Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Mukhopadhyay, Anis Chandra (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

The study of optimality of block designs formally began with wald (1943) proving a very important optimality property, designated as D-optimality of Iatin Square. Desi gns in a given physical set up. Next Ehrenfield (1955) proved the E-optimality property of Latin Square Designs. Since then, the work in: the area has primarily consisted in evolving a number of useful opti- mality oriteria for oomparing block design in a reasonable set up given and oharacterizing and/or constructing desi gns saiisfying the so called optimality criteria developed. But it was not until 1958, when the theory was given a proper and precise formu- lation with the probleme defined in a syst ematic and standardized manner by Kiefer. Restricting to the class of connected block designs, Kief er (1958, 1959) considered the general problem of est im at ion of a set of orthonormal treatment contrasts n = PT %3! and gave precise definitions of a number of standard optimality criteria including A, D- and E-optimality, for measuring the performances of the least square estimates of these treatment contrasts from the design. Given the parameters b= number of blocks, k constant block size and v = number of treatments, a BI ED, when it exists , was proved to be optimal with regard to all Felevant optimality criteria defined by Kiefer (1956). the same time Mote (1958) proved the E-optimality and Khirsagar (1958) the A- and D-optimality of the BI BDs independently in the same set up.Roy (1958) took up the problem of estimation of - 2 - all elementary treatment contrasts and proved that the most efficient design for this problem for given b,k and withefficient design for this problem for glven b,k and with regard to A-optimality criterian is a BIBD, when it exists Kiefer (1975) defined a very general class of optimality critoria including the previously considered and widely used A-, D and E-optimality criteria. He also generalised the BIBDs to define BBDs (which reduced to the former in the special case k< v) and proved their universal optimality property, i.c. their optimal property with regard to every optimality criterion belonging to the general class defined by him. The major part o this paper was devoted to the optimality st udy of designs in a two-way heterogeneity setting, which we discuss in a laite paragraph.First result on the optimality of asymmetrical designs was established by Takeuchi (1961). Using a new technique, le proved the B-optimality of the GDDs with λ2=λ1+1 Laler, Contrfe and Stone (1974) derived an improved upper bound to the efficien of a block design with regard to A-optimality criterion. Cheng (1978b) sed a technique similar to that used by Conniffe and Stone (1974) to establish the optimality of the MBGDDs of typo 1.e. the DDs with m = 2 andλ2=λ1+1 with regard to a general class of optimality criteria termed by him t- optimali criteria of type 1, which included the age old A-, D and E-optimality criteria montioned before.


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