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This article proposes a simple nonparametric estimator of quantile residual lifetime function under left-truncated and right-censored data. The asymptotic consistency and normality of this estimator are proved and the variance expression is calculated. Two bootstrap procedures are employed in the simulation study, where the latter bootstrap from Zeng & Lin (2008) is 4000 times faster than the former naive one, and the numerical results in both methods show that our estimating approach works well. A real data example is used to illustrate its application. Keywordsleft-truncated, right-censored, quantile residual lifetime, empirical process MSC(2010) 62N01, 90B25Citation: Wang Y X, Liu P, Zhou Y. Quantile residual lifetime for left-truncated and right-censored data.

Abstract:Nonparametric and semiparametric regression models are useful statistical regression models to discover nonlinear relationships between the response variable and predictor variables. However, optimal efficient estimates for the nonparametric components in the models are biased which hinders the development of methods for further statistical inference. In this paper, based on the local linear fitting, we propose a simple bias reduction approach for the estimation of the nonparametric regression model. The new approach does not need to use higher-order local polynomial regression to estimate the bias, and hence avoids the double bandwidth selection and the design sparsity problems suffered by higher-order local polynomial fitting. It also does not inflate the variance. Hence it can be easily applied to complex statistical inference problems. We extend our new approach Statistica Sinica: Newly accepted Paper (accepted version subject to English editing)to varying coefficient models, and to estimate the variance function and to construct simultaneous confidence band for the nonparametric regression function.Simulations are carried out for comparisons with existing methods, and a real data example is used to investigate the performance of the proposed method.