Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model

Consider the generalized growth curve model Y = ∑ i = 1 m X i B i Z i ′ + U E subject to R ( X m ) ⊆ ⋯ ⊆ R ( X 1 ) , where B i are the matrices of unknown regression coefficients, and E = ( ε 1 , … , ε s ) ′ and ε j ( j = 1 , … , s ) are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ . We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function tr ( C Σ ) with C ≥ 0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator y ′ A y with y = V ec ( Y ′ ) of tr ( C Σ ) to be the UMVNNQUE are obtained as well.