Estimation of the mean function with panel count data using monotone polynomial splines

Abstract

We study nonparametric likelihood-based estimators of the mean function of counting processes with panel count data using monotone polynomial splines. The generalized Rosen algorithm, proposed by Zhang & Jamshidian (2004), is used to compute the estimators. We show that the proposed spline likelihood-based estimators are consistent and that their rate of convergence can be faster than n 1/3. Simulation studies with moderate samples show that the estimators have smaller variances and mean squared errors than their alternatives proposed by Wellner & Zhang (2000). A real example from a bladder tumour clinical trial is used to illustrate this method.

Asymptotic Normality of Nonparametric Z-estimators with Applications to Hypothesis Testing for Panel Count Data

“…Conditions C1-C5 and C7 are required by Lu, Zhang and Huang (2007); condition C6 is required by Balakrishnan and Zhao (2009). Conditions C8 and C9 are similar to C11 in Wellner and Zhang (2007).…”

“…For estimation of the smooth function Λ 0 (t), we use B-spline function approximation (Lu, Zhang, and Huang (2007)). Let T = {s i , i = 1, .…”

“…Following Lu, Zhang and Huang (2007), the estimatorsΛ ps n andΛ n are the values that maximize l ps n (Λ) and l n (Λ) with respect to Λ ∈ Ψ n , respectively.…”

“…Here, the nonparametric maximum likelihood estimatorsΛ l,mle , Λ 0,mle were computed by using the modified iterative convex minorant algorithm in Wellner and Zhang (2000); the spline likelihood estimatorsΛ l andΛ 0 were computed by using the algorithm in Lu, Zhang and Huang (2007). The results reported here are based on 1000…”

“…Sun and Kalbfeisch (1995), Wellner and Zhang (2000), Lu, Zhang, and Huang (2007), and Hu, Lagakos, and Lockhart (2009a, b) studied the nonparametric estimation of the mean function of the underlying counting process with panel count data by using isotonic regression techniques, the likelihood approach, the spline likelihood approach, 2 the estimating equation approach, and the generalized least squares method, respectively; Thall and Lachin (1988), Sun and Fang (2003), Zhang (2006), and Balakrishnan and Zhao (2009) presented some nonparametric tests for nonparametric comparison of mean function of counting process with panel count data. For a comprehensive review about the analysis of panel count data, see Sun and Zhao (2013).…”