The L2 Rate of Convergence for Event History Regression with Time-dependent Covariates

Consider repeated events of multiple kinds that occur according to a rightcontinuous semi-Markov process whose transition rates are in ̄uenced by one or more timedependent covariates. The logarithms of the intensities of the transitions from one state to another are modelled as members of a linear function space, which may be ®niteor in®nitedimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen ®nite-dimensional approximating spaces. It is shown that the L2 rates of convergence of the maximum likelihood estimates are determined by the approximation power and dimension of the approximating spaces. The theory is applied to a functional ANOVA model, where the logarithms of the intensities are approximated by functions having the form of a speci®ed sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). It is shown that the curse of dimensionality can be ameliorated if only main effects and low-order interactions are considered in functional ANOVA models.