Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model Y = ∑ i = 1 m X i B i Z i ′ + U E subject to R ( X m ) ⊆ R ( X m - 1 ) ⊆ ⋯ ⊆ R ( X 1 ) , where B i are the matrices of unknown regression coefficients, X i , Z i and U are known covariate matrices, i = 1 , 2 , … , m , and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ . An unbiased invariant minimum norm quadratic estimator ( MINQE ( U , I ) ) of tr ( C Σ ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE ( U , I ) of tr ( C Σ ) to be a uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator ( MINQLE ( U , I ) ) of tr ( C Σ ) + ∑ i = 1 m tr ( D i ′ B i ) is also given. To compare with the existing maximum likelihood estimator (MLE) of tr ( C Σ ) , we conduct some simulation studies which show that our proposed estimator performs very well.