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)


Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

The study of admissible, minimax, and Bayes procedures has been of primary importance ever since the pioneering work of Wald (50). Since early seventies, new directions have been opening up, and not merely new techniques, completely new interpretations and interrelations have come to be known.To prove admissibility of estimates the most commonly used technique is to show that it is extended Bayes and approximate its risk by the risk of the corresponding Bayes estimates. This technique is due to Blyth (51). A sort of conver se result, which cssentially shows that this technique must work for all admissible est imates is due to Farrell and Stein who dorive a necessary and sufficient c ondition for admissibility.Karlin (58) was among the foremost statisticians to have evolved a general technique of proving admissibility in one dimen- sion. In the one parameter exponential fanily with a density p(x,w) = exw(w)dll(x)< w o, is admissiblo for Ew(X), ifa ∫(w)β λ dw=∞= S(w)dw, β λ for w < a,b< wKarlins conject ure that the converse of this result is also truc is open till this day. In this the sis, we shall use Karlins tech- nique quite extensively in Chapters 2 and 3, for deriving sufficient conditions of adnissibility of probably non-linear estinates in the one paraneter regular cxponential and non-regular fanilies. i unified proof of tho adnissibility of cone standard gonoralizod Bayos estinates of the nean in the oxponential fanily has rocently been given by Brown and Hwang (81) in the lines of Blyth (51).Fron late sixties, interest in admissibility work shifted from particular problens to vory general prablens. Same leading st ones in this direction were Brown (66), Brown (71) and a sorico of articles duc to Berger (76). Brown (66) showed that the beat inväriant estimate of a location paraneter is, under very goneral conditions, adnissible in dinension 1 and inadnisaible for dimen sions 3 and more. Subsequent ly, in the cont ext of simultencous ostim tion of independent normal means, Brown (71) discovered a novel relation between admissibility and the rocurrence of a rclated diffusion process. Brown (71) also practically char acterisod all th admissible estima tes of the multivariate normal mean in thc goneral zcd Bayes class. Some important work in the spirit of Brown (71) was subsequently done by Srinivasan (81) and in the contexi of control rablems by Srinivasa. (82). While rown (66) dwolt, in broad gonerality, on the problem of estimating a full location vector, Berger (76a, 76b) considered the question of admissibility or otharwiac of generalized Bayes estimators of cuordinatos of a location vector.The most popular and clegant tool now in use of proving inadmissibility or improving upon inadmissible estimat ors is the nethod of solving differential inequalities. The tool should bo primarily att ibuted to Stein who achieved a najor brealkthrough by proving what is por larly known as Stcin's identity. It was later goneralized by Hudson (7), Berger (80), Hwang (). Sone very impe tant works in nultijaramcter inadn. seibility and differen- tial incqualitios were also done by Brown (79), Brown (81), and Ghosh and Parsian (80).


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