Biomedical Engineering Reference
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of t
0
and that f(t
0
);g(t
0
) > 0. Let F
n
denote the NPMLE of F that can
be obtained using PAVA (Robertson et al. (1988)). Then, Theorem 5.1 of
Groeneboom and Wellner (1992) shows that
4 F(t
0
) (1 F(t
0
)) f(t
0
)
g(t
0
)
1=3
n
1=3
( F
n
(t
0
) F(t
0
)) !
d
Z;
(3.1)
whereZ= arg min
t2
R
fW(t) +t
2
g, with W(t) being standard two-sided Brow-
nian motion starting from 0. The distribution of (the symmetric random vari-
able)Zis also known as Cherno's distribution, having apparently arisen rst
in the work of Chernoff (1964) on the estimation of the mode of a distribution.
A so-called \Wald-type" condence interval can be constructed, based on the
above result. Letting f(t
0
) and g(t
0
) denote consistent estimators of f(t
0
) and
g(t
0
), respectively, and q(Z;p) the p-th quantile ofZ, an asymptotic level 1
CI for F(t
0
) is given by
h
F
n
(t) n
1=3
C q(Z;=2) ;
i
F
n
(t) + n
1=3
C q(Z;=2)
(3.2)
where
4 F
n
(t
0
) (1 F
n
(t
0
)) f(t
0
)
g(t
0
)
!
1=3
C
consistently estimates the constant C sitting in front ofZin Equation (3.1).
One of the main challenges with the above interval is that it needs consistent
estimation of f(t
0
) and g(t
0
). Estimation of g is possible via standard density
estimation techniques because an i.i.d. sample of G is at our disposal. However,
estimation of f is significantly more dicult. The estimator F
n
is piecewise
constant and therefore nondifferentiable. One therefore has to smooth F
n
. As
is shown in Groeneboom, Jongbloed, and Witte (2010), a paper I will come
back to later, even under the assumption of a second derivative for F in the
vicinity of t
0
, an assumption not required for the asymptotics of the NPMLE
above, one obtains only an (asymptotically normal) n
2=7
consistent estima-
tor of f. This is (unsurprisingly) much slower than the usual n
2=5
rate in
standard density estimation contexts. Apart from the slower rate, note that
f in a finite sample can depend heavily on bandwidth
the performance of
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