An exact bootstrap approach towards modification of the Harrell–Davis quantile function estimator for censored data

A new kernel quantile estimator is proposed for right-censored data, which takes the form of , where w j(u, c) is based on a beta kernel with bandwidth parameter c. The advantage of this estimator is that exact bootstrap methods may be employed to estimate the mean and variance of [Qcirc](u; c). It follows that a novel solution for finding the optimal bandwidth may be obtained through minimization of the exact bootstrap mean squared error (MSE) estimate of [Qcirc](u; c). We prove the large sample consistency of [Qcirc](u; c) for fixed values of c. A Monte Carlo simulation study shows that our estimator is significantly better than the product-limit quantile estimator [Qcirc] KM(u)=inf{t:[Fcirc] n (t)≥u}, with respect to various MSE criteria. For general simplicity, setting c=1 leads to an extension of classical Harrell–Davis estimator for censored data and performs well in simulations. The procedure is illustrated by an application to lung cancer survival data.

[1]  W. J. Padgett.  A Kernel-Type Estimator of a Quantile Function from Right-Censored Data , 1986 .

[2]  E. Parzen,et al.  Unified estimators of smooth quantile and quantile density functions , 1997 .

[3]  Mei Ling Huang,et al.  On a distribution-free quantile estimator , 2001 .

[4]  W. J. Padgett,et al.  Smooth nonparametric quantile estimation under censoring: simulations and bootstrap methods , 1986 .

[5]  W. J. Padgett,et al.  On the Asymptotic Properties of a Kernel-Type Quantile Estimator from Censored Samples. , 1986 .

[6]  Sana S BuHamra,et al.  Inference concerning quantile for left truncated and right censored data , 2004, Comput. Stat. Data Anal..

[7]  James Stephen Marron,et al.  Kernel Quantile Estimators , 1990 .

[8]  Stephen M. Stigler,et al.  Fractional Order Statistics, with Applications , 1977 .

[9]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[10]  QUANTILE ESTIMATION FOR LEFT TRUNCATED AND RIGHT CENSORED DATA , 2000 .

[11]  H. A. David,et al.  Order Statistics (2nd ed). , 1981 .

[12]  J. Faraway,et al.  Bootstrap choice of bandwidth for density estimation , 1990 .

[13]  Xiaojing Xiang.  Deficiency of the Sample Quantile Estimator with Respect to Kernel Quantile Estimators for Censored Data , 1995 .

[14]  Frank E. Harrell,et al.  A new distribution-free quantile estimator , 1982 .

[15]  Alan D. Hutson,et al.  The exact bootstrap mean and variance of an L‐estimator , 2000 .

[16]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[17]  Song-xi Chen,et al.  Beta kernel estimators for density functions , 1999 .

[18]  D. Klippenstein,et al.  Semiquantitative Visual Approach to Scoring Lung Cancer Treatment Response Using Computed Tomography: A Pilot Study , 2009, Journal of computer assisted tomography.