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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 703-728

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.093

\vspace{3pc} \normalsize \centerline{\large\bf QUANTILE REGRESSION
FOR COMPETING RISKS DATA} \vspace{2pt} \centerline{\large\bf WITH
MISSING CAUSE OF FAILURE} \vspace{.25cm}
\author{Yanqing Sun$^1$, Huixia Judy Wang$^2$ and Peter B. Gilbert$^{3,4}$}
\vspace{.4cm} \centerline{\it $^1$University of North Carolina,
Charlotte, $^2$North Carolina State University,} \centerline{\it
$^3$University of Washington and $^4$Fred Hutchinson Cancer Research
Center} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} This paper considers generalized linear
quantile regression  for competing risks data when the failure type
may be missing. Two estimation procedures for the regression
coefficients, including an inverse probability weighted
complete-case estimator and an augmented inverse probability
weighted estimator, are discussed under the assumption that the
failure type is missing at random. The proposed estimation
procedures utilize supplemental auxiliary variables for predicting
the missing failure type and for informing its distribution. The
asymptotic properties of the two estimators are derived and their
asymptotic efficiencies are compared. We show that the augmented
estimator is more efficient and possesses a double robustness
property against misspecification of either the model for
missingness or for the failure type. The asymptotic covariances are
estimated using the local functional linearity of the estimating
functions. The finite sample performance of the proposed estimation
procedures are evaluated through a simulation study. The methods are
applied to analyze the `Mashi' trial data for investigating the
effect of formula- versus breast-feeding plus extended infant
zidovudine prophylaxis on HIV-related death of infants born to
HIV-infected mothers in Botswana.\par \vspace{9pt} \noindent {\it
Key words and phrases:} Augmented inverse probability weighted,
auxiliary variables, competing risks, double robustness,
 efficient estimator, estimating equation, inverse probability weighted, local functional linearity,
 logistic regression, Mashi trial, missing at random, quantile regression. \par
\end{quotation}
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