Necessary and sufficient conditions for the existence of UMRU estimators in growth curve model (Q1320590)

The growth curve model proposed by \textit{R. F. Potthoff} and \textit{S. N. Roy} [Biometrika 51, 313-326 (1964; Zbl 0138.143)] is defined as \[ {\underset p\times {n} Y} = {\underset p\times {m} X} {\underset q\times {n} \tau} {\underset q\times {n} Z}+ {\underset p\times {n} \varepsilon}, \tag{1} \] where \(\tau\) is unknown; \(X\) and \(Z\) are the known design matrices of ranks \(m<p\) and \(q<n\), respectively; the columns of \(\varepsilon\) are independent \(p\)-variate normal with the mean vector 0 and unknown common covariance matrix \(V>0\). We discuss the uniformly minimum risk unbiased (UMRU) estimators of the parameters in the model (1).