Date of Submission

2-28-1994

Date of Award

2-28-1995

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Supervisor

Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

The asymptotic approach to statistical estimation is frequently adopted be cause of ita general applicability and relative simplicity. The modern study of asymptotic theory, initiated in Le Cam (1953), has undergone a vigorous devel- opment through the classic works of Le Cam, Hájek, Bahadur, Ibragimov and Has'minskii (Khas'minskii), Bickel, Pfanzagl, Millar and many other scholars; see Le Cam (1986), Le Cam and Yang (1990), Ibragimov and Has'minskii (1981) and the review article by Ghosh (1985) for an account of this development.Most of the results in asymptotic theory of estimation are obtained under the classical Cramér-Rao type regularity conditions or their substantial generalizations like LAN, LAMN etc. While redoubtably these are the most important cases, they are by no means the only cases of interest. It is well known that quite different and interesting phenomena occur when regularity conditions are violated. For example, in the case of a family of discontinuous densities, the best rate of convergence is n-1 instead of,n-1/2 in the regular cases. Also in contrast to the regular cases, the MLE is not efficient (but Bayes estimates are) and actually the notion of efficiency depends on the loss function involved. These "non-regular" cases attracted the attention of researchers from an early period and often were used to produce counterexamples. Convergence rates of estimates (particularly the MLE) were considered in several papers, see Polfeldt (1970) and Woodroofe (1972, 1974) for instance. In Polfeldt (1970a,b), the questions related to the order of variance of minimum variance unbiased estimators were investigated. A general theory of nonregular (as well 'as regular) cases was first attempted in Weiss and Wolfowitz (1974) who showed that the MLE may not be efficient, but maximum probabilty estimators (which are basically Bayes estimators depending on the chosen loss func- tion) are always efficient. In fact, in general, the posterior distribution and Bayes procedures may behave well even if the MLE in not well behaved, see Schwartz (1965) in this context. A modern treatment of both regular and nonregular cases appeared in the works of Ibragimov and Has minskii (1970, 1971, 1972a, b, 1973a, b, 1974, 1975a, b, 1976, 1977) which are now collectively available in Ibragimov and Has minskii (1981). Since this will be the main reference throughout the present work, it is convenient to abbreviate the above as IH. The general formulation of IH views the (normalized) likelihood ratios as a stochastic process in the following way:procedures may behave well even if the MLE in not well behaved, see Schwartz (1965) in this context. A modern treatment of both regular and nonregular cases appeared in the works of Ibragimov and Has minskii (1970, 1971, 1972a, b, 1973a, b, 1974, 1975a, b, 1976, 1977) which are now collectively available in Ibragimov and Has'minskii (1981). Since this will be the main reference throughout the present work, it is convenient to abbreviate the above as IH. The general formulation of IH views the (normalized) likelihood ratios as a stochastic process in the following way:

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843004

Control Number

ISILib-TH241

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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