Non-orthogonal projections and their application to calculating the information in a partly linear Cox model

Information bounds are obtained for a generalization of the Cox model that replaces the linear prognostic index by a partly linear form: that is a proportional hazards model with a relative risk function that is parametric for certain covariates and non-parametric for others. The efficient score for the parametric component is calculated by a two step procedure. First the score is projected orthogonally to the tangent space for the baseline hazard function. Next it is projected orthogonally to the tangent space of the non-parametric regression function. Using the same techniques it is also shown that the estimators studied by Prentice & Self (1983) for general relative risk functions are efficient. An overview of projections onto sum spaces from the perspective of statistical models is included.