The $L_2$ Rate of Convergence for Event History Regression
with Time-Dependent Covariates
by Jianhua Z. Huang and Charles J. Stone
Abstract:
Consider repeated events of multiple kinds that occur according to a
right-continuous semi-Markov process whose transition rates are
influenced by one or more time-dependent covariates. The logarithms
of the intensities of the transitions from one state to another are
modeled as members of a linear function space,
which may be finite- or infinite-dimensional. Maximum likelihood
estimates are used, where the maximizations are taken over suitably
chosen finite-dimensional approximating spaces.
It is shown that the $L_2$ 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 specified 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.