Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 68

the characterization, we introduce some notation: Let F 03 (t) = 1 F + (t),
a k = (F 0k (t 0 )) 1 for k = 1; 2; 3, and for a nite collection of functions
g 1 ;g 2 ;:::, let g + denote their sum. Groeneboom et al. (2008b) show that
there exist almost surely a unique pair of convex functions ( H 1 ; H 2 ) with
right continuous derivatives ( S 1 ; S 2 ) satisfying

a i H i (h) + a 3

H + (h) a i V i (h) + a 3 V + (h) ; i = 1; 2 and h 2R;

(1)

Z

fa i H i (h)+a 3

H + (h)a i V i (h)a 3 V + (h)gd F i (h) = 0 i = 1; 2 ;

(2)

and

(3) For each M > 0 and i = 1; 2, there are points 1i < M and 2i > M
such that a i H i (h) + a 3

H + (h) = a i V i (h) + a 3 V + (h) for h = i1 ; i2 .

The self-inducedness is clear from the above description as the defining proper-
ties of H 1 and H 2 must be written in terms of their sum. The random functions
S i are the limits of the normalized sub-distribution functions as Theorem 1.8
of Groeneboom et al. (2008b) shows:

fn 1=3 ( F i (t 0 + hn 1=3 ) F 0i (t 0 ))g i=1 ! d (S 1 (h);S 2 (h))

in the Skorohod topology on D(R) 2 . Here, D(R) is the space of real-valued

cadlag functions onRequipped with the topology of convergence in the Sko-

rohod metric on compact sets. In particular, this yields convergence of finite-
dimensional distributions; thus, n 1=3 ( F i (t 0 )F 0i (t 0 )) ! d S i (0) for each i. The
proof of the above process convergence requires the local rate of convergence

of the MLEs of F 01 and F 02 discussed earlier.
It is somewhat easier to characterize the asymptotics of the naive estima-
tor. Let H i denote the GCM of Vi i and let S˜i i denote the right derivative of
H i . Then,

fn 1=3 ( F i (t 0 + hn 1=3 ) F 0i (t 0 ))g i=1 ! d ( S 1 (h); S 2 (h)) :