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Statistica Sinica 21 (2011), 901-921



Jiancheng Jiang$^1$ and Xuejun Jiang$^{2,3}$

$^1$University of North Carolina, Charlotte, $^2$Zhongnan University of
Economics and Law and $^3$The Chinese University of Hong Kong

Abstract: The partly linear additive Cox model is a useful tool for modeling failure time data with multiple covariates. The global smoothing method based on polynomial splines has been demonstrated as an efficient estimation approach for this model in the sense that it achieves the semiparametric information bound. However, there is no method available for consistently estimating the asymptotic variance matrix of the resulting estimators of finite parameters, which hampers inference for the model. This motivates us to propose a bootstrap method for estimating the distributions of the estimators; it is shown to be consistent. Moreover, to test linear hypotheses on the finite parameters, we propose a new test statistic and obtain its asymptotic null distribution. We show that the test is consistent and can detect alternatives nearing the null hypothesis at a rate of $\sqrt{n}$. Our results enable inference about the model based on the efficient polynomial splines estimation. Simulations are conducted to demonstrate nice performance of the proposed method. A data example is also given.

Key words and phrases: Bootstrap, conditional hazard rate, hypothesis testing, partial likelihood, polynomial spline.

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