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STS353 H. Zhao et al.
maximum likelihood estimator (NPMLE) of by using the self-consistency
algorithm (Turnbull, 1976).
Next, we will generalize the estimation procedure above to clustered
interval-censored data and present the WCR estimation procedure. The idea
is to randomly select one subject from each of clusters with replacement
and estimate unknown parameters based on the set of sampled
independent subjects. By repeating this process, one can then perform the
estimation by using the average of the resample–based estimates. More
specifically let be a pre-specified positive integer, the number of the
resampling process described above, and = { , , ; = 1, … , } the
̂
independent sample generated from the ℎ resampling process. Also let
denote the estimator of given by the solution to estimating equation (4)
based on the sample , = 1, … , .
Note that in practice, in addition to estimation of , one may also be
interested in or need estimating the function (). For this, first note that
0
for the corresponding to ( , ], by the assumptions, we have
( < ≤ )
( ≤ | = , = , < ≤ , ) = ∫ () + ( > )
where is defined similarly as = 1, … , . Furthermore, under model
,
(1), one can show that ( ≤ | ) = ( () − ) and
0
0
| {( ≤ )|} = ,| { |,, [( ≤ )|, , ]|}. These naturally
suggest the following estimating equation
( < ≤ )
(2) ((); ) = ∑ { ∫ ̂ () + ( > ) − (() − )} = 0
=1 ̂
̂
for estimating (). Here and below, we take (∙)to be the NPMLE of
(∙), which can be easily obtained by the self-consistency algorithm
(Turnbull, 1976) among other methods. Let ̂ (; )denote the resulting
estimator of ().We propose to estimate and () by the following
0
0
0
WCR estimators
1 1
̂
̂
̂
= ∑ and ̂ () = ∑ ̂ (; ).
=1 =1
3. Results
About the theoretical properties of the estimators, we can show that as
̂
→ ∞ and under some regularity conditions, and ̂ () are consistent
̂
estimators of and (), respectively. Furthermore we have √( −
0
0
) → (0, ∑ ) in distribution.
0
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