Multiple additive models

The models constituting a multiple model will correspond to d treatments of a base design. When we have individual additive models Y(l)=X0β(l)+∑i=1wXiZi(l),l=1,…,d, with Z1(l),…,Zw(l), independent, with c1,…,cw independent components, with χr,1,…,χr,w, the r−th order cumulants, r=2,3,4, the multiple model will be additive. Using a classic result on cumulant generation function we show how to obtain least square estimators for cumulants and generalized least squares estimators for vectors β, l=1,…,d, in the individual models. Next we carry out ANOVA-like analysis for the action of the factors in the base design. This is possible since the estimators β˜(l) of β(l) l=1,…,d, have, approximately, the same covariance matrix. The eigenvectors of that matrix will give the principal estimable functions ϵi⊤β(l) i=1,…,k, l=1,…,d, for the individual models. The ANOVA-like analysis will consider homolog components on principal estimable functions. To apply our results we assume the factors in the base design to have fixed effects. Moreover if w=1, and Z(1) has covariance matrix σ2In, our treatment generalizes that previously given for multiple regression designs. In them we have a linear regression for each treatment of a base design. We then study the action of the factors on that design on the vectors β(l),l=1,…,d. An example of application of the proposed methodology is given.