  Thus far, likelihood-based interval estimates for quantiles have not been
studied in the literature on interval censored case 2 data and partly interval
censored data, and, in this context, the use of smoothing has not been
considered for any type of censored data. This article constructs smoothed
weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles
in a unified framework for various types of censored data, including right
censored data, doubly censored data, interval censored data and partly interval
censored data. The fourth order expansion of the weighted empirical
log-likelihood ratio is derived and the theoretical coverage accuracy equation
for the proposed WELRCI is established, which generally guarantees at least
`first order' accuracy. In particular, for right censored data, we show that
the coverage accuracy is at least $O(n^{-1/2})$ and our simulation studies show
that in comparison with empirical likelihood-based methods, the smoothing used
in WELRCI generally provides a shorter confidence interval with comparable
coverage accuracy. For interval censored data, it is interesting to find that
with an adjusted rate $n^{-1/3}$, the weighted empirical log-likelihood ratio
has an asymptotic distribution completely different from that obtained by the
empirical likelihood approach and the resulting WELRCI perform favorably in the
available comparison simulation studies.
