#### Date of Submission

2-28-1980

#### Date of Award

2-28-1981

#### 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)

In one of his fundamental papers Le Can (1960) introduced what is now called locally asymptotically normal (LAN) families of distributions and obtained several basic results regarding the asymptotic theory of estimation and testing. Roughly speaking, a sequence of families is said to satisfy the LAN condition if the corresponding sequence of appropriately normalised log-likelihood function is locally approximated with probability tending to one by the sum of two expressions, the first one being a sequence of rand om linear functions of the normalised parameter and the second one being a non-random quadratic form of the normalised parameter, and the sequence of rand om vectors involved in the linear term of the approximation converges weakly to the normal distribution with mean vector zero and the covariance matrix being the matrix involved in the quadratic form of the approximation. Actually LeCam (1960) considered a more general approximation in the sense that he allowed the above mentioned second term of the approximation to be any non-rand om function of the normalised parameter and the limit of the random vectors of the first term to be any arbitrary distribution and then he showed that if one further assumes the contiguity condition, which is impossible to avoid in a large part of statistical theory, the given families satisfy the LAN condition. Dhis is indeed a remarkable result since it implies that if one could approximate by linearly indexed expential families one could also approximate by normal familiea.An important thing to observe regarding the basic and restricted assumptions of TeCam (1960) is that a large part of a asymptotic theory depends only on the approximating form of the likelihood function, and any specific property such as i.i.d or any other special form of dependence is irrelevant. Thus, to mention only a small fraction of the results of LeCam (1960), Le Cam presented, under the LAN condition, a far-reaching generational of Wald's (1943) a sympathetic theory of testing and showed that this testing problem can be simply treated as if it were regarding the normal distribution.Based on LeCam (1960), more importantly based on the above mentioned observation, Hjek (1970, 1971 and 1972) further obtained several basic results regarding the asymptotic theory of estimation.Though the LAN condition covers a large part of statistical theory associated with asymptotic normality, there are problems in which thee assertions cannot be made in terms of asymptotic normality. Therefore, Lecam (1972 and 1974) further developed his theory and obtained quite general and more forceful result in a more general framework which amount to the following. If one is interested in the asymptotic Properties such as local asymptotic minimaxity and admissibility for the given sequence of families it is just enough to obtain the results for the limit of the given sequence of families and then the corresponding limiting statements for five results, even when the limits of the families are remote from the usual normal families.

#### Control Number

ISILib-TH44

#### 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

Jeganathan, P. Dr., "Asymptotic Theory of Extimation when the Limit of the Log-Likelihood Ratios is Mixed Normal." (1981). *Doctoral Theses*. 123.

https://digitalcommons.isical.ac.in/doctoral-theses/123

## 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:28842899