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INTERNATIONALIMS-ILAS WORKSHOP ONMATRIX METHODS FOR

STATISTICS4 & 5 Decemb er \992

University of AucklandNew Zealand

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INTERNATIONAI. IMS-ILAS WORKSHOP

ON MATRIXMETHODS

FOR STAT'ISTICS

Auckland, New Zealand

4-5 December 1992

ABSTRACTS

Characterizing Invariant Convex Functions of Matrices

James V. Bondar and Alastair J. Scott

Carleton University ancl University of Auckland

Some years ago, C. Davis (1957) showed that the invariant convex functions on theclass of symmetric matrices are precisely the functions that are convex increasing functionsof the eigenvalues. (Here "invariant" rneans invariant under the similarity transformationO --+ OMO'where O is orthogonal.) In this talk we show that the invariant convexfunctions on the class of all matrices are precisely those func.tions of the singular values

that are both convex and weak Schur convex. (Here "invariant" means invariant underthe transformation M -+ OtMOz where the O; are orthogonal.)

It has been known for some tirne that increasing symrnetrix convex func.tions of thesquares of the singular values are rnatrix convex. Bondar (1985) strengthens Schwartz'results to increasing sylnmetric convex functions of the singular values. Here we show

that to be a convex, weak Schur convex function of the singular values is necessary andsufficient for matrix convexity. We also derive sufficient conditions that are simpler toapply in practice.

ReferencesJ. Bondar (1985). Convexity of acceptance regions of multivariate tests.Linear Algebra Appl.,67: 195-199.

C. Davis (1957). Al1 convex invariant functions of Hermitian matrices. Arch. Math.,8: 276-278.

Application of a Generalizecl Matrix Schwarz hrequalityEvaluation of Biasecl Estimation in Linear Regression

J. S. Chiprnan

unive rs i\o l}fi['..' ", "

In the general linear regression rnoclel, three estiurators are considered: (1) the Gauss-

Markoff estimator; (2) the Gauss-Marlioff estirnator subject to a set of linear restrictionson the regression-coefficient vector; (3) the Theil-Goldberger estirnator with uncertainlinear restrictions corresponding to the certain ones of (2). This third estimator is styiedtlre generalized ridge estimator. It is shown, generaliztng a result of Toro-Vizcarrondoand Wallace, that a sufficient condition for both estimators (2) and (3) to have lowermatrix-mean-square error than estimator (1) is that the noncentrality parameter arisingin the F-test for the restrictions in (2) should be less than 1. A generalization is also

obtained of the Hoerl-Kennard result that for sufficiently small but positive values of theirl.-parameter, estimator (:3) has lower matrix-mean-square error than estimator (1).

References.L S. Chipmarl, "On least squares with insufficient observations," J.A.S.A.59 (1964),

1078-1111.

A. E. Hoerl and R. W. Kennard, "Ridge regression: biased estimation for nonorthogonal

problerns," Techtrometric-s 12 (1970), 55-67.C. Toro-Vizcarrondo and T. D. Wallac€, "A test of the rnean square error criterion for

restrictions in linear regression," J.A.S.A. 63 (1968), 558-572.

Statistical Contribr.rtions to Matrix Methocls

Richard Wiiliani Farebrother

University of Manchester

The Formalisation of Matrix Algebra in the 1850s and 1860s was preceded by a numberof significant developments, sorne of which (*) are to be found in statistics works:

1. Cauchy's use of rnodem subscript notatiou for matrix eiements.

2. Gauss's use of matrix rnultiplication when trausforming variables in a quadraticform.

3.* Mayer's prototype of the elirnination procedure.

4.* Gauss's identification of a iinearly dependent systern of equations.

5.* Gauss's elirnination procedure and the implicit LU decomposition of a square matrix.

6.* Clauss's definition of a generalised inverse of a tnatrix.

7.* Clauss's derivation of a LDL' decomposition of a symmetric matrix.

8.* Gauss's least squares updating fortuulas.

9.* Gauss's proof that the inverse of a symrnetric matrix is itself syrnmetric.

10. Cauchy's latent value decomposition of a 3 x 3 rnatrix.

11.* Laplace's derivation of an orthogonalisation procedure subsequently rediscovered byGram and Schmidt.

12.* Donkin's definition of an orthogonal basis for the orthogonal courplement of a rna-trix.

13. Jacobi's characterisation of the least squares solution as a weighted sum of subsetestimates. This ciraracterisation of the Least Squares solution as a weighted sum ofsubset estimates by C)laisirer and Subrahrnaniarn.

Later Contributions Include:

14.* Thiele's discussion of the canonical form of the linear model.

15.* The Singular value decomposition derived by Eckart and Young.

i6.* The so-callecl Gur.r-Mu.kov theorem is clue to Ciauss alone.

17.* The generalisation of this theoren usually attributed to Aitken would seem to be

due to Plackett.

Stationary Distributions and Mean First Passage Times inMarkov Chains using Generaltzed Inverses

Jeffrey J. Hunter

Massey University

The determination of the mean first passage tirnes in finite irreducible discrete tirneMarkov chains requires the computation of a generalized inverse of .I - P, where P is thetransition matrix of the Markov chain, and also knowledge of the stationary distributionof tlie Markov chain. Generalized inverses can be used to find stationary distributions.Tire linking of these two procedures and the cornputation of stationary distributions usingvarious multi-condition generalized inverses is investigated.

Cornbining Inclependent Tqsts For A Cotllmon Mean-lA" Ap'irlication of the Parailel Surn of Matrices

Thomas Mathew

Departrnent of Mathematics and Statistics' UniversitY of MarYlanclBaltimore CountY

Baltimore, MarYl an d 21228U.S.A.

I. the co.text of the recovery of inter-block inforrnation in a BIBD, the problem of

conrbining two inclepenclent tests is addressecl in cohen and sackrowitz (1989, Journ'al of

the. Ameico, Stotirtical Association). Apart from the intra- and inter-block F-tests, their

cornbi.ecl test uses a correlation type statistic, and this statistic has a positive expected

value whe' the null hypothesis of equality of the treatment effects is not true' We show

that a similar statistic .or b" definecl in the context of hypotiresis testing for a common

lnean i' several indepenclent linear models. Using the properties of the parallel surn of

matrices, we also sirow that this statistic has a positive expected value wheu the null

hypothesis is not true.

Invariant Preorcleriirgs of Matrices ancl Approximation Problernsin N4ultivariate Statistics ancl MDS

Renate Meyer

Institute of Startistics and DocurnentationAachen University of Technology, German.y

Tlre singular value decomposition of a complex n x k -matrix A : Dl=, og4(A)u;vf, witho1r1(A) 2 "pt(A)the Hennitian matrix A * A, plays an important role in various multivariate descriptivestatistical nethods, as for exarnple in principal components and analysis, canouical corre-

lation analysis, discriminant analysis, and correspondence analysis. It is applied to solve

the key problem of approximating a given rz x k-rnatrix (1 < k < ,) by a matlix of lowerrank r < k. Define A1,; : Di=rop4(A)u;vf . The minimum norm rank r approximationresult

,b@ - At,l) < 'i(A - G) for all G with rank (G) ( r, (0 1)

was obtainecl by Eckart ancl Young (1936) for the Eucliclean norm $(A: {tr,O.e.1}. Thesolution A1,; turned out to be "robust" with respect to the choice of the approxiurationcriterion, as it was extended by Mirsky (1960) to the class of unitarily invariant norms.

The objective in the first part of this paper is to generalize this result to an even largerclass of real-valued loss functions, encolrlpassing the unitarily invariant norurs. Certainpreorderings { of complex rnatrices, that occur naturally in the context of multivariatestatistics, will be considering in proving uniuersal optimality of A1,;, i.e.

A-Ar,r < A-G forall Gwith rank (G) Sr. (0.2)

Of course, this irnmediately entails (1) for ail real-valued {-monotone functionsTy'.The second part treats the problem of approximating a Herrnitian n x n matrix by

a positive-sernidefinite rnatrix of given rank, which is of major relevance in the contextof multidimensional scaling (MDS). Thereby, the hitherto rnost general result of Mathar(1985) is extended and further universally optirnal properties of the classical MDS solu-

tion are provided.Iiey uords: singular value decornposition, group induced preorderings, invariant order-

ings, weak majorization, principal components analysis, rnultidimensional scaling.

References

Eckart, C., Young, G. (1936). The approximation of one matrix by another of lower rank.P.sy ch o m etrik a L, 2Il -218.'

Jensen, D.R. (1984). Invariant orderingIrtequalities in,9tat'istics an d Probability.,Ca.,26-ii4.

and order preservation. In: Y.L. Tong, Ed.,histitute of Mathematical Statistics, Hayward,

6

Mathar, R. (1985). The best Euclidean fit to a given distance matrix in prescribed dimen-sions. Linear Algebra Appl. 67, 7-6.

Meyer, R. (1991) Multidirnensional Scaling as a frarnework for Correspondence Analysisand its extensions, in: M. Schader, Ed., Analyzirtg and Mod.elin.g Data and l{nowledge,

Springer, 63-72.

Mirsky, L. (1960. Symmetric gauge functions and unitarily invariant norms. Quart. J.

Math. Oxford, ,9er. 2,11, 50-59.

A Cornpact Expression For Variance of Sample Second-orderMoments in Multivariate Linear Relations

Heinz Neudecker

Department of Actuarial Science and EconometricsUniversity of Amsterdam"

The Netherlands

Satorra (1992) considered the random (prl) vector

, :iBr;; * tr,i=7

where the random (n;rl) vectors 51 are independent with rnean E(6) : 0 and varianceD.(5;), and B; are LL are coustants

(i:t...nr).

He derived the variance of u(zz'), where u(.) is the short version of vec(.).It is the aim of this note to give a compact expression for the variance and subsequently

derive Satorra's result.

Matrix Tiicks Related to Deleting an Observation in the GeneralLinear Model

Markliu Nurhonen and Simo Puntanen

Departrnent of Mathematical Sciences- UniversitY of TarnPere

Consicler the linear motlel U, X0,V, where X has full colurnn rank and V is positive

definite. when estimating the pararneters of the model, it is natural to consider the

consequences of sorne changes or perturbations in the data on the estimates: regression

diag'ostics rneasure these consequencies. One fundamental perturbation is omitting one

o, ,",r"rul of the observations from the model. In this paper our interest focuses oll some

helpful matrix forrnulas while studying regression diagnostics.

The full CS-cleconrposition of a partitionecl orthogonal tnatrixl

Chris C. Paige

Sc;hool of Computer Science, McGill UniversityMontr6al, Qu6bec, Canacla H3A 2A7

The C,9-decomposition (CSD) of a 2-block by 2-block partitioned unitary matrix Q :

(fl:rt, 3::r) reveais the relationships betwee' the si*gular value c{ecornpositions (svDs)

of eaclr of the 4 subblocks of Q. The CSD shows each SVD has the form Ql; : LI;D;iVf ,

lor i, j :7,2, where each [.I; and I/i is unitary, and each D;1 is essentially diagonal. Here

we give a simple proof of this which has no restrictions ou the dirnensions of Q11.

The CSD was originally proposed by C. Davis and W. Kahan, and is important infinding the principal angles between subspaces (Davis ancl Kahan, Bjorck and Golub),such as in cornputing canonical correlations between two sets of variates. It aiso arises

in, for example, the Total Least Squares (TLS) problern. The relationships betrveen the 4subblock SVDs (in particular the way that each unitary matrix U; or V1 appears in 2 dif-ferent SVDs) has macle the CSD a powerful tool for providing sirnple and elegant proofs

of many useful results invoiving partitioned unitary matrices or orthogonal projectors.Here we also show the CSD makes several nice rauk relations obvious, aud can be used

to prove some interesting results involving general nonsingular matrices.I(e.y words: CS decornposition, unitary tnatrices, tank relations.

lJoint Research withShanghai 200062, China.

Musheng Wei, Departmenl

10

of Mathematics, East China Normal University,

How Not to use Matric,es when Teacliing Statistics

D.J. SavilleNZ Pastoral Agriculture Research lnstitute Ltcl

andG.R. Wood

Department of Mathematics, university of canterbury

Hic,lclen in the appenclices of some statistical texts is a geometric apProaclt to analysis

of variance ancl r"gr"rsion. This approach has been large un-used in teaching for two

reasons. Firsi, the approach has been considered too hard, and second, this century has

been a period in which algebraic methods have been dominant. It is the aim of this paper

to show that the first reason offered is nonsense: the geornetric approach can make things

easy.

How cioes the geometric methocl rnalie things easy? The matirematical framework

necessary for the geometric cievelopment of analysis of variance and regression can be

reducecl to a halclful of straightforward vector ideas. Once these are learnt a concrete

methocl is followecl i1 any situation. Three objects are isolated: the observation vector'

the 'roclel

space ancl hypothesis clirections. All lie in a finite-climensional Euclidearl space.

Two routine processes then serve to fit the rnodel and test hypotheses: to fit tire rnodel we

project the olservation vector onto the model spac.e) while to test hypotheses we cornpare

averages of squared lengths of projections'

Why bother to clo ii.ingr clifferently? Conventional uretirods fail to convey a satis-

factory unclerstalcling of the principies which unify these basic statistical methods. The

cookbook approach r"n uin, mysterious, while the matrix approach is accessible only to

tlrose with lrathernatical rnaturity. The geometric approacir provides at elemen'tary b:ut

rigorouspath through the rnaterial. It has been used successfuily by the atrthors to teach

,"'.ond-y"ur stuclelts in statistic.s as well as postgraduate stuclents in the applied sciences'

These icleas will be cliscussecl, ancl an example of the method in actiou presented.

1l

Problerns with clirect soiritions of the normal eqrrations forrlon-paratnetric aclditive uro clels

M. G. Sc:hintek

Medical Biornetrics Group, University of Graz Medical Schoois,A-8036 Graz, Austria

To stucly the funciioual clependence between variables of a multivariate regression

problem in a data-driven fashion, non-pal'alletric techniques like Generalized AdditiveModels (Buja, A., Hastie, T. and Tibshirani, R.: Linear smoothers and additive rnodels.

Artn. Statisf. 1989, 17,453-555) are very useful. Instead of solving the associated uonnalequations a Jacobi-type iterative procedure called backfitting is usually appiied. This

is prirnarily done for the purpose of computational efficiency and to avoid singularityproblens. But little is known from a theoretical and even less frorn a practical point ofview about the quality of the obtained results.

Let us observe (d + l)-dirnensioual data (r;,X) with z; : (r0r,...,r;a). The

r1j,...tr,,.j represent indepenclent observations drawn frorn a randorn vector X :(*r,,...,i0)anclther;anclXfuifilY;:g(r;)*e;forI<i(zr,wireregisallurl-known smooth function from Rd to ft, and €1r... )en are independent errors. An additiveapproximation to g

d

s6)veo+fsi,xi)j=1

is aimed at, where 96 is a constant and the !;s are any srnooth functions. To make these

functions g; identifiable it is required that E(g16)): 0 for I < i ( d. For instance we

can apply thepopular cubicpolynomial srnoothing splines. Let Sr: (1+)A.1ifr)-1 bethesnroother matrix (operator) and 1(4 a penaity rnatrix of such a spline. For l. :1,2,...,dthe normal equations form a (nd) x (n d) system

1) [) :I:::,9a ,94 ,94

L9, ,91

szlS,SflSza

Sau

denoted by Pg: QA, where P and Q are block matrices of smoothing operators ^9r.

One way to solve the system wouid be to apply the rnethod of successive overrelaxatiotl,

but for d > 2 we do not l<now how to choose the relaxation parameter. Instead we propose

taking advantage of the specific block structure (especially the position of the /s) of P. Itallows us to clerive a recursive scheme for the calculation of P-1. IJnder non-singularitya direct solution can be obtained by 0 : P-t Qy. This approach is cheap but depends on

tlre chosen scatterplot smoolher. 12

In case of singularity we propose another approach based on a specific type of Tichonowregularization assuming measurement errors of tire dependent variable. Let us have thesingular systern Pr : Qa. Let a be a regularization parameter, then ihe disturbed systemtakes tlre forrn (P*P * al)r - P*Qy.The deviation lli - rll can be estimated in termsof 7/a and the measurernent error of the right-hand side of the equation. The system can

be solved by standard techniques but is cotnputationally expensive.

Research supported by Austrian Science Research Fund trant P8153-PIIY,

i3

On the Efficiency of a Linear Unbiased Estimator and on aMatrix Version of the Cauchy-Schwarz Inequality

George P.H. Styan

Department of Mathematics and Statistics,McGill lJniversity

Montr6a1, Qu6bec, Canada H3A 2A7

We consider the efficienc.y of a linear unbiased estirnator in the general linear model and

its conlection to a determinant version of the Cauchy-Schwarz inequality; we illustrate

our results with an example from simple linear regression.

I4

Alexander Craig Aitken: 1895-1967

Garry J. Tee

Departrnent of Mathernatics ancl StatisticsUniversity of Aucklancl

Alexander Craig Aitken was bom at Dunedin on 1895 April 1, and he attended Otago

Boys High Scirool. On holiday at his grandparents dairy falm on Otago Peninsula in

1904, he discoverecl the norv-famous breecling colony of the Royal Albatross at Taiaroa

Head. His Calvinist grandparents punished hirn for telling such. an unlikely tale but later

o1e of his uncles was appointed as a Ranger to protect the colony. After 2 years at the

University of Otago he joined the army and was severely wounded at the Battle of the

Somme.

.He completed his studies at the University of Otago, and graduated NI.A. in 1919 .

In 1920 he rnarriecl Mary Betts, who lec.tured in Botany at Otago l-lniversity. He taught

at Otago Boys High School until 1923, when Professor R. J. T. Bell persuaded him tostudy at the tiniversity of Edinburglt, where he spent the rest of his life. Professor E. T.

Whittaker asigned hint the problem of smoothing of data, which had practical iurportance

i1 actuarial work. His thesis was of suc.h merit that he was awarded the degree of Doctor

of Science, rather than Ph.D.Aitken s mathentatical rvork was devoted mainly to nurnerical analysis, statistics and

linear algebra. He founded the renowr ed Oliver & Boyd series of textbooks and wrote the

firsb tlvo himseif: both Determinants and Matrices and Statistical Mathematicsare recognized as classic textbooks. In numerical analysis he devised many methods which

exploited the capabilities of the calculating machines which were then available, and which

have proved to be funclamental to rnuch later work in scientific cornputing. He gairrecl

wide fame as the greatest mental calculator for whom detailed and reliable records exist.

lVhittaker retired in 1946, and Aitken, without any move on his part, was elected to

the Chair of Vlathematics at Edinburgh. He was elected Fellow of the Royal Societies of

London and of Edinburgh, and Honorary Fellow of the Royal Society of New Zealand, of

the Society of Engineers, and of the Paculty of Actuaries of Edinburgh. He was awarded

Honorary Degrees by the Universities of New Zealand and of Glasgow, and he was awarded

several rnajor prizes in mathematics. He was an inspiring lecturer and an extremely gentle

person, intensely devoted to music, and ire wrote sorne poetry of distinction.In 1963 he published his mernoir Gallipoli to the Somme: Recollections of a

New Zealand Infantryman, which was acclairned as a classic account of death and lifei1 the trenches. The Royal Society of Literature elected hiur as a Fellow, in recognition

of his achievernent in writing that memoir.Aitken retired in 1965 (in poor health), and ire died at Edinburgh on 1967 November

3, aged 72.

15

Use of permatlettts ancl their analogues.inmr.rltivariate galnltta, binonlial and negative

the represeutatiou ofbinonrial distriltutious

D. Vere-,Jotres

Institute of Statistics and Operations ResearchVictoria University of Wellington

Wellington, New Zealancl

Porvers of cleterminantal expressions of the form det[1 -TA],7: diag(tt,t2,...,t,,)

occur i1 expressiols for the moment or probability generating functions for several types

of nultivariate clistribuiion - gamrna, negative binomial, binomial at least- The funda-

'rental ideltity relating the logarithm of the determinant to the trace of the logarithrn of

the matrix allows the coefficients in the determinantal expressions to be characterised as

multilinear forms in the elements of A, reducing to the detemrinant of A rvhen the power

is *1, to its permanent when tire power is -1, and to an extension of both concepts

(,,alpha-permalents" in the tenninology of Vere-Jones (1988)), for general porvers. This

pup"'. will review some properties of these representations and the associated multivariate

,lirtrib.rtions, inclucling extensions to nixtures of binornial or negative biuomial distribu-

tio1s, alcl to stochastic processes. Some open questions wiil be urentioned, including the

problem of developing effective numerical techniques.

ReferenceVere-Jones, D., "A generalisation of pertlauents and detertninants,"

Li,near Algebra an'd lts Application-s 111 (1988)' 199-124'

16

Andrew Balemi Richard William FarebrotherMathematics and Statistics Department of Econometrics and Social StatisticsUniversity of Auckland University of Manchester38 Princes St ManchesterPrivate Bag 92019 M13 gPL

Auckland United KingdomNew Zealand

m s rbs rf @cms. mcc. ac. u k

balemi@mat.aukuni.ac.nz

Rod Ball Arthur R. GilmourHort Research NSW Agriculture CentreHort Research lnstitute NZ Forest Road

P.O. Box 92169 OrangeAuckland NSW2800New Zealand Australia

rod@marc.am.nz

John S. Chipman Harold V. HendersonDepartment of Economics Statistics and Computing CentreUniversity of Minnesota Ruakura Agriculture Centre1035 Management and Economics Hamiiton271 19lh Avenue South New ZealandMinneapolis, MN 55455U.S.A. henderson@ruakura.cri.nz

jchipman@ uminnl.bitnet

Ronald Christensen Jefirey J. HunterDepartment of Mathematics and Statistics Department of Mathematics & StatisticsUniversity oi New Mexico Massey UniversityAlbuquerque Palmerston NorthNM 87131 New ZealandU.S.A.

jhunter@massey.ac. nz

f letcher@gopher.unm.edu OR

Peter Clifford Snehalata HuzurbazarOxford University Department of StatisticsJesus College University of GeorgiaOxford Athensox1 3DW GA 30602-1952UnitedKingdom USA

clifford@vax.oxf ord.ac.uk lata@rolf .stat.uga.edu

Richard JarrettDepartment of StatisticsUniversity of AdelaideDepartment of StatisticsGPO 498Adelaide 5001Australia

rjarrett@stats. adelaid e. ed u.au

Murray A. JorgensenWaikato Centre for Applied StatisticsUniversity of WaikatoPrivate Bag 3050HamiltonNew Zealand

maj@waikato.ac.nz

Zivan KaramanLimagrain GeneticsB.P. 1 1563203-Riom CedexFrance

Alan J. LeeDepartment of Mathematics and StatisticsUniversity of AucklandPrivate Bag 92019AucklandNew Zealand

lee@math.aukuni.ac.nz

W. Geoffrey MacClement

John MaindonaldHorticultural Research lnstitutePrivate Bag 92169Mt AlbertAucklandNew Zealand

Thomas MathewDepartment of Mathematics and StatisticsUniversity of Maryland-Baltimore CountyCatonsvilleMD 21228U.S.A.

mathew@umbc2.u mbc. ed u

Renate Meyerlnstitute of Medical Statistics and DocumentationAachen University of TechnologYGermanyPauwelsstr. 3051 00 AachenGermany

renate@tolkien. imsd. rwth-aachen

Gita MishraMathematics and StatisticsUniversity of Auckland38 Princes StPrivate Bag 92019AucklandNew Zealand

mishra@mat.aukuni.ac.nz

Chris C. PaigeMcGill U niversityMont16alMontr6al

Simo Puntanen George A.G. SeberDepartment o{ Mathematical Science Mathematics and StatisticsUniversity of Tampere University of AucklandP.O. Box 607 38 Princes St

SF-33101 Private Bag 92019Tampere AucklandFinland New Zealand

sjp@uta.fi (il 12t92 seber@mat.aukun i.ac. nz

Ken RussellDepartment of MathematicsUniversity oi WollongongP.O. Box 1144WollongongNSW 25OO

Australia

kgr@its. uow.edu.au

Jai Pal SinghDepartment of Agricultural EconomicsHaryana Agricultural UniversityHisarHaryana 125 004lndia

Garry J. TeeMathematics and StatisticsUniversity of Auckland38 Princes StPrivate Bag 92019AucklandNew Zealand

tee@mat.aukuni.ac.nz

Michael G. Schimek Jon SteneDepartment of Psychology lnstitute ol StatisticsMcGill University University of Copenhagen1205 Dr. Penfield Avenue Studiestrcede 6

Montreal DK -1455

Quebec CopenhagenCanada H3A 181 Denmark

ramsay@ramsay2.psych. mcgill.c

Alastair J. Scott George P. H. StyanMathematics and Statistics Department of Mathematics and StatisticsUniversity of Auckland McGill UniversityPrivate Bag 92019 Burnside Hall

Auckland 805 ouest, rue Sherbrooke Street WestNew Zealand Montreal, Quebec

Canada H3A 2K6

scottmt56@m usica.mcgill.ca

Shayle R. SearleBiometrics UnitCornell University337 Warren HallIthacaNY 1 4853-7801U.S.A.

btry@cornella.bitnet

John M. D. Thompson Graham R. WoodCot-death Department of MathematicsUniversity of Auckland University of CanterburyPrivate Bag 92019 ChristchurchAucklandNew Zealand grw@math.canterbury.ac.nz

cotdeath@ccu 1 .auku n i.ac. nz

Christopher M. Triggs Thomas W. YeeMathematics and Statistics Mathematics and StatisticsUniversity of Auckland University of Auckland38 Princes St 38 Princes St

Private Bag 92019 Private Bag 92019Auckland AucklandNew Zealand New Zealand

triggs@mat.auku n i.ac.nz yee@mat.aukuni.ac. nz

Alain C. VandalDepartment of Mathematics and StatisticsMcGill UniversiiyBurnside Hall805 ouest, rue Sherbrooke Street WestMontreal, QuebecCanada H3A 2K6

alain@zaphod. math. mcgill.ca

David J. Vere-Joneslnstitute of Statistics and Operations ResearchVictoria UniversityWellingtonNew Zealand

dvj@isor.vuw.ac. nz

Christopher J. WildMathematics and StatisticsUniversity of Auckland38 Princes StPrivate Bag 92019AucklandNew Zealand

wild@mat.aukun i.ac.nz