Biomedical Engineering Reference
In-Depth Information
i
's are conditionally independent. Consider now the simple RSS for testing
H
0
: F(t
0
) =
0
. The least squares criterion is given by
X
X
[
i
F(U
i
)]
2
=
[
(i)
F(U
(i)
)]
2
:
LS(F) =
i=1
i=1
Equations (3.3) and (3.4) show that F
n
is the least-squares estimate of F
under no constraints apart from the fact that the estimate must be increasing
and that F
n
is the least squares estimate of F under the additional constraint
that the estimate assumes the value
0
at t
0
. Hence, the RSS for testing
H
0
: F(t
0
) =
0
is given by
X
X
[
i
F
n
(U
i
)]
2
[
i
F
n
(U
i
)]
2
:
RSS RSS(
0
) =
i=1
i=1
Before analyzing RSS(
0
), we introduce some notation. Let I
n
denote the set
of indices such that F
n
(U
(i)
) 6= F
n
(U
(i)
). Then, note that the U
(i)
's in I
n
live
in the set D
n
and are the only U
(i)
's that live in that set. Next, letP
n
denote
the empirical measure of f
i
;U
i
g
i=1
. For a function f(;u) dened on the
support of (
1
;U
1
), byP
n
f we mean n
1
P
i=1
f(
i
;U
i
). The function f is
allowed to be a random function. Similarly, if P denotes the joint distribution
of (
1
;U
1
), then by Pf we mean
R
f dP. Here, we use operator notation,
which is standard in the empirical process literature. Now,
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