#### Date of Submission

2-28-1979

#### Date of Award

2-28-1980

#### 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

Mitra, Sujit Kumar

#### Abstract (Summary of the Work)

Our interent will be centred around the Gaunn Markov model (Y,X6,2A), where Y ls a randon variable asnuning values in R" with expectation and dispersion natrix given byE(Y) = XA (1.1) (1.2) * X in Rnxm and A In the subset of nonnegative definite (n.n.d.) natrices of RAre known. Unless apecifled to the contrary, A will be assuted to be positive definite (p.d.). The unknown paraneter vector B varies in Ag, a subset of Rand o? in, will alvays be the positive half of the real line, unless otherwine specifled, and 1ikewfse satiafien the minimum requirement dim (n,) = m.Historically, the first contribut ion towards estinating 1inear functionals of 8 wan by Caunn f16] who in 1821 showed that the nethod of least nquares provides the BLJE (best linear unbiased estirator) of 8, when R(X) = n and A = 1. Karkov [22] in 1912 and David and Keynan ti2) in 1930 gave a systenatie presentation of the theory under the sane conditiona. In 1934 Aftken (1] considered the metip where R(X) - m. but A could be any positive definite natrix. Bose (61 in 1944 was the first to conaider defieieneies in R(X). In his nodel R(X) - ren and A- 1, while Rao [28) in 1945 generalised this to any positive definite A. In all of these log was Rm.Bose [6) defined a linnar paramotrie functional pR as Nolne estinat a If It poscases an untlased astimatruhieh is limsar in Y. Any paranetrie fonctlonal without this property wan nonestimable. 1E din nb) = n, then it is not difficult to sne that p is (1linearly) estinable if and anly ir there exists a he R, sch that p = X, b;Y being an unbiased estinabr of pin such cases, Clenrly nonstinahle functlonals are preeinely thone for which pt Mex). estinstion of netveral lincar parantric functionals, re (P heing a mateix ie estinable ir Mcp M(x), vlojation of shich will ruke it Fer slmultaneous nonestinable, It n(x1) - m, then, in partiealar A is nomestimable.While estimable functionals have been studied in denth, very Feu theoretical Investigations are coneerned with nenestirable linuar parare tric functionals. However, experinenters usine a fractional roplicate of a factorial design are of ten reqaired to ertinste important factorial effects which aru nonentinable on account of incomplete replieation. Tven if one starts with a full rank design matrix (), or at least ene htch allows for undiased ent Iration of all the rioranotric functionals of interst, do: to factore heyond ones control, data might only ba avallahla for a subset of the experinental points at the end of the exper Inent, to renter Important paranetric functionala nonestinable. Thin ts the well known problen of mising obnervat lons. Since it ie penerally Impossthle to repeat the experiment(with the hope of hetter outeomes), it in of grent inportance to device ways to Infer about nonentinable functionals. This work will be directed towards this goal. Put, before laumching upon the task, let us examine the difficulties which may arise, how we might he at the overcome them.

#### Control Number

ISILib-TH98

#### 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

Majumdar, Dibyen Dr., "Statistical Analysis of Nonestimable Functionals." (1980). *Doctoral Theses*. 41.

https://digitalcommons.isical.ac.in/doctoral-theses/41

## 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:28842817