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

Abstract

In semiparametric and nonparametric statistical inference, the asymptotic normality of estimators has been widely established when they are √ n-consistent. In many applications, nonparametric estimators are not able to achieve this rate. We have a result on the asymptotic normality of nonparametric M -estimators that can be used if the rate of convergence of an estimator is n −1/2 or slower. We apply this to study the asymptotic and are easy to implement. Simulation studies show that the proposed tests perform well even for small sample sizes. We find that a new test is always powerful for all the situations considered and is thus robust. For illustration, a data analysis example is provided.

Semi‐supervised inference for nonparametric logistic regression

“…Motivated by Theorem 3.3.1 of van der Vaart and Wellner 58 and Theorem 1 of Zhao and Zhang, 44 we need to verify the following conditions.…”

“…In particular, Lu et al 42,43 studied nonparametric likelihood estimators of the mean function with panel count data by monotone B‐spline. Zhao and Zhang 44 proved the asymptotic normality for a general class of smooth functionals of splines estimators. A computationally efficient algorithm was provided by Zhang and Jamshidiam 45 …”

Semiparametric estimation and testing for panel count data with informative interval‐censored failure event

“…In practical situations, the problem of comparing counting processes with panel count data across multiple samples frequently arises. While nonparametric tests for panel count data have been discussed in many literatures, such as Balakrishnan and Zhao 17 and Zhao and Zhang, 18 in this study, we focus on the same question for the conditional semiparametric mean model of panel count data, assuming that the distributions of the failure event time among different groups are equal. It can be easily extended to the case of unequal distributions by following the approach outlined in Subsec.…”

“…A number of methods have been developed for the analysis of panel count data without associated interval-censored data. They include: parametric approaches (e.g., Hinde; 6 Breslow; 7 Kalbfeisch and Lawless; 8 Thall 9 ), nonparametric estimation approaches (e.g., Sun and Kalbfeisch; 10 Wellner and Zhang; 11 Lu et al; 12 Hu et al 13 ), nonparametric tests (e.g., Thall and Lachin; 14 Sun and Fang; 15 Zhang; 16 Balakrishnan and Zhao; 17 Zhao and Zhang 18 ), and semiparametric regression (e.g., Sun and Wei; 19 Hu et al; 20 Huang et al; 21 Wellner and Zhang; 22 Lu et al; 23 Ma and Sundaram; 24 Zeng and Lin 25 ). Sun and Zhao 1 and Chiou et al 26 provided comprehensive review on existing methods for panel count data analysis.…”