The additive and block decompositions about the WLSEs of parametric functions for a multiple partitioned linear regression model

The necessary and sufficient conditions for the weighted least-squares estimators (WLSEs) of parametric functions K1β1+K2β2+⋯+KmβmK1β1+K2β2+⋯+Kmβm under a multiple partitioned linear model ℳ={y,X1β1+⋯+Xmβm,σ2Σ}ℳ={y,X1β1+⋯+Xmβm,σ2Σ} to be the sum of the WLSEs of KiβiKiβi under the mm small models ℳi={y,Xiβi,σ2Σ},i=1,2,…,m, and for the WLSEs of parametric functions KiβiKiβi under the linear model ℳℳ to be equal to the WLSEs of parametric function KiβiKiβi under its small model ℳi,i=1,2,…,m are derived. Some statistical properties about the WLSEs of parametric functions are also described.