Biomedical Engineering Reference
In-Depth Information
which allows for variable degrees of smoothing at dierent time points and
implementation of boundary kernels. Here the bandwidth b
t
and the ker-
nelK
t
() depend on the time point t, where the estimate of h(t) is to be
computed.K
t
() is a kernel if t is in the interior region; it is a polynomial
boundary kernel if t is in the boundary regions. See [102] for details of con-
struction of polynomial boundary kernels. Other specic boundary kernels
have been considered by Gasser, Muller, and Mammitzsch
57
, Muller
101
, and
Messer and Goldstein
99
. Additionally, removing boundary eects has been
proposed by Muller
100
, Hougarrd
73
, Hougaard, Plum, and Ribel
74
, and by
Hall and Wehrly
67
.
Liebscher
96
derived uniform strong convergence rates of kernel estima-
tors of the density and hazard functions when the failure times form a
stationary -mixing sequence; his results represented an improvement over
Cai's
28
.
2.2. Spline-based Estimation
Let y
1
< y
2
<< y
s
denote the distinct uncensored and censored times,
and let m
i
and c
i
be the uncensored and censored numbers, respectively,
at y
i
. The log-likelihood of h() can be expressed as
Z
X
n
y
j
`(h) =
j
log(h(y
j
))
h(u)du
(7)
0
j=1
"
#
Z
X
s
y
i
=
m
i
log(h(y
i
))(m
i
+ c
i
)
h(u)du
:
(8)
0
i=1
To get a nonnegative estimate of h(t), by developing Anderson and Senthil-
selvan's approach
8
to estimating the baseline hazard function in the PH
model, Senthilselvan
127
made the substitution (t) =
p
h(t) and applied the
penalized likelihood technique that was introduced by Good and Gaskins
60
in the context of nonparametric probability density estimation, by adding
the roughness penalty functional
R
I
f
0
(u)g
2
du to the log-likelihood `(h)
(8) to obtain the penalized log-likelihood
"
2m
i
log((y
i
))(m
i
+ c
i
)
#
Z
s
X
y
i
2
(u)du
`
p
() =
0
i=1
Z
f
0
(u)g
2
du:
(9)
I
Here is the smoothing parameter to be used throughout this chapter. It
regulates the trade-o between smoothness and goodness-of-t. The
0
() is
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