(N) and

then

Qjl j5kl...kt//k~~~~~~ .. Jsl ki...k ktllkTl + Mj (jk T)il ... iski..)t

For the products * and ** we have the following differentiation rules. Let M CS"r(T) and M'QeY"u(T) then

(M * M)il irkl kt/lk - (M'l__//k M'[K+ M QK M',['K2/,k) . (4.10)

Setting (M,Jk)k,... kt = M&k i ..kt//kl

(4.10) may be written succinctly as

(M* M'),,k = MI/k * M +MirMk. (4.11)

Similarly, (Mb**yM')k = MI/k * TM' + Me * M;,k. (4.12)



Strings 167

In particular, if M and M' are strings of derivatives one has

(M*M )JIi, k = (M*-M/)IJiJ Mk MJkMJo k-M M'I' JJ -k - ... ktllk JJ'k, ..... kt k Jk- J'k,...kt -

kl ... ktM Jk (4.13)

and (M ** M')Y9k1 kt//k (M ** M')YJJk1 kt k -MJ k M '[kl - k ktM

(4.14)

5. INTERTWINING

The concept of intertwining gives a correspondence between strings and sequences of tensors. In statistical contexts it has, as will be illustrated in ?6, proved its importance as a tool in efforts to obtain parametrization invariant quantities. The concept of intertwining may also be applied to p-double strings as indicated at the end of the present section, where also strings of derivatives are characterized by means of this concept.

Let jr denote the following set of sequences of tensors

gr= {T- {T.. ~r 00 TV1..r Cj- S s1 ..1 J1 ... In}n= 1 .i . 3s11 ... In s+n}

Given a connection string Fe/'1(oo) with Ti = 8i we establish in theorem 5.1 a one-to-one correspondence between -rY and "r(co) via the relations

mlli"I=r Y Tir Ji Pr1.. I

Si .. sr k1 ... kt T 1KI Tii --S I1 -- T K1 KT (.)

where K = {kV * * kt}

There is little loss of generality in assuming Fikl = 8k. In fact, let F be any connection string with non-singular first element, i.e. with [Ft-] a non-singular matrix, and define F' with elements F' by

kl... kt = L rkl... kt,

where [L>] is the inverse matrix of [Jj]. Then F' is also a connection string and its first element is atk.

THEOREM 5.1. Let FeYl(oo) with Fi = 8k1.

Then (i) if MeC?r#(oo) the formula (5.1) determines recursively a sequence

T - {T>53 that is an element of -rs and conversely

(ii) if Ter.gr the formula (5.1) determines a sequence M = { M.i,rk kr)}t1 that is an element of Yr'(oo).

From the point of view of statistical applications (i) seems to be the most important statement of theorem 5. 1. We speak of the elements of T as the tensorial components (or just the components) of M with respect to F. (In a special case considered by McCullagh & Cox (I 986) the elements of T were referred to as the Mobius derivatives of M.) Furthermore, we say that M, T and F are related by intertwining and we express their relation by writing T = M A r or M = TW F-. The latter formula is just a condensed version of (5.1).




The concept of intertwining was introduced and the conclusion of (i) was proved in Barndorff-Nielsen (I986b) for the case Me YO(oo) and with the additional assumption that M and F were both symmetric. The proof of (i) given here besides being more general is also simpler because of the introduction of the * product of strings and the subsequent observation that r(* r) is a string (cf. subsection 3 e).

For notational convenience, throughout the rest of this section we let A = al a B-b ...b, I = il jr = J=j... j *IA= Va a and O/B = ?/b ... &s 7 etc. With this notation (5.1) becomes

M&klkt= ~~ ~2T&z1 F'I *.*j'T (5.2) MJikl ....... ..... kt E /TIJI, ..IT 1rK- Kr ............. K5.2 ,r= 1K/TK1 K

and by (3.17) we obtain

M?k1kt = ~~~ I(*T~) Iill, I (5.3) MI kl. kt = ETJ1 IT k ...kt

Proof of theorem 5.1. Suppose that Me yr(oo) is given. To prove the first statement in the theorem we shall show that for t = 1, 2, ... one has

=TTA1 AkVB/OIJl.k wOkt (5.4) TBC, . .. ct = TJikl .. . kt */ I I)B /C1 ...l * I Cctt (54

This is done by induction after t, the number of elements of C = {cl, ..., ct}. Clearly (5.4) is correct for t = 1. Rewriting (5.3) and using the induction hypothesis and the fact that p(*T) is a (T,O) string (cf. ?3) one obtains

t-1 TA - A _ vTA -(* T) di. d, BC1 ..Ct = BC1 .. Ct Bell--- dTr C1 ............ CtT

T =1

t t-1

= E E MIJkl kT/1 */ Tl TIC1 IT N** i /T

T= 1 CIT T

t~~~~~- ( rkl ... #CkT 4))C /r . h ..?k l . ER* ?(5. (5)

=- E \ E MJkl ....... . kt, TI1Jl, .. IT kl... klC T /Ci I /C */I I/B ..........................(5)

By (3.17) the term in the parentheses in (5.5) becomes min (C, t-i)

MJkJ ... kc E T'i... IT pkli... kg

which, according to (5.3), vanishes for = 1, ..., t- 1 and is equal to Tkl. kt for = t. This proves (5.4). To prove the second conclusion in the theorem suppose Te gr is given. From

(3.17), (5.3) and (2.1) applied to elements of r(*T) it follows that

MA - d T) di...c d MBC1 . .. Ct =E TABdl . .. dT rC1 . .. Ct

= TJI I z rl .... T @/C @ @ @ (I)IC 71>/l @/B =- II . 7 kp T...klT (okicw...kOC rVfIAwJ

T = 1 = a



Strings 169

t t

aE-1 Ea rz TJI1 I-VT Ik .kT (Oki . .. (O/ k, A

(s/B t

a-1 C/r {r-1 Jll . . . IT rkl ) kl *.* IT} k Ck A B CT{T=1 &1 T1C1 1 T W... 0<)

V<' ?w(

a-1lClo- T=l *** ... ko)/ IC IC *l II)k IB

M iki .. 0<Ih I ~j = 1 C/er

which shows that the elements MIk1 kt' defined by (5.2), constitute a string. U Formula (5.2) (or (5.3)) gives a direct expression for M in terms of T and F. The

following theorem provides an inversion of this formula, i.e. it gives a direct expression for T in terms of M and r. To formulate the result we introduce the following quantities for an ordered set of integers ml < M2 < ... < Mff

T-i

r(M 'MT-)kll ... km71 = J7 F(*mp) kp, kpmp , (5.6)

where F(*mp) is given by (3.17). Thus one has, for instance,

="1 ,3 n1 ni IF(* 2)Mlm2JI'(*3)111213. ki k2 k3 k4 ml M2 11213 k 1k2 k3k4

With this notation we have THEOREM 5.2. T = MA r is given by

t-1 t-r

TJI1tt=MJ... It it... It + EJkl kTE(......... k) r(lY-mT) l - - T 1= (r=Ml<M2...<MIT<t)

(5.7)

where the last sum is over all sets of integers (ml, ..., m,) such that

T =ml < m2 < ...< m <t.

In the proof of theorem 5.2 we use the notation Z[,T;,t] for such sums. Proof. Formula (5.7) is shown by induction after t. Clearly (5.7) is correct for

t = 1. From (5.3) one has

t

TII ..It I = MIJI1 . It I- TIJI'.. I, T *T 1 ,. It I

and from the induction hypothesis and (5.6) it follows that

i 1t, .. 1 t I M&kl .... k r . It T = 1

t T-1 r-C - z M Jkl ....... kC E (-1)i ..T E "(Ml, Mjpr) k kC (5.8) T = 2 1 1T = 1 [7T; ,T]




Interchanging the order of summation the last term on the right-hand side of (5.8) turns into

t-1 t r-C

E M- kl .. . kC E E (-1 ) J E jmi, M, mT) ki... kc =T=C+1rT= 1 [7r;9,r]

t-l1t- t

MI1kl ..kCrE1(_ )Y E z n(m,909 m7pr) ki III..... kc C- r-1 r=7T+[7T; ,Tr]

MIJ M .. k C (1)+ pi1,mn..)....

t-l (t+l)- = E M kI k (-... k p(m ...

n= 2 [;, t+1] i

The proof is now easily completed. a

As a corollary of theorem 5.2 we have that if M and r are symmetric then the same is true of the tensorial components of M with respect to r.

Let Me gr(T) be a string of finite length. Then the elements of M are of the form (5.2) for t = 1, ..., T and for some finite sequence of tensors {T&I1 .T.

Extending the latter sequence arbitrarily to an element in gS one obtains, by (5.2), a (r, s) string of infinite length whose first T tensorial elements are those of M, i.e. M may be extended in a variety of ways to an element in rc(oo).

Let P denote the tensorial components of Mc Yr with respect to P - 91 and let G denote the components of P with respect to another connection string F, for instance a canonical string. It is then easy to show the following relation between T and T, the components of M with respect to r:

Vill..It = E TJdlE dGrL1 CTGLT (5.9) r = 1 LTr

where L = {11, ..., lt}. Comparing formulae (5.2) and (5.9) it is tempting to say that T is obtained by intertwining the two sequences of tensors T and G and to write P = T O a. With this terminology the result may be expressed in the following way:

T=MAF, G=FAr=:T=TAG (5.10) or, equivalently,

MAF= (MAr)A(FAr). (5.11)

By formulae (4.7) and (4.8) and theorem 5.1 strings of derivatives, introduced in ?4, may be characterized as follows. A string M E gr (oo) is a string of derivatives if and only if the tensorial components of M with respect to the canonical string r equals the sequence of tensors obtained by covariant differentiation of the tensor

M1 To see this, let

MJkl . kt z S/ T1 PI K1 PT (5.12)



Strings 171

From (4.7) and (4.8) it follows that

t MIJk I ... kt//kt+l 1 IT{ Jl .*--IT//kt+l K1 KT

+ S TJI *--IT K, (Kli- U kt+,) KT ( )

Writing

TJI... T1kt+1 Ti JI... IT//IT+1 kT+1

it follows that M?kl...kt+l = MJkl...ktl/kt+l t = 2 ... if and only if TJ1...1T//IT+1 =TJ1...1T+1T = 1, 2, *.. Because TJI1 = M-I1 one has that M is a string of derivatives if and only if TJI1 I = MJI1//I2 I .

We conclude this section with some remarks concerning intertwining of p-double strings. Let M be a double string of degree (r, s) and length (oo, oo) and let r and rbe two connection strings both having the Kronecker tensor as the first element. Then the formula

t t

r-1 K/T ..1 t/T K. .. I K K l (5.14)

determines uniquely a sequence TJI1 .I .It, t, = 1, 2, ..., of elements in +t+t This sequence T is said to be obtained by intertwining M and (r, 7V) and we write T = M A (F, F). Similarly, a p-double string may be intertwined with p connection strings. Finally, we note the obvious results that

T= MA F, T = M'A'= T?(T' = (M?M')A(F, Tv), (5.15)

T=MAr, T'=M'AF= *T'=(M*M')AF (5.16) and

T=MAF, T'=M'AF=T**T'= (M**M')Ar. (5.17) Here

(T * T')II11 = T4 T'fL, (* JJ'Il ...It = E TJL1T JL2 L/2

= T1 T'IL, (T** )JJ/... It = E; TIJ L1 L21 L/2

where L = {l, It}.

6. STATISTICAL EXAMPLES

We shall now illustrate the concepts and results discussed above in the particular context of likelihood based statistical inference. For this we adopt the notations used in ??3 and 4 of Barndorff-Nielsen (I986b). Thus v and A will indicate, respectively, joint moments and joint cumulants of derivatives of the logarithmic




likelihood function I of a parametric statistical model X# with model function p(x; l). For instance, we have

vi j= E{li I}. -E{lij} =vi,

Pi jk= E{li ljk} Vi, j, k, I = E{li lIj k 11} and

Ai J = K{li, lj} = Vi, Ai, jk = K{li, tjk}

A. j, k, I = K{Ji ' lk ll} ( = Vi j k I-Vij Vkl[3]),

where E and K indicate mean value and joint cumulant, respectively, and where

'k, ... kt =llk, ... kt =

ki . .. akt 1

We then (Barndorff-Nielsen I986 b) have that for each real a a costring ",v e? is defined by

I + + v

2 2 where

Pi .s ki. kt js, ki... kt (6.1)

and k

i,...s ki...kt K/ T (6.2)

with K = {k1, ..., kt} and K1, ..., KT constituting a partition of K into T blocks. In particular, (6.1) and (6.2) are costrings. For s = 1 and t = 2 the 'v are equal to the Riemann-Christoffel symbols of the a-connections introduced by Chentsov and Amari, see Amari (I985).

As a new example we now show that the same statement is true if throughout we substitute v by A, i.e. moments by cumulants, thereby obtaining strings aA.

For this we need the fact that cumulants have the following well-known multilinearity property. Let y, = a4 xji,, j = 1, ..., m, be m linear combinations y. of random variables xii, the a4 being non-random coefficients. (As indicated by the notation, ,i is an index whose range may depend onj. Furthermore, by the Einstein summation convention, in the expression a3i xji. summation over the index ij is understood whereas j is kept fixed.) The joint cumulant of y1, ..., Ym then satisfies the multilinearity relation

K{all x11i, ..., aim xmim} = ali4 ... ai Kx .., x (6.3) Now, let

1Ajl.1s8ki ..kt = Ajl js kt ...kt

-1'i . ski ... kt = z Aiq... js8Ki1...9KT r K/Tr

and

2 1A+ 22-1A. Because E{ll} = 0 we have

and hence 1A E 9 0. Thus it suffices to show that - 1A E OS.



Strings 173

First we note that the arrays of cumulants of logarithmic likelihood derivatives given by

AK', ..., KP,

where the K", r = 1, ..., p, are sets of indices, constitute a p-double string. This follows immediately from (6.3).

From this we find, in an obvious notation,

t

1bi...bsc.l . ct E A..bit ... bs, C1 ... , CT

{

t Ici ICTI

Clr -1 CllA 'Tr 1 CT/,p 1 1* k

x w)kii ~kip i kA.. Wtk T99 (0) 1 ...w(1 s /C11

... /C199 1 I /CTl IC },J lb * lbs

b=lIKl

{ E E E E K ..., KY /CI /CC lb1

lbs 1II= 9Clgy=1 K/yl j

w1)ki .. w)kc w01i &s) { ( z -1Ajll . J,s, kl ... kC /C * /CJ /b.. Ibs'

which proves that -'A e? Now, let a be an arbitrary statistic such that (c2), a), where (2i denotes the

maximum likelihood estimator of ,), is a smooth one-to-one transformation of the minimal sufficient statistic for the model S. Viewing the logarithmic likelihood function I as a function of w) and (Ct, a), i.e. I = l(w); Ct), a), and letting aj = 8/a

we shall consider the mixed logarithmic model derivatives

A 1 ... ip;j.. tail ...

aip aji...3

a I

where t is the operator that substitutes c by c). As stated in Barndorff-Nielsen (I986b) the particular mixed logarithmic model

derivatives

Fki ... kt ;j (6.4) and

tIj; k, ... kt (6.5)

each constitute a costring of degree one, so that for any real a a costring aJof degree one may be defined by

tjk ... kt F'ki ... kt; j+ 2 j k... kt (6.6)

The three types of string in 9J0 given by yv, YA and ayJare rather similar. However, -A is more closely analogous to ayJthan is yv; more precisely -A and ayJare the same, but different from av except for a = 1, when X is a (d, d) exponential model (in the sense of Barndorff-Nielsen (I980)).

There is no quite immediate extension of (6.6) to costrings of arbitrary degree



174 0. E. Barndorff-Nielsen and P. Blhesild

s. However, a rather natural generalisation may be obtained as follows. Let [i 3;]

be the inverse matrix of [%KA;j] the latter being the observed information tensor on./X. Intertwining the connection string

klc...kt rFi;ki... kt

with the scalar string of logarithmic likelihood derivatives 1k. kt we obtain the tensorial components of the latter that are denoted by a'hjl = .. (Ci; ), a) (where (7) is the maximum likelihood estimator and a is an auxiliary statistic). We now define multiarraysXA, ... iski ... kt by

L'x- ask kh tjkl kthi1...18. (6.7) These arrays constitute a costring of degree s.

In particular, for s = 2 we have

hjl j2 = ljl 2 ;j2; jt 1

and hence ijlj2 ki ... kt ki ... kt = tV1J2 J Vi ki ... kt

For t = 1 this yields

aj)02 ki 2 'Til 2 ki n

where Ji1 j2 ki =

/lJ2; kiF kl; jilJ2

In fact, T is a symmetric covariant tensor of degree three, the so-called observed skewness tensor (cf. Barndorff-Nielsen (i986a, I987)). For t = 2

Kjlii2 ki k2 = ijl2; ki k2 j2; A; k, k2. (6.8)

For a = 1 this equals '1l 12; ki k2' a covariant tensor of degree four which occurred, in Barndorff-Nielsen (i 986 b), as part of a decomposition of the 'observed Bartlett adjustment' into four invariant terms. While the tensorial nature of 4Mil2; k, k2

originally appeared to be a rather fortuitous fact this can now be explained in terms of intertwining of double strings and connection strings, as discussed in ?5. Specifically, the mixed logarithmic model derivatives

,Yki . .. kt ; 11. ... Ilu

constitute a double string, of degree (0, 0), and intertwining of this with the pair of connection strings

( Xfkl k2 ...- kt;1 fi t j; kl . .. kt)

shows that as a four index array (6.8) is a covariant tensor. Also occurring in the observed Bartlett adjustment are the tensors frl J2 h

and

A(h h2 jdefined as the a = 1 and s = 3 respectively s = 4 cases of



Strings 175

For general a one finds, after some algebra,

1 1'2 = 0 2,

a_3a- otyjl 2 2 1l32 33'

oyl J2J3>4= (2a- 1) (Yjilj23 j4 + jljj2 j3;j4[4]) + (x-1 ),Vjlj2;134 [6]

+ ;j,(1-a) (5a-l)} 3] +(I +a) (3-5a) J3 l 4 ri;JlJ22yi';j3J4[ 4 161 J2 ;tth J4;t

5ax2 - 2a +1I + 4 ti;j j2 'yj3; J6]

In particular, it may be noted that y1lJ2j3

= 7TjlJ

23 and that

31J2 j3 = 0;

the value a'-= is related to the question of logarithmic likelihood symmetrization, cf. Kass (I984).

Because of the particular simplicity and importance of the costring ,Vk1 kt ;j it is pertinent to ask whether this is obtainable by successive generalized covariant differentiation as defined by (4.2). To settle this question we enquire whether 1k1k2k3; J can be obtained from Ykl k2;j by applying (4.2) with some connection F, i.e. we wish to know if there exist Riemann-Christoffel symbols 1 k, such that

Ykl k2 k3; j =k3ykl1k2;j }k3Pklk2;i

or, equivalently,

Jkj k2; jk3 =Fk3J ki k2; P

It is evident that this is not the case. In conclusion we propose a statistical example of an infinite string of degree

(0, 0) and of weight one, cf. the definition of strings of weight w, given at the end of ?2. Consider the model function p(4, co I a) for the conditional distribution of the maximum likelihood estimator 6 given an ancillary statistic a, and let

ik ... kt(), a) tki ... ktp(w; w I a). (6.9) where t is the operator that substitutes co by w. For fixed value of the ancillary a, the model derivatives ik. kt1 t = 1, 2, ..., constitute an infinite (0, 0) string of weight one. In particular, ik is a (0, 1) tensor of weight one.

7. EXTENSORS

The concept of extensors was referred to at the end of ?1. As a prototypical example of this concept we may consider an extensor of excontravariant order one, excovariant order one, contravariant order one and covariant order one, which is a multiarray Ti AIk satisfying the transformation law

b. -d 3 Al (K) kIK) (A i A-)




here y, 8, K, A denote non-negative integers, and Vf,(n) and W,(n) denote the nth order derivatives of Vlfc and wd with respect to the parameter 0 of a one-dimensional smooth curve W in the manifold X, the quantities V/fc and )d being considered as functions on that curve.

Example 7.1. Suppose that Tk is a (1, 1) tensor on X and define T 'Kk by Kk (K) Tk(KA)

where

Tlk(n) =_Tk(c)0 I=o T(0w(0)).

Then the quantities T AIk constitute an extensor of excontravariant order one, excovariant order ohe, contravariant order zero and covariant order zero.

There is a formal similarity between extensors and strings in that both kinds of quantities possess two types of indices, one type behaving as tensorial indices and the other relating to repeated differentiation. Moreover, an important function of extensors as well as of strings lies in the construction of useful tensors (cf. Craig (I943), McCullagh & Cox (I986), Barndorff-Nielsen (I986b) and also ??5 and 6 of this paper). However, a more substantial link between the two concepts appears not to exist.

REFERENCES

Amari, S.-I. I985 Differential-geometric methods in statistics. (Lecture notes in statistics, vol. 28.) Heidelberg: Springer-Verlag.

Barndorff-Nielsen, 0. E. 1980 Conditionality resolutions. Biometrika 67, 293-310. Barndorff-Nielsen, 0. E. i986a Likelihood and observed geometries. Ann. Stati8t. 14,

856-873. Barndorff-Nielsen, 0. E. i986b Strings, tensorial combinants, and Bartlett adjustments. Proc.

R. Soc. Lond. A 406, 127-137. Barndorff-Nielsen, 0. E. I987 Differential and integral geometry in statistical inference.

Differential geometry in 8tati8tical inference, IMS Monograph. Institute of Mathematical Statistics, Hayward, California. (In the press.)

Craig, H. V. I937 On tensors relative to the extended point transformation. Am. J. Math. 59, 764-774.

Craig, H. V. I943 Vector and ten8or analy8i8. New York: McGraw-Hill. Eisenhart, L. P. 1926 Riemannian geometry. Princeton University Press. Kass, R. E. I 984 Canonical parametrizations and zero parameter-effects curvature. Jl R. statist.

Soc. B 46, 86-92. Kawaguchi, A. I939 Eine Verallgemeinerung von Extensoren. Mh. Math. Phy8. 48, 329-339. Kawaguchi, A. 1940 Die Differentialgeometrie hoherer Ordnung I. Erweiterte Koordinaten-

transformationen und Extensoren. Jour. Fac. Sci. Hokkaido Univ. 9, 1-152. McCullagh, P. & Cox, D. R. I986 Invariants and likelihood ratio statistics. Ann. Stati8t. 14,

1419-1430. Morey, P. S. & Craig, H. V. 1972 A generalization of extensors. Ten8or, N.S. 26, 243-249. Thomas, T. Y. I96I Concept8 from ten8or analy8i8 and differential geometry. London: Academic

Press.



Documents

Strings: Mathematical Theory and Statistical Examples