Admissibility of linear estimate of regression coefficients in growth curve model under matrix loss (Q1335396)

Consider the Growth Curve Model: \[ X = ABC + E, \quad E(X) = ABC, \quad \text{Cov} \bigl( \text{Vec} (X) \bigr) = I_ n \otimes \sigma^ 2G, \tag{1} \] where \(X\) and \(E\) are \(p \times n\) random matrices. \(A_{p \times q} (q \leq p)\) and \(C_{k \times n}\) are known as design matrices. \(B_{q \times k}\) is an unknown parameter matrix and \(\sigma^ 2 > 0\) is an unknown parameter. Suppose that the linear function \(KBL\) is estimable, that is, \(\mu (K') \subseteq (A')\), \(\mu (L) \subseteq \mu (C)\). Let the matrix loss function of the estimate \(d(X)\) for \(KBL\) be \[ \bigl( d(X) - KBL \bigr) \bigl( d(X) - KBL \bigr)'. \tag{2} \] We consider necessary and sufficient conditions that \(DXF\) is admissible for \(KBL\) within the class \[ {\mathcal T} = \{DXF : D_{t \times p} \quad \text{and} \quad F_{n \times l} \quad \text{are known}\}. \] For model (1) and loss function (2), we obtain a necessary condition and the sufficient condition for \(DXF\) to be admissible for \(KBL\) in the class \({\mathcal T}\) (written \(DXF {\overset {\mathcal T} \sim} KBL)\), respectively. We also establish a necessary and sufficient condition of \(DXF {\overset {\mathcal T} \sim} KBL\) under certain conditions.