#### Date of Submission

2-28-1978

#### Date of Award

2-28-1979

#### 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)

There are many important statistical problems of the following kind. The family of probability measures O is parametrized by a vector parame ter n varying in a q-dimen- sional domain. P can be represented as an exponential family of probability distributions with k canonical para- me ters where k is greater than q. The canonical parameters do not vary in a domain in R, but are restricted by polyno- mial or analytic equations. They vary on a curved surface defined by the polynomial or analytic equations within the natural parame ter space of the exponential family. The present work is concerned with the problem of point estimation of para- metrio functions in such statistical problems. This work is done in the spirit of R.A. wijsman, Ju. V. Idnnik and A. M, Kagan.In Chapter 1 we present the basic facts of the theory of exponential families, Several example s are given to indi- cate the importance of the kind of exponential families we study.In Chapter 2 we prove a theorem of A,M. Kagan and V.P. Palamodov characterizing the class of uniformly minimum ii variance unbiased estimators in an exponential family. dominated by the Lebesgue measure when the canonical para- meters are restricted by polynomial equations. The proof we give brings about substantial simplifications in the original proof of Kagan and Palamodov. We use this the orem to prove two conjectures of J. K. Ghosh.ators in a family of normal distributions with an unkmown integer mean. In Chapter 3 the variance components models, under the normality assumption, are treated as exponential families to characterize the uniformly minimum variance unbiased esti- mators, We consider this as a very important application of the theorem of Kagan and Palamodov. Explicit likelihood equations are also derived.In Chapter 4 we extend and strengthen a result of A,M. Kagan on the inadmissibility of certain estima tors which are functions of the minimal sufficient statistio. This result has an important application to a special type of location parame ter family.In Chapter 5 we prove an interesting the orem charac- terizing the uniformly minimum variance unbiased estimators in a family of normal distributions with an unkmown integer mean.As a corollary, the mean itself is shown to have no uniformly minimum variance unbiased estimator,.In Chapter 6 we discuss the problem of unbiased esti- ma tion in a censored gamma family of distributions.

#### Control Number

ISILib-TH115

#### Creative Commons 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

#### Recommended Citation

Unni, Krishnan Dr., "Theory of Estimation in Algebraic and Analytic Exponential Families with Applications to Variance Components Models." (1979). *Doctoral Theses*. 314.

https://digitalcommons.isical.ac.in/doctoral-theses/314

## 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:28843385