Biomedical Engineering Reference
In-Depth Information
Ramlau-Hansen
116
, Tanner and Wong
136
, and Yandell
157
investigated
the asymptotic properties of the following kernel estimator of the hazard
function h(t) with dierent techniques:
n
n
X
X
(j)
nj + 1
K
#
i
nR
i
+ 1
K
#
(tY
i
) : (4)
h
#
(t) =
tY
(j)
=
j=1
i=1
Here Y
(1)
;:::;Y
(n)
are the ordered Y
i
's;
(1)
;:::;
(n)
are the correspond-
ing censoring indicators; R
i
is the rank of Y
i
; and # is either a positive
valued bandwidth (smoothing parameter) or bandwidth vector. For the
kernel estimator of h(t), # = b and K
#
(u) = b
1
K(u=b) forK(), which
is a symmetric nonnegative kernel with integral
R
K(u)du = 1. The ker-
nel estimator h
#
(t) = h
b
(t) is referred to as a 1-parameter estimator and
can be regarded as a convolution smoothing of the formal derivative of the
empirical cumulative hazard function
P
H(t) =
Y
i
t
i
=(nR
i
+ 1) that
is an N-A estimator of the cumulative hazard function H(t). The kernel
estimator h
b
(t) is a generalized version of the kernel estimator of Watson
and Leadbetter
149
for the uncensored case.
Tanner and Wong
137
proposed a 3-parameter estimator h
#
(t) (4) with
# = (b
1
;b
2
;k), and K
#
(tY
(j)
) = (b
1
d
jk
)
1
K((tY
(j)
)=b
2
d
jk
) for d
jk
being the distance to the kth-nearest failure neighbor in the sample from
the point Y
(j)
. The d
jk
will be large (small) and the kernel will be at
(peaked) in data-sparse (-dense) regions; thus, the 3-parameter estimator
is a variable kernel estimator. They developed a data-based algorithm with
modied-likelihood criterion for bandwidth selection by employing the idea
of cross-validation and showed via a simulation study that the performance
of the data-based 3-parameter estimator is superior to that of the data-
based 1-parameter estimator. Tanner
135
studied the asymptotic properties
of the following variable kernel estimator of h(t):
n
X
1
2d
k
(j)
nj + 1
K
tY
(j)
2d
k
h
d
k
(t) =
;
(5)
j=1
where d
k
is the distance to the kth-closest failure neighbor from t.
To tackle the problems of boundary eects near the endpoints of the
support of h(t) and a substantial increase in the variance from left to right
over the range of abscissas where h(t) is estimated, Muller and Wang
102
modied the kernel estimator h
b
(t) as follows:
n
X
1
b
t
j
nj + 1
K
t
tY
(j)
b
t
h
b
t
(t) =
;
(6)
j=1
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