
126 Statistical inference, based on Kaplan–Meier estimator
but what this value of θ is the hypothesis does not prescribe – it re-
mains unknown and we replace it in our testing procedures by its esti-
mator
b
θ. As a consequence, the process
M
n
(t,
b
θ) = N
n
(t) −
Z
t
0
Y
n
(y)µ(y,
b
θ)dy,
unlike the process (9.6), is not a martingale any more and the limit
theorem (10.4) for M
n
(·,
b
θ) is not true.
The possibilities to resolve this problem were investigated in
Maglapheridze et al. [1989] in a relatively general framework. Namely,
using the method suggested in Khmaladze [1981], the authors pro-
posed a transformation of M
n
(·,
b
θ), similar to the one which we used
in Lecture ...