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Biometrika 1973 60(3):525-534; doi:10.1093/biomet/60.3.525
© 1973 by Biometrika Trust
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Estimation of a linear transformation

LEON JAY GLESER* and GEOFFREY S. WATSON

The Johns Hopkins University
Princeton University

* Now at Purdue University.

A problem arising in the Earth Sciences can be formulated as follows. Pairs (xi, yi) of independent p-dimensional normal random vectors having common covarianoe matrix {sigma}2{Sigma}are observed (i = 1, ..., n). It is assumed that xi and yi have respective mean vectors {xi}iandB{xi}i, where the p × 1 vectors {xi}1, ...{xi}n, the p×p matrix B and {Sigma}2 are unknown, and the p×p matrix {Sigma} is assumed known. Maximum likelihood estimators of the unknown parameters are obtained when n ≥ 2p, and their behaviour is studied both theoretically and in simulations. Some other approaches to the estimation of functional relationships are also studied, and are shown to yield the same estimators as maximum likelihood.

Key Words: Structural and functional relationship • Linear transformation • Maximum likelihood • Errors in variables • Multivariate regression • Multivariate normal distribution


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