Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model subject to R(X_m)⊆R(X_m-1)⊆?⊆R(X₁), 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(CΣ) 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(CΣ) 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 is also given. To compare with the existing maximum likelihood estimator (MLE) of tr(CΣ), we conduct some simulation studies which show that our proposed estimator performs very well.