A Review of Various Models for Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 33

use of the independence or conditional independence assumption between T

and the censoring variables.

2.2.1 Full Likelihood

For a random variable, say R, let F R and f R be the cdf and density function,
respectively, and let S R = 1F R . Hereafter, abusing notation, we write F =
F T . Let f I and F I be the \density function" and \cdf" of the random set I. If I
takes countably many values, then f I (I) = P(I = I). Notice that L(FT T ) does
not involve censoring distribution, and according to the generalized likelihood
function defined by (Kiefer and Wolfowitz, 1956), the full likelihood of these

I i 's is really

Y

= f I (I i ):

i=1

Proposition 2.1 The full likelihood can be simplied as L(F) i

8
<

f(t)G(I) if I = ftg

f I (I) = (2.3)

R I dF(t)G(I)

:

if I = (l;r] for each value I of I;

where

G is a function of I and G is non-informative(about F); (2.4)

that is, G and F do not share a common parameter.

Various models have been proposed for the IC data since 1955 so that

condition (2.3) and (2.4) hold. These model assumptions are getting weaker

and weaker, so that they are getting more realistic to the real applications

and people can determine whether the statistical procedures are applicable

by checking the model assumptions. A good model also provides a valid way

to carry out simulation studies to examine the properties of certain statisti-

cal procedures when the theory on them has not been resolved. Notice that

in Example 1, Equation (2.2) implies that T 6? (U;V ). However, if one as-
sumes T ? U, instead of assuming Equation (2.2), then by setting G(I) =