
136 Statistical inference, based on Kaplan–Meier estimator
Under the first choice, the martingale M
m,n
becomes
M
m,n
G
(t) =
1
p
mn(m + n)
Z
t
0
[Y
2m
(s)dN
1n
(s) −Y
1n
(s)dN
2m
(s)],
and its quadratic variation becomes
hM
m,n
G
i(t) =
Z
t
0
Y
1n
(s)
n
Y
2m
(s)
m
Y
1n
(s) +Y
2m
(s)
m + n
µ(s)ds.
We approximate this quadratic variation by
A
m,n
G
(t) =
Z
t
0
Y
1n
(s)
n
Y
2m
(s)
m
dN(s),
which is also computationally simpler – cf. Breslow and Crowley
[1970], where A
m,n
G
(∞) was considered.
To verify the conditions of the central limit theorem for M
m,n
G
is
easy. The resulting statement is
if the hypothesis (11.13) is true and m,n → ∞, then
M
m,n
G
d
→W = w ◦A
G
,
where
A
G
(t) =
Z
t
0
[1 −G
1
(s)][1 −G
2
(s)][2 −G
1
(s) −G
2
(s)][1