Sufficient dimension reduction on the mean and rate functions of recurrent events

The counting process with a Cox-type intensity function has been extensively applied to analyze recurrent event data, which assume that the underlying counting process is a time-transformed Poisson process and that the covariates have multiplicative or additive effects on the mean and rate functions of the counting process. The existing statistical inference, however, often encounters difficulties due to high-dimensional covariates, such as in gene expression and single nucleotide polymorphism data that have revolutionized our understanding of cancer recurrence and other diseases. In this paper, a technique of sufficient dimension reduction is applied to the mean and rate function for the number of occurrences of events over time. A two-step procedure is proposed to estimate the model components: first, a nonparametric estimator is proposed for the baseline, and then the basis of the central subspace and its dimension are estimated through a modified slicing inverse regression. On the basis of the estimated structural dimension and on the basis of the central subspace, we can estimate the regression function by using the local linear regression. A simulation is performed to confirm and assess the theoretical findings, and an application is demonstrated on a set of chronic granulomatous disease data.