Variance Estimation of Statistics Arising from Replicated Temporal Point Processes and Applications in Residual Analysis and Robust Rate Regression

Abstract This paper is concerned with variance estimation for statistics that are computed from a set of inhomogeneous temporal point processes. Unlike statistics that are de- flned on the whole set of processes, statistics considered here are based on a single process from the set, which are important in diagnosis for each single process. The proposed method only assumes a semiparametric form for the flrst-order structure of the processes but not for the second-order (i.e., dependence) structure. The new vari- ance estimator is shown to be consistent for the target parameter under very mild conditions. The estimator can be used in many,applications in semiparametric rate regression analysis of recurrent event data such as outlier detection, residual diagno- sis as well as robust regression. A simulation study and application to two real data examples are used to demonstrate the use of the proposed method. KEY WORDS: Inhomogeneous Point Process, Recurrent Event Process, Robust Re- gression, Variance Estimation.