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

Consider the generalized growth curve model Y=∑i=1mXiBiZi′+UE subject to R(Xm)⊆⋯⊆R(X1)R(Xm)⊆⋯⊆R(X1), where BiBi are the matrices of unknown regression coefficients, and E=(ε1,…,εs)′E=(ε1,…,εs)′ and εj(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 ΣΣ. We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function tr(CΣ) with C≥0C≥0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator y′Ay with y=V ec(Y′) of tr(CΣ) to be the UMVNNQUE are obtained as well.