Radhakrishna Rao, C.
(1968)
*A note on a previous lemma in the theory of least squares and some further results*
Sankhya - Series A, 30
.
pp. 259-266.
ISSN 0581-572X

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Official URL: http://sankhya.isical.ac.in/search/30a3/30a3026.ht...

## Abstract

Let Y be a vector of random variables such that E(Y)=Xβ where β is a vector of unknown parameters and Σ be the covariance matrix of Y. A linear function L'Y is said to be best linear unbiased estimator (BLUE) of a parametric function p'β with respect to Σ if L'ΣL is a minimum subject to p'=L'X. The paper deals with necessary and sufficient conditions that, for every estimable parametric function or for a given subset, the BLUE with respect to Σ is the same as the BLUE with respect to Σ =I (identity matrix) or the same as the BLUE with respect to Σ=Σ_{0} (a given matrix). Let Z be a matrix of maximum rank such that X'Z=0. It is shown that when Σ=Σ_{0} is non-singular, or rank (X:Z) =rank (X:Σ_{0}Z), then a NAS condition for the equality of BLUE's of all estimable functions for Σ and Σ_{0} is that Σ is of the form Σ=XθX'+Σ_{0}ZΓZ'Σ_{0}+Σ_{0} where θ, Γ are arbitrary. The representations of Σ in other situations where Σ_{0} is singular have also been obtained.

Item Type: | Article |
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Source: | Copyright of this article belongs to Indian Statistical Institute. |

ID Code: | 54751 |

Deposited On: | 12 Aug 2011 13:00 |

Last Modified: | 12 Aug 2011 13:00 |

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