A Review of Various Models for Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 31

difference in the properties of the estimators, due to these conventions. Thus

for convenience, we make use the convention L < T R in this chapter. Then

8
<

(L;R] if L < R

the resulting interval I is of the form I = where (a;1) is

:

[T;T] if L = R;

treated as (a;1]. The common likelihood function for interval-censored data

is given by Peto (1973) as

Y

(f(Ti)) if )) if (F(R if ) F(L if )) 1 if L(F) = (2.1)

if

where if is the indicator function of the event fL if = R if g, F is the cdf of T, f
the density function of T under the parametric setup but f(t) = F(t)F(t)
under the nonparametric setup. No further assumptions are made in Peto

(1973). The derivation of the nonparametric maximum likelihood estimator

(NPMLE) under L does not need any further assumptions. Peto supposes

that the MLE or NPMLE under L is consistent and asymptotically ecient

without further assumptions. However, this is not true (see a counterexample

as follows).

Example 1 Suppose that a random sample of IC data are of the forms:

(L if ;R if ) = (1; 2) or (2; 3) or (1;1), each having n if replications. Assume
a parametric setup that fT T (3) = p and fT T (2) = 1 p. By Equation (2.1),
the likelihood based on (L if ;R if )'s is L = p n 2 (1 p) n 1 (1) n 3 . The MLE of p is
p = n 2 =(n 1 + n 2 ).
We shall show that without proper assumptions, the MLE is inconsistent.

Let (U;V ) be a random censoring vector, V = U + 1,

8
<

3=4 if t = 2

if UjT (0jt) = andf UjT (2jt) = 1 f UjT (0jt) (2.2)

:

1=2 if t = 3

8
<

(1;U) if T U

and the observable random vector (L;R)=

(V;1) if T > V

:

(U;V ) if U < T V: