Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 56

sum over the i's in a single block. We have

X

( F n (U (i) ) 0 ) 2 X
i2I n

( F n (U (i) ) 0 ) 2

RSS( 0 ) =

i2I n

h f( F n (u) F(t 0 )) 2 ( F n (u) F(t 0 )) 2 g1fu 2 D n g i
= n 1=3 (P n P) h f(n 1=3 ( F n (u) F(t 0 ))) 2
(n 1=3 ( F n (u) F(t 0 ))) 2 g1fu 2 D n g i

= nP n

h f(n 1=3 ( F n (u) F(t 0 ))) 2
(n 1=3 ( F n (u) F(t 0 ))) 2 g1fu 2 D n g i

+ n 1=3 P

:

Empirical processes arguments show that the first term is o p (1) because the
random function that sits as the argument to n 1=3 (P n P) can be shown to
be eventually contained in a Donsker class of functions with arbitrarily high
preassigned probability. Hence, up to a o p (1) term,

= n 1=3 Z f(n 1=3 ( F n (u) F(t 0 ))) 2 (n 1=3 ( F n (u) F(t 0 ))) 2 gdG(t)

RSS( 0 )

D n

= n 1=3 Z

f(n 1=3 ( F n (u) F(t 0 ))) 2 (n 1=3 ( F n (u) F(t 0 ))) 2 gg(t)dt

D n

Z

n
(n 1=3 ( F n (t 0 + hn 1=3 ) F(t 0 ))) 2
(n 1=3 ( F n (t 0 + hn 1=3 ) F(t 0 ))) 2 o g(t 0 + hn 1=3 )dh

=

n 1=3 (D n t 0 )

Z

(X n (h) Y n (h)) g(t 0 ) dh + o p (1) :

=

n 1=3 (D n t 0 )

Now, using Equation (3.5) along with the fact that the set n 1=3 (D n t 0 ) is
eventually contained in a compact set with arbitrarily high probability, we

conclude that

Z

f(g a;b (h)) 2 (g a;b (h)) 2 gdh: RSS( 0 ) ! d g(t 0 )

There are some nuances involved in the above distributional convergence which
we skip. The next step is to invoke Brownian scaling to relate g a;b and g a;b to
the \canonical" slope-of-convex-minorant processes g 1;1 and g 1;1 and use this
to express the limit distribution above in terms of these canonical processes.