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

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 modelled 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.     

Epidemics in heterogeneous communities: estimation of R0 and secure vaccination coverage

A stochastic multitype model for the spread of an infectious disease in a community of heterogeneous individuals is analysed. In particular, estimates of R ₀ (the basic reproduction number) and the critical vaccination coverage are derived, where estimation is based on final size data of an outbreak in the community. It is shown that these key parameters cannot be estimated consistently from data; only upper and lower bounds can be estimated. Confidence regions for the upper bounds are derived, thus giving conservative estimates of R ₀ and the fractions necessary to vaccinate.     

L p –Estimators as Estimates of a Parameter of Location for a Sharp–pointed Symmetric Density

We study the asymptotics of L _p estimators, p > 0, over a sample having a symmetric density with a sharp–point at the centre of symmetry of the distribution. The rates of convergence of the L _p estimators in this situation depend on p and on the shape of the density. To obtain some of the limit distributions, we present new results in the asymptotics of M–estimators. We extend the delta method to the case when the Euclidean norm of the conveniently normalized M–estimators converge to a power of the Euclidean norm of a (possibly Gaussian) stable distribution.