Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 52

by connecting the points f(x i ;y i )g i=0 successively, by means of straight lines.
Next, consider the so-called CUSUM (cumulative sum) \diagram" given by
fi=n ; P i=1 (i) =ng i=0 . Then,

i X

fv i g i=1 = slogcmfi=n ; (j) =ng i=0 ;

j=1

0

while

i X

fv i g i=1 =

@ slogcmfi=n ;

(j) =ng i=0 ^ 0 ;

j=1

1
A :

i X

(m+j) =ng nm slogcmfi=n ;

i=0 _ 0

j=1

The maximum and minimum in the above display are interpreted as be-
ing taken component-wise. The limiting versions of the MLEs (appropriately

centered and scaled) have similar characterizations as in the above displays.

It turns out that for determining the behavior of LRS( 0 ), only the behavior
of the MLEs in a shrinking neighborhood of the point t 0 matters. This is a
consequence of the fact that D n ft : F n (t) 6= F n (t)g is an interval around
t 0 whose length is O p (n 1=3 ). Interest therefore centers on the processes

X n (h) = n 1=3 ( F n (t 0 + hn 1=3 ) F(t 0 )) and

Y n (h) = n 1=3 ( F n (t 0 + hn 1=3 ) F(t 0 ))

for h in compacts. The point h corresponds to a generic point in the interval

D n . The distributional limits of the processes X n and Y n are described as
follows: For a real-valued function f dened onR, let slogcm(f;I) denote

the left-hand slope of the GCM of the restriction of f to the interval I. We

abbreviate slogcm(f;R) to slogcm(f). Also dene

slogcm 0 (f) = (slogcm(f; (1; 0])^0)1(1; 0] + (slogcm(f; (0;1))_0)1(0;1) :

For positive constants c;d, let X c;d (h) = cW(h) + dh 2 . Set g c;d (h) =
slogcm(X c;d )(h) and g c;d (h) = slogcm 0 (X c;d )(h). Then, for every positive K,

(X n (h);Y n (h)) ! d (g a;b (h);g a;b (h)) in L 2 [K;K] L 2 [K;K] ; (3.5)