Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 73

Some minor modification is needed for 0 < t < h; but as h is the band-

width and will go to 0 with increasing n, the modification is on a vanishing
neighborhood of 0. These smoothed versions of theG n;i 's lead to a smoothed
version of the empirical measure given by

d P n (u;) = d G n;1 (u) + (1 ) d G n;0 (u) :

This can be used to define a smoothed version of the log-likelihood function,

namely,

Z

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

The MSLE, denoted by F M n , is simply the maximizer of l n over all sub-
distribution functions and has an explicit characterization as the slope of a

convex minorant as shown in Theorem 3.1 of Groeneboom et al. (2010). The-
orem 3.5 of that paper provides the asymptotic distribution of F M n (t 0 ) under
certain assumptions, which, in particular, require F and G to be three times
differentiable at t 0 ; under a choice of bandwidth of the form h h n = cn 1=5 ,
it is shown that n 2=5 ( F M n (t 0 )F(t 0 )) converges to a normal distribution with
a non-zero mean. Explicit expressions for this asymptotic bias as well as the
asymptotic variance are provided. The asymptotics for f M n , the natural es-
timate of f which is obtained by differentiating F M n , are also derived; with
a bandwidth of order n 1=7 , n 2=7 ( f M n (t 0 ) f(t 0 )) converges to a normal
distribution with non-zero mean.

The construction of the SMLE simply alters the steps of smoothing and

maximization. The raw likelihood, l n (F) is first maximized to get the MLE
F n , which is then smoothed to get the SMLE:

Z

F S n (t) =

K h (tu) d F n (u) :

Again, under appropriate conditions and, in particular, twice differentia-
bility of F at t 0 , n 2=5 ( F M n (t 0 ) F(t 0 )) converges to a normal limit with a
non-zero mean when a bandwidth of order n 1=5

is used and the asymptotic