Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 65

of failure. However, data are often available not only on the status of an

individual{that is, whether they have failed or not at the time of observation|
but also on the cause of failure. A classic example in the clinical setting is that

of a woman's age at menopause, where the outcome of interest is whether

menopause has occurred, U is the age of the woman, and the two competing

causes for menopause are either natural or operative.

More generally, consider a system with K (finite) components that will

fail as soon as one of its component fails. Let T be time to failure, Y be

the index of the component that fails, and U be the (random) observation

time. Thus, (T;Y ) has a joint distribution that is completely specied by
the sub-distribution functions fF 0i g i=1 , where F 0i (t) = P(T t;Y = i).
The distribution function of T, say F + , is simply P i=1 F 0i and the survival
function of T is S(t) = 1 F + (t). Apart from U, we observe a vector of
indicators = ( 1 ; 2 ;:::; K+1 ), where i = 1fT U;Y = ig for i =
1; 2;:::;K and K+1 = 1 P j=1 j = 1fT > Ug. A natural goal is to
estimate the sub-distribution functions, as well as F + . Competing risks in
the more general setting of interval-censored data was considered by Hudgens

et al. (2001) and the more specific case of current status data was investigated

by Jewell et al. (2003). In what follows, I restrict to two competing causes

(K = 2) for simplicity of notation and understanding; everything extends

readily to more general (finite) K but the case of infinitely many competing

risks, the so-called \continuous marks model," is dramatically dierent and I

touch upon it later.

Under the assumption that U is independent of (T;Y ), the likelihood func-
tion for the data that comprise n i.i.d. observations f j ;U j g j=1 , in terms of
generic sub-distribution functions F 1 ;F 2 is

Y

F 1 (U i ) j 1 ;F 2 (U i ) j 2 S(U i ) j 3 ;

L n (F 1 ;F 2 ) =

j=1

this follows easily from the observation that the conditional distribution of

given U is multinomial. Maximization of the above likelihood function is