Biomedical Engineering Reference
In-Depth Information
wherefjgdenotes the label of the individuals failing at time t
(j)
, j =
1;:::;m, and K
b
(u) = b
1
K(u=b) forK() being a symmetric nonnegative
kernel function and b a given bandwidth. Let
^
= (
^
0
;:::;
^
p
)
T
maximize
the local log partial likelihood (20). Then
^
()
(x
0
) = !
^
is an estimator
of
()
(x
0
). Note that the local log partial likelihood (20) does not involve
the intercept
0
= (x
0
) because it cancels out. Therefore, the function
value (x
0
) is not directly estimable. The identiability of (x) is ensured
by imposing the condition (0) = 0. The function (x) =
R
x
0
0
(u)du can
be estimated by
Z
x
^
(x) =
^
0
(u)du:
(21)
0
For practical implementation, Tibshirani and Hastie
143
suggested approxi-
mating the integration by the trapezoidal rule.
Fan,
estimator H(t;
^
)
Gijbels,
and
King
suggested
the
=
P
j=1
^
j
I(t
(j)
t) for the cumulative baseline hazard function H(t). Here
m
P
i2R
j
exp(
^
(X
i
))]
1
, which is the Breslow-
type estimator of the baseline hazard function
22;23
^
= (
^
1
;:::;
^
m
)
T
, and
^
j
= [
for
j
in the nonpara-
P
m
j=1
j
I(t
(j)
t) for H(t) and can be obtained
by maximizing `(h; ), given in (17) with respect to = (
1
;:::;
m
)
T
.
H(Y
i
; ) and
^
(X
i
) replace H(Y
i
) and (X
i
), respectively. One can em-
ploy a kernel smoothing technique to obtain an estimate of h(t) via
h(t;
^
) =
metric model H(t; ) =
R
W
g
(tx)dH(x;
^
), where W
g
(u) = g
1
W(u=g) forWbeing
a given kernel function and g a given bandwidth. An alternative approach
to estimating H() and h() is the locally approximated H(t) and h(t) as
follows:
H(t)exp(b
0
+ b
1
(tt
0
)); and h(t)exp(b
0
+ b
1
(tt
0
))b
1
; (22)
where t is in a neighborhood of t
0
. For a given estimator
^
() such as the
one in (21), the local version of the log-likelihood (17) corresponding to the
local linear models (22) can be expressed as
n
n
i
h
i
X
^
(X
i
)
W
g
(Y
i
t
0
)
b
0
+ b
1
(Y
i
t
0
) + log b
1
+
i=1
o
exp[b
0
+ b
1
(Y
i
t
0
)] exp[
^
(X
i
)]
: (23)
Let b
0
and b
1
maximize the local log-likelihood (23). Then H(t
0
) =
exp(b
0
), and h(t
0
) = exp(b
0
)b
1
are smoothed type estimators of H(t
0
) and
h(t
0
), respectively. Consequently, one can have smoothed type estimators
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