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\noindent Statistica Sinica {\bf 13}(2003), 519-537
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\centerline{\large\bf EFFICIENT ESTIMATION FOR THE PROPORTIONAL}
\vspace{2pt}
\centerline{\large\bf HAZARDS MODEL WITH LEFT-TRUNCATED AND}
\vspace{2pt}
\centerline{\large\bf ``CASE 1'' INTERVAL-CENSORED DATA}
\vspace{.25cm}
\centerline{Jong S. Kim}
\vspace{.4cm}
\centerline{\it Portland State University}
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\begin{quotation}
\noindent {\it Abstract:}
The maximum likelihood estimator (MLE) for the proportional hazards model with
{\it left-truncated and ``Case $1$'' interval-censored data} is studied. Under
appropriate regularity conditions, the MLE of the regression parameter is shown
to be asymptotically normal with a root-n convergence rate and achieves the
information bound, even though the difference between left-truncation time
and censoring time of the MLE of the baseline cumulative hazard function
converges only at rate $n^{1/3}$. Two methods to estimate the
variance-covariance matrix of the MLE of the regression parameter are
considered. One is based on a generalized missing information principle and the
other is based on the profile information procedure. Simulation studies show
that both methods work well in terms of bias and variance for samples of
moderate sizes. An example is provided to illustrate the methods.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Asymptotic distribution, left-truncated and ``Case 1'' interval-censored data,
proportional hazards model, variance estimation.
\end{quotation}
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