The Existence Of UMVIQUE For An Unknow Parameter in Multivariate Linear Model

Let Y=X₁BX'₂+ Uε,Eε=0,be a multivariate linear model, whereX₁ andBX'₂ are known matrics, B is an unknown matrix and e is a random matrix. Assume thatε=（ε₁,…,ε_n)',\vec\epsilon=\widehat=\left(\varepsilon_1^\prime, \cdots, \varepsilon_n^\prime\right)^\prime and\mathrmE(\vec\varepsilon \vec\varepsilon)=I \otimes \Sigma, where \Sigma \geq 0 is an unknown covariance matrix. This paper gives a sufficient and necessary condition for \vecY^\prime（MAM）\vecY to be the uniformly minimum variance invariance quadratic unbased estimation （UMVIQUE） of tr（CΣ）, where M=I-X_1 X_1^+ \otimes X_2 X_2^+ and C≠0 is an arbitrary symmetric matrix. As corollary, the sufficient and necessary conditions for UMVIQUE of tr（CΣ） to be exist and tr（CΣ^*） to be the UMVIQUE of it（CΣ） are given.