Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Bhimasankaram, P.

Abstract (Summary of the Work)

We consid er the poneral lingar nodel Y = X v e, here Y io an nx1 rundor voctor taking valacs in 2, x is an nXu ratrix (the deni gn ratrix), ip an 1X1 vectur uf unknown partro tere varying in R and 1s an nx1 voctor of errors with E(e) = 0 and E(ee') - o2v,o2 boing a positive scalar (known or unknown) and is an nXn non-negative definite Ta trix. It in assurmod that n < n. Such a nodel (also known as the Gauss-Markov nodel) is usually denoted by (Y, Xβ,α2v). he defini tions ot un catinable lincar paranctric function, sinple least squires estimator (SLSE), best linear unbia sed estinator (BLUE), linsar ninir:un bias cstinator (IIMBE) and best Iinear nininum bias estinator (BIIMBE) under the nodel (Y, Xβ,α2v arc we 11. known and we retor to Rao and Mitra (1971, Chaptere 7 and B) for the details.Early contributions towärde estinating linear functionals of β are due to Logenire (i806), Gauss (1609) and Varkov (1912), where attention was concentrated on the case where R(X)= n and V = I, the identity natrix. Aitken (1934) considercd the problen of bost lincar unbianed catiration under the setup vhere R(x) = n and V is any positive definite natrix. Bose (1944) conside the casc where R(X) < n and V = 1, while Rao (1945) genera d this to any positive defini te V. Seal (1967) ives a good histo- rical account of the linear model upto 1935 and Plackott (194 9) givus a shorl histurical nute un the raothori uf Jount aquarce. WEE the covariance atixv is nunaingnlar ilh v is nunainglar . la V known und tiho nXn 15 Lrix X is of tnll ruk, i.c. o? runit n and when fur thor the culca ut the ratri are li orthonor- Lial igenvectcre ut v, tun it is at yasil wririahle fact that ie icontical1 ita its S13. his ract was first pointed aut by Andernon (1948) and nutice of it was lakon soon af ter by Durbir und watecn (1950). Fro thin tire cnards, the problen of deriving necessary and sufficient conditions under which the SISE't are e also corresponiing ELUE a hae received con- siderable attontion, mainly due to the norputationl advantage of the S1.SR over the ILUE. The present work is devoted to the study of the robuatness of cstiration and testiag: prucordarcs in linar codels with in- correct desim und dispersion aaa brices. Betore giving a surary of the probleris considUred we shull prosent a brief rovicw of th 1iterature in this area.A atatonant on vorious noceary anl nfricient ountitionn for the equalily of tha iaa aid correapotsiin BIJRse de by Zyakind (1962) On: of the cunlatiuna ota tod huru 1a that thore exista a nahuut of r udiponveetoro of V that forma banio uf the vector spucu spied by tho colunna of the design ratrix X. A proot thut the cáguvuotur ounition is both neoessary and auf- firient for the correapuiliny HTE arsl SLSE to huve the ae cuvarianeu natri: du oresoutua vdth X n all at ai 7 aonningular hy Ma, maa ant aGuáro (1962).


ProQuest Collection ID:

Control Number


Creative Commons License

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


Included in

Mathematics Commons