Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 72

conditions. If F is twice differentiable around t 0 , it is not unreasonable to ex-
pect that appropriate estimators of F(t 0 ) will converge faster than n 1=3 . This
is suggested, firstly, by results in classical nonparametric kernel estimation of

densities and regression functions where kernel estimates of the functions of
interest exhibit the n 2=5 convergence rate under a (local) twice-differentiability

assumption on the functions and, secondly, by the work of Mammen (1991) on

kernel-based estimation of a smooth monotone function while respecting the

monotonicity constraint. In a recent paper, Groeneboom et al. (2010) provide

a detailed analysis of smoothed kernel estimates of F in the current status

model.

Two competing estimators are proposed by Groeneboom et al. (2010): the

MSLE (maximum smoothed likelihood estimator), originally introduced by

Eggermont and LaRiccia (2001) in the context of density estimation, which

is a general likelihood-based M estimator and turns out to be automatically

smooth, and the SMLE (smoothed maximum likelihood estimator), which is

obtained by convolving the usual MLE with a smooth kernel. IfP n denotes the
empirical measure of the f i ;U i g's, the log-likelihood function can be written
as

Z

l n (F) = f log F(u) + (1 ) log(1 F(u))gdP n (;u) :

For i 2f0; 1g, dene the empirical sub-distribution functions

X

1
n

G n;i (u) = 1 [0;u]fig (T j ;U j ) :

j=1

Note that dP n (u;) = dG n;1 (u) + (1 ) dG n;0 (u). Now, consider a proba-
bility density k that has support [1; 1], is symmetric and twice continuously

differentiable onR; let K denote the corresponding distribution function; and

let K h (u) = K(u=h) and k h (u) = (1=h) k(u=h), where h > 0. Consider now
kernel-smoothed versions of theG n;i 's given by

G n;i (t) = R

g n;i (u) du for

[0;t]

i = 0; 1, where

Z

g n;i (t) = k h (tu) dG n;i (u) :