Table Of ContentContributions to Statistics
Barry C. Arnold
Narayanaswamy Balakrishnan
Carlos A. Coelho Editors
Methodology
and Applications
of Statistics
A Volume in Honor of C.R. Rao
on the Occasion of his 100th Birthday
Contributions to Statistics
The series Contributions to Statistics contains publications in theoretical and
appliedstatistics,includingforexampleapplicationsinmedicalstatistics,biometrics,
econometricsandcomputationalstatistics.Thesepublicationsareprimarilymono-
graphs and multiple author works containing new research results, but conference
andcongressreportsarealsoconsidered.
Apartfromthecontributiontoscientificprogresspresented,itisanotablecharac-
teristicoftheseriesthatpublishingtimeisveryshort,permittingauthorsandeditors
topresenttheirresultswithoutdelay.
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/2912
· ·
Barry C. Arnold Narayanaswamy Balakrishnan
Carlos A. Coelho
Editors
Methodology
and Applications of Statistics
A Volume in Honor of C.R. Rao on
the Occasion of his 100th Birthday
Editors
BarryC.Arnold NarayanaswamyBalakrishnan
DepartmentofStatistics DepartmentofMathematicsandStatistics
UniversityofCalifornia McMasterUniversity
Riverside,CA,USA Hamilton,ON,Canada
CarlosA.Coelho
NOVASchoolofScienceandTechnology
NOVAUniversityofLisbon
Caparica,Portugal
ISSN1431-1968
ContributionstoStatistics
ISBN978-3-030-83669-6 ISBN978-3-030-83670-2 (eBook)
https://doi.org/10.1007/978-3-030-83670-2
MathematicsSubjectClassification: 62F10, 62F03, 62E10, 62H15, 62E20, 62E15, 62H11, 62H20,
62JXX,62KXX,62PXX
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2021
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Professor C. R. Rao has continued his productive research career through his
tenth decade, and now begins the eleventh. A celebration of his 100th birthday is
completely appropriate. There has been no difficulty in encountering individuals
enthusiasticaboutcontributingtothisfelicitationvolume.Thatitwillappearintime
forProf.Rao’s101stbirthdayisaconsequenceofthecareinvestedinpreparingsuit-
ablecontributionsbythemanyscholarswhoseliveshavebeendirectlyorindirectly
influenced by him and his many contributions to Statistics and related fields. We,
theeditors,consideritanhonortohavebeenabletocontributetotheorganization
and development of this tribute volume. If ever there was a statistician who needs
nointroduction,itisC.R.Rao.Thisistrueeventhoughmanystudentsmaywonder
if his first name is Cramer, or perhaps that his last name might be Blackwell. Be
thatasitmay,theydefinitelyknowandrespect(aswedo)Prof.Raoandhismany
accomplishments!
We present this volume to Prof. Rao with our best wishes for a continued
productiveandenjoyablelife.
Wewishtotakethisopportunitytothankallthosewhohavecontributedpapers
to this volume. In addition, we express our sincere gratitude to the scholars who
providedrefereeingservices.Theirconstructivecommentarieswere,inmostcases,
muchappreciatedbytheauthors.
We also wish to thank our Springer editor, Ms. Veronika Rosteck, who has
patientlyhelpedusbyguidingusthroughthenecessarystepstobringthisprojectto
fruition.
HuntingtonBeach,CA,USA BarryC.Arnold
Hamilton,Ontario,Canada NarayanaswamyBalakrishnan
Caparica,Portugal CarlosA.Coelho
May2021
v
Dedication
ThepresentvolumeisintendedtohonorProf.CalyampudiRadhakrishnaRaoonthe
occasionofhis100thbirthday,whichoccurredonSeptember10,2020.
Professor Calyampudi Radhakrishna Rao is an Eberly Professor Emeritus of
StatisticsatPennsylvaniaStateUniversityandbeforehisretirementhealsoserved
asDirectoroftheCentreforMultivariateAnalysisatPennStateUniversityandas
anAdjunctProfessorattheUniversityofPittsburgh.
Calyampudi Radhakrishna Rao was born on September 10, 1920 in Huvvina
Hadagalli,thenintheintegratedMadrasProvince,nowinthestateofKarnataka,to
C.D.NaiduandA.Laxmikanthammaastheireighthchild.Ifweareallowedtodo
so, we will hereon address him just as C. R. Rao, for short, and also actually as a
statementoffriendshipandrespect.
C. R. Rao’s father was a reputed police Inspector at the Criminal Investigation
Department,whosejobrequiredthefamilytomovefromplacetoplaceeverytwo
or three years. For this reason, C. R. Rao completed his education in classes two
andthreeatGudur,classesfourandfiveinNuzvidandthefirstandsecond forms
inNandigrama,allnowadaysinthestateofAndhraPradesh.Afterretirement,C.D.
Naidu decided to settle in a coastal city in Andhra Pradesh called Visakhapatnam
and C. R. Rao finished his high school and obtained his first college degree B.A.
(Hons.), with a first class and first rank, in Visakhapatnam. In 1943, C. R. Rao
receivedandMScinMathematicsfromAndhraUniversityandonJanuary1,1941,
hejoinedtheIndianStatisticalInstitute(ISI)inCalcuttaandafewmonthslaterhe
becameastudentatCalcuttaUniversity,inthenewlystartedMAcourseinStatistics.
Heobtained his M.A.degree inStatisticsfromCalcutta Universityin1943 witha
first class, first rank, gold medal, and record marks unbeaten till now. Rao has the
distinctionofstandingfirstinthefinalexaminationofallclasseswherehestudied
fromPrimarySchooltoUniversity.
In 1948, C. R. Rao, after working under the direction of R. A. Fisher, obtained
his Ph.D. degree from the Cambridge University, King’s College, with a Thesis
describingnewmethodsforanalyzingmultivariatedatathathedevelopedduringhis
twoyearsstay.
vii
viii Dedication
Afewyearslater,CambridgeUniversityawardedhimthehigherdoctoratedegree
Sc.D.,basedonpeerreviewofhiscontributionstostatisticsandKing’sCollegemade
himalifefellow,ararehonor.
C. R. Rao completed his 100th birthday on September 10, 2020, and along his
longandveryproductivelifehegathered,sofar,animpressivelistof39Honorary
DoctorateDegreesfromnineteencountriesinsixdifferentcontinents:
(cid:129)
in Asia: 14 from India (Andhra University, 1967; University of Delhi, 1973;
Osmania University, 1977; Indian Statistical Institute, 1989; University of
Hyderabad, 1991; Sri Venkateswara University, 1993; Visva Bharati Univer-
sity, 2001; Calcutta University, 2003; University of Madras, 2007; Jawaharlal
NehruTechnicalUniversity,2011;KarnatakaUniversity,2012;RashtriyaSanskrit
Vidyapeeth University, 2014; Indian Institute of Technology, Kharagpur, 2014;
JamiaHamdardUniversity,2018),1fromthePhilippines(UniversityofPhilip-
pines,Manila,1983),1fromSriLanka(UniversityofColombo,2012);
(cid:129)
inAustralia:1(UniversityofWollongong,2001);
(cid:129)
inAfrica:1(UniversityofPretoria,SouthAfrica,2004);
(cid:129)
inEurope:1fromCyprus(UniversityofCyprus,Nicosia,2001),1fromFinland
(UniversityofTampere,1985),1fromGermany(UniversityofMunich,1995),2
fromGreece(UniversityofAthens,1976;AthensUniversityofEconomicsand
Business, 1994), 1 from Poland (University of Poznan, 1991), 1 from Portugal
(NOVAUniversityofLisbon,2006),1fromRussia(LeningradUniversity,1970),
1fromSlovakia(SlovakAcademyofSciences,1994),1fromSpain(University
ofBarcelona,1995),1fromSwitzerland(UniversityofNeuchatel,1989);
(cid:129)
in North America: 2 from Canada (University of Guelph, 1996; University of
Waterloo, 1997), 6 from U.S.A. (Ohio State University, 1979; Colorado State
University, 1990; Kent State University, 2000; Oakland University, Rochester,
2002;UniversityofRhodeIsland,2007;UniversityatBuffalo,StateUniversity
ofNewYork,2013);
(cid:129)
in South America: 1 from Brazil (University of Brasilia, 1982), 1 from Peru
(UniversidadNacionaldeSanMarcos,Lima,1982).
C.R.RaowasmarriedtoBhargaviRaofor69years.Shewasapsychologistand
professor at Jadavpur University, India. He has a daughter, Tejaswini, a professor
of Nutrition and classical Indian dancer, and a son, Veerendra, an electrical engi-
neerandcomputerscientistwhosewifeMaliniisaspeechpathologist.Hehastwo
grandchildren:Amar,acomputerscientistandRohith,inbusinessadministration.
Dedication ix
SomeofC.R.Rao’sMostNotableContributions
toStatistical ScienceandOtherAreas
C. R. Rao is an extremely prolific author. He is the author of 476 research papers
publishedinprestigiousjournalsand16books,oneofwhich,“StatisticsandTruth”,
originally published in English, was translated into French, German, Japanese,
Mandarin and Taiwanese, Turkish, and Korean. One other book, “Linear Statis-
ticalInference”,whichhasbeeninthemarketforover50years,isavailable,besides
English,alsoinGerman,Czech,Polish,Chinese,JapaneseandRussian.
C. R. Rao has also edited 42 volumes of the “Handbook of Statistics”, dealing
withthelatestmethodologiesinStatistics.
SomeofC.R.Rao’sinfluentialpaperswerethecornerstoneforwhatnowadays
are well-known techniques and terms used in Statistics and other areas. His 1945
paperintheBulletinoftheCalcuttaMathematicalSociety,writtenattheyoungage
of25(Rao,1945),togetherwiththe1946bookbyH.Cramér(Cramér,1946)estab-
lishedthebasisforthewell-known“Cramér-Raolowerbound”andthe“Cramér-Rao
inequality”,whichsetthelowerboundforthevarianceofunbiasedestimators.This
same paper, together with another paper from Blackwell (Blackwell, 1947) also
establishedthefoundations ofthewell-knownRao-BlackwellTheoremandofthe
methodcommonlyknownasRao-Blackwellization,throughwhichonemaybeable
toobtain,byconditioningonasufficientstatistic,amuchbetterestimatorfromwhat
might originally be just a crude estimator, and which combined with some results
from the Exponential Family allows for an easy way to obtain UMVU (Uniform
MinimumVarianceUnbiased)estimators.
This1945C.R.Raopaperestablishedyetthebasisforwhatarecalledthe“Fisher-
Rao metric” and “Rao distance”, which are measures of the change induced in a
probabilitydistributionbysmallchangesinitsparameters(AitkinsonandMitchell,
1981).Theywereintroducedinconnectionwithstatisticalproblemsofclassification
andclusteranalysisandarebasedonadifferential-geometricalapproach,buthave
found applications in different areas as in some problems in quantum mechanics
(BrodyandHughston,1998)andinthedetectionofstructuresinimages(Maybank,
2004,2007).Amari(1985)refersthat“ItwasRao(1945),inhisearlytwenties,who
first noticed the importance of the differential-geometrical approach” in Statistics
related problems. This Rao (1945) paper also gave rise to the term “Fisher-Rao
metric”,whichkeepsbeingatopicofresearchindistributiontheory(Brigantetal,
2021).
C. R. Rao’s contributions to Multivariate Analysis originated from a study of
skeletonsfromtheJebelMoyaarchaeologicalsiteinSudan.JebelMoyaisanarchae-
ological site in the southern Gezira Plain in Sudan, approximately 250 km south
southeast of Khartoum, where still nowadays the University of Cambridge carries
onexcavations.ThesiteisoneofthelargestpastoralistcemeteriesinAfrica,where
thousandsofburialshavebeenexcavatedthusfar.Intheearly1940’s,theAnthro-
pology Department of Cambridge University sent an expedition to Jebel Moya, to
digoutancientgravesandbringtheskeletonsofpeopleburiedthereforstudy.The
x Dedication
Departmentwantedtoanalyzethemeasurementstakenontheskeletonstodetermine
therelationshipofthepeoplewholivedtherewithpeoplecurrentlylivingthereorin
nearbyareas.However,therewerenomultivariatestatisticalmethodsavailableatthat
timetotreatsuchproblems,involvingmultiplemeasurements.C.R.Raohaddevel-
opedsomenewmultivariatemethodsforanalyzingsuchmultivariatedataattheISI,
inCalcutta,togetherwithProf.Mahalanobis,andhadtheexperienceofaddressing
suchproblems.InJuly1946,Dr.J.C.Trevor,ProfessoroftheAnthropologyDepart-
ment at Cambridge University, sent a telegram to Prof. Mahalanobis asking him
to send someone from ISI to analyze the measurements on Jebel Moya skeletons
using the methodology developed at ISI. C. R. Rao was selected along with R. K.
Mukherji,ananthropologist,togotoCambridgetotakemeasurementsonskeletons
andanalyzethedata.TheywenttoCambridgeandworkedintheDuckworthLabo-
ratoryoftheAnthropologyMuseumfortwoyears(1946–1948)asvisitingscholars
withpay.Raohadanywaytodevelopsomenewmultivariatemethodstoanalyzethe
data.Theresultsbasedontheanalysisofmeasurementswerereportedinthebook,
Ancient Inhabitants of Jebel Moya, published by the Cambridge University Press
in 1954 under the joint authorship of C. R. Rao and the two anthropologists J.C.
TrevorandR.K.Mukherji.AttheinvitationoftheRoyalStatisticalSociety,C.R.
Raopresentedadiscussionpaperdescribingthenewmultivariatemethodshedevel-
opedinsolvingtheJebelMoyaproblem.Thepaperwithdiscussionwaspublished
intheJournalofRoyalStatisticalSociety,Utilizationofmultiplemeasurementsin
problemsofbiologicalclassification,J.Roy.Statist.Soc.10:159–203(1948).Some
newmultivariatetestsandmodelsthatC.R.RaousedinanalyzingtheJebelMoya
datasuchasMANOVA(multivariateanalysisofvariance)aredescribedinhispaper
Tests of significance in multivariate analysis, Biometrika, 35: 58–9 (1948). These
twopapersarefurtherdiscussedinRao’sbook,AdvancedStatisticalMethods,John
Wiley(1952)andtheyprovidedsomeofthefoundationsofMultivariateAnalysis.
Yetin1948,C.R.Raopublishedapaper(Rao,1948)whereheworksoutallthe
detailsofwhatisknownas“Rao’sScoreTest”,analternativetoLikelihoodRatioand
Waldtests.Thethreetestsareequivalenttothefirstorderofasymptotics,beingall
threeasymptoticallyoptimal,butdiffertosomeextentinthesecondorderproperties.
Relativetothelikelihoodratiotest,Rao’sscoretesthastheadvantageofrequiring
onlytheestimationofthelikelihoodfunctionunderthenullhypothesis,and,unlike
theWaldtest,itisinvarianttotransformationsoftheparameters.Also,incontrastto
mostlikelihoodratiotests,itmaybeusedwhentheparameterslieontheboundary
oftheparameterspace.Rao’sScoreTestfoundapplicationsinawiderangeofareas,
from genetics to econometrics (Rao, 1950, 2005; Bera and Ullah, 1991, Anselin,
2011).
ButthisextraordinarilyproductiveandfruitfuleraofC.R.Raohasalsoproduced
three papers (Rao, 1946, 1947, 1949) which would be the basis for the so-called
“Rao’sOrthogonalArrays”,atermactuallycoinedin1950byBush(Bush,1950).
Rao’sOrthogonalArraysare“aversatileclassofcombinationalarrangementsuseful
for conducting experiments to determine the optimum mix of a number of factors
in a product to maximize the yield, and in the construction of a variety of designs
foragricultural,medicalandotherexperiments”,asreferredbyC.R.Raohimself
