#### Date of Submission

2-22-1991

#### Date of Award

2-22-1992

#### 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)

Neyman and Scott (1948) were the first to point out that the method of maximum likelihood fails to provide elfficient estimates when the number of parameters grows with the sample size n. Consider the following examples introduced by them:Ezample 1.1 Let { Xi } be a sequence of independent random vectors in I", components Xij of Xi being independent normal with mean u, and variance o2 Ilere o2 is the parameter of interest. It is casy to sce that the maxinum likelihood estimate for o2 is not even consistent. It is also known (see Lindsay (1980), Pfanzagl (1982), van der Vaart (1987)] that if p >2 , the maximum partial likelihood estimate based on Xij - Xi is efficient.Ezample 1.2 This is similar to Example 1.1 except that the components of Xi being independent normal with mean and variance of. Here u is the parameter of interest. It can be shown that the maximum likelihood estimate i is consistent and asymptotically normal provided p 2 3 and no is bounded away from zero, but it is not efficient. For p = 1, Bickel and Klaassen (1986), and for general p, van der Vaart (1987) show how an efficient, asymptotically normal estimate can be constructed.Lindsay (1980) and Bickel and Klaassen (1986) provide an extremely useful general discussion of such problems. See also van der Vaart (1987, 1988).

#### Control Number

ISILib-TH183

#### 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

Bhanja, Joydeep Dr., "Contributions to the Neyman-Scott and Mixture Problem." (1992). *Doctoral Theses*. 340.

https://digitalcommons.isical.ac.in/doctoral-theses/340

## 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:28843435