Optimal Covariate Designs: Theory and Applications

231 Pages·2015·2.32 MB·English

by  Premadhis Das

Premadhis Das · Ganesh Dutta
Nripes Kumar Mandal
Bikas Kumar Sinha
Optimal
Covariate
Designs
Theory and Applications
Optimal Covariate Designs
Premadhis Das Ganesh Dutta
(cid:129)
Nripes Kumar Mandal Bikas Kumar Sinha
(cid:129)
Optimal Covariate Designs
Theory and Applications
123
Premadhis Das NripesKumar Mandal
Department ofStatistics Department ofStatistics
University of Kalyani University of Calcutta
Kalyani Kolkata
India India
Ganesh Dutta BikasKumar Sinha
Department ofStatistics Indian Statistical Institute
Basanti Devi College(Affiliated to Kolkata
University of Calcutta) India
Kolkata
India
ISBN978-81-322-2460-0 ISBN978-81-322-2461-7 (eBook)
DOI 10.1007/978-81-322-2461-7
LibraryofCongressControlNumber:2015942803
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The greatest gifts we ever had were the gifts
from god we call them parents
In memory of My Father Late Jadunandan
Das and My Mother Late Sankari Das
Premadhis Das
Dedicating to My Father Shri Amarendra
Prasad Dutta and to My Mother Shrimati
Sankari Dutta for their Love, Affection &
Blessings
Ganesh Dutta
In Living Memory of My Parents Late
Jatindranath Mandal and Late Bidyutlata
Mandal
Nripes Kumar Mandal
Remembering My Parents Late Birendra
Nath Sinha and Late Jogmaya Sinha for their
Love, Affection & Blessings
Bikas Kumar Sinha
Foreword
Itisastandardclassroomexercisetoassertthatinasimplelinearregressionmodel
involving only one regressor [or, covariate] x, viz., y¼αþβxþerror, the
covariate-values (x), assumed to be continuous and to lie in a finite nondegenerate
interval a(cid:2) x (cid:2)b, should allow for maximum dispersion in order that the
regression parameterscanbe estimated with thehighestefficiency.Thissuggestsa
50–50splitofthetotalnumberofobservations,i.e.,thesetofobservationsaretobe
generated by setting the covariate (x) at the two extreme values, viz., x¼a and
x¼b, equally often.
Goingbeyondthis,therearebasicresults,whenmorethanonecovariatelikethis
are involved. On the other hand, in the absence of any such covariates, we have
available standard ANOVA models involving ‘design parameters’.
TheANCOVAmodelsintroducedinthetextbooksandintheliteraturearebased
on the study of models in situations wherein regression parameters and design
parameters are both present.
Naturally the question of the most efficient estimation of the regression
parameter(s) in the presence of design parameters needs to be studied in very
general terms, and also under very specialized experimental settings.
Lopes Troya initiated this study and BKS (Bikas Kumar Sinha) followed it up
with his research collaborators [Kalyan Das (KD), Nripes Kumar Mandal (NKM),
PremadhisDas(PD),GaneshDutta(GD), S.B.Rao,P.S.S.N.V.P.Rao,G.M.Saha
(GMS)].Itisamazingtonotethatsomuchwashiddeninthistopicofresearch,and
that their successful collaboration over these years had culminated in a Research
Monograph.
I had an opportunity to collaborate with BKS several years back and I am
thankful to the authors for approaching me to write this Foreword.
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viii Foreword
Thismonographaimsatprovidinganup-to-dateaccountoftheresearchfindings
in various experimental settings. As the authors describe and admit, mostly they
confine to ‘idealistic scenarios’ in order to develop and apply tools and techniques
for the study of optimal estimation of covariates’ parameters. In the introductory
chapter as also in Chapter 9, they discuss about ‘real life’ examples and provide a
detailed study of optimality.
Theauthorshavetakenupathoroughstudyoftheproblemsassociatedwiththis
area of research. I personally thank them for their tremendous efforts and con-
gratulate them for this remarkable achievement.
June 2015 Gour Mohan Saha
Retired Professor of Statistics
Indian Statistical Institute
Kolkata
India
Preface
Three of us are ‘Senior Citizens’ in the context of ‘Statistics Learning’ and we are
ever-grateful toourreveredpostgraduateteachersforhighlightingthefundamental
and basic contributions of R.A. Fisher and Frank Yates in such areas as Design of
Experiments[DoE].Wehadtheopportunitytoreadtheirbooks,somuchsothatwe
went through Fisher’s original book published in the 1930s. These are indeed
‘TreasuredCollections’!OurfascinationforDoEstartedfromthatpointoftimeand
it has continued to be intriguing for more than 40 years! We thoroughly enjoy
reading, learning and discussing all aspects of DoE—theory and applications.
There are two incidences to be told in real-time experience underlying this
project.
First,around2002oneoftheco-authorswastryingtomakea‘dent’intoapaper
on Optimal Covariates Designs [OCDs] with a colleague of him with very little
successprimarilybecausethenotationsweredifficulttofollow.Fortunatelyforthe
rest of us and for the optimal design community at large, they did not give up
altogether. Instead, at the earliest opportunity they approached one of the other
co-authors for looking into this paper. That was one positive development indeed
and together, they could digest the paper and go forward as a ‘high speed jet’! On
another occasion around 2003, again one of the co-authors was struggling with a
constructional probleminvolvingOCDsandthis time hewas accompanied byone
enthusiasticgraph-theoristandonematrix-specialist.Whiletheywerein‘seemingly
deep’ trouble and in a ‘confused state of mind’, one of their colleagues—a design
specialist—suddenly ‘peeped in’ and made a very casual observation, ‘it seems …
you are discussing some aspects of Mixed Orthogonal Arrays’ and that was it to
give again another big push to this work.
Inanutshell,thesetwoincidencesgaveaboosttoourgroupandwedidnothave
tolookbackanymore!Wehaveenjoyedworkingonthisproject.Wehavederived
muchpleasureworkinginagroupdiscussing,arguingandcounter-arguing,tillthe
time that we thought we came to understand enough of this fascinating topic of
research to prepare a Research Monograph.
We must hasten to add that the youngest member of our group [GD] kept the
others in toe with his frequent ‘claims’ and ‘counter-claims’ and ‘proofs’ and
ix
x Preface
‘counter-examples’! Working with him was a matter of great pleasure for us. His
enthusiastic and provocative statements/claims frequently served as ‘make-belief’
prophecieswhichweretobeverifiedbytheotherthree;itwasnoteasyallthetime
anyway.
Finally, we are here with a comprehensive account of what we believe to be a
Treatise on OCDs, more from the viewpoint of ‘Idealistic Scenarios’ in different
experimentalsituations.Theemphasisallthroughisabout‘optimal’choiceofwhat
are called ‘controllable covariates’ in continuous domain(s). Only in the last
chapter, we dwell on ‘realistic experimental situations’ and provide solutions to
some well-posed problems.
Confusion continued to follow us and it gave us a scope for generating argu-
ments and counter-arguments till we reached Clarity with our own understanding
of the findings.
Kolkata, West Bengal, India Premadhis Das
June 2015 Ganesh Dutta
Nripes Kumar Mandal
Bikas Kumar Sinha
Acknowledgment
During the course of our work on this fascinating topic, we had the opportunity to
interactwiththreeenthusiasticresearchers:Prof.KalyanDas[CalcuttaUniversity],
Profs. S.B. Rao and Prasad Rao [both from Indian Statistical Institute, Kolkata].
Their collaboration enriched our thoughts and we are grateful to them for their
enthusiasm and encouragement.
Duringthisperiod,wehadtheopportunitytobeinvitedtosomeconferencesand
make technical presentations of our papers before learned gatherings. We were
benefited by the remarks/comments/criticisms received therefrom.
We are thankful to the authorities at Indian Statistical Institute, Kolkata [BKS],
Department of Statistics, Calcutta University [NKM], Department of Statistics,
Kalyani University [PD] and Basanti Devi College, Kolkata [GD] for providing
excellent research opportunities for our team.
We are thankful to Prof. Gour Mohan Saha of the Indian Statistical Institute,
Kolkata for his insightful comments during a critical phase of this study and for
complying with our request for writing the Foreword of this monograph.
We have freely consulted available published literature and books and journals
and benefited immensely from the collective wisdom of researchers worldwide on
topics such as Hadamard Matrices, Mutually Orthogonal Latin Squares [MOLS],
Orthogonal Arrays [OAs], Mixed Orthogonal Arrays [MOAs], Linear Models,
ANOVA Models, Regression designs and ANCOVA models.
We fondly hope this monograph will be received enthusiastically by the sta-
tistical design theorists in general and combinatorial design theorists in particular
and they will identify open problems in the broad area of Optimal Covariates’
Designs.
Kolkata Premadhis Das
June 2015 Ganesh Dutta
Nripes Kumar Mandal
Bikas Kumar Sinha
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