Biomedical Engineering Reference
In-Depth Information
See pages 1724{1725 of Banerjee and Wellner (2001) for the exact nature of
the scaling relations, from which it follows that
Z
f(g
a;b
(h))
2
(g
a;b
(h))
2
gdh
d
a
2
Z
f(g
1;1
(h))
2
(g
1;1
(h))
2
gdh:
It follows from the definition of a
2
that
RSS(
0
) !
d
0
(1
0
)D:
Thus, RSS=
0
(1
0
) is an asymptotic pivot and confidence sets can be ob-
tained via inversion in the usual manner. Note that the inversion does not
involve estimation of f(t
0
). Now, the RSS is not quite the LRS for testing
F(t
0
) =
0
although it is intimately connected to it. First, the RSS can be
interpreted as a working likelihood ratio statistic where, instead of using the
binomial log-likelihood, we use a normal log-likelihood. Second, up to a scal-
ing factor, RSS(
0
) is asymptotically equivalent to LRS(
0
). Indeed, from the
derivation of the asymptotics for LRS(
0
), which involves Taylor expansions
one can see that
RSS(
0
)
0
(1
0
)
= LRS(
0
) + o
p
(1) ;
the Taylor expansions give a second-order quadratic approximation to the
Bernoulli likelihood, effectively reducing LRS(
0
) to RSS. The third-order
term in the expansion can be neglected as in asymptotics for the MLE and
likelihood ratios in classical parametric settings. I should point out here that
the form ofDalso follows from considerations involving an asymptotic test-
ing problem where one observes a process X(t) = W(t) + F(t), with F(t)
being the primitive of a monotone function f and W being standard Brown-
ian motion onR. This is an asymptotic version of the \signal + noise" model
where the \signal" corresponds to f and \noise" can be viewed as dW(t) (the
point of view is that Brownian motion is generated by adding up little bits of
noise; think of the convergence of a random walk to Brownian motion under
appropriate scaling). Taking F(t) = t
2
, which gives Brownian motion plus
quadratic drift and corresponds to f(t) = 2t, consider the problem of testing
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