No. 1118

Title:

Uniform rate of convergence of the smoothed Nelson-Aalen estimator

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

Kernel estimator, smoothing, Nelson-Aalen estimator, uniform consistency

Abstract:

    In the multiplicative intensity model for counting processes, Ramlau-Hansen (1983, {\em Ann. Statist.}) derived the uniform consistency of the smoothed Nelson-Aalen estimator for the hazard function. We extend this result up to that the rate of the uniform consistency is $ o_{P}(n^{-1/2}b_n^{-1/2-\gamma}) $ for some $ \gamma \in (0,1/2] $, where $ b_n $ is the bandwidth, by using the weak convergence theory for $ \ell^\infty $-valued martingales given by Nishiyama (2000, {\em Ann. Probab.}).