Elimination of the type B uncertainty (Q2905566)

A linear model is called here a two-stage model if the response vector \(Y\) can be partitioned into two uncorrelated subvectors, say \(\mathbf Y_1\) and \(\mathbf Y_2\), so that the expectation of \(\mathbf Y_2\) is modeled by a linear combination of two vectors, one of which is a linear function of the mean parameter of \(\mathbf Y_1\), say \(\beta \), and the second is a linear function of an additional, maybe nuisance, parameter. The author investigates a two-stage linear model with a full column rank model matrix and a known positive definite block diagonal covariance matrix. The covariance matrix of \(\beta\) and its estimable function is of primary interest in this paper. Based on the response vector \(\mathbf Y\), it can be decomposed into two parts, which are referred to as uncertainties of type A and type B. The author derives a necessary and sufficient condition (on the model matrices and covariance matrices of both \(\mathbf Y_1\) and \(\mathbf Y_2\)), under which the uncertainty of type B vanishes, which translates into the following property: the best linear unbiased estimator (BLUE) of a linearly estimable function of \(\beta\) based on the second stage response \(\mathbf Y_2\) and on the first stage BLUE of \(\beta\) (based on \(\mathbf Y_1\)), coincides with the BLUE based on the transformed \(\mathbf Y_2\) that eliminates the nuisance parameter.