Estimation of a linear transformation

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 σ2Σare observed (i = 1, …, n). It is assumed that xi and yi have respective mean vectors ξiandBξi, where the p × 1 vectors ξ1, …ξn, the p×p matrix B and Σ2 are unknown, and the p×p matrix Σ 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.