Biomedical Engineering Reference
In-Depth Information
where L
2
[K;K] is the space of real-valued square-integrable functions de-
ned on [K;K] while
s
F(t
0
) (1 F(t
0
))
g(t
0
)
!
and b =
f(t
0
)
2
a =
:
These results can be proved in dierent ways, by using \switching relation-
ships" developed by Groeneboom (as in Banerjee (2000)) or through contin-
uous mapping arguments (as developed in more general settings in Banerjee
(2007)).
Roughly speaking, appropriately normalized versions of the CUSUM di-
agram, converge in distribution to the process X
a;b
(h) in a strong enough
topology that renders the operators slogcm and slogcm
0
continuous. The MLE
processes X
n
;Y
n
are representable in terms of these two operators acting on
the normalized CUSUM diagram, and the distributional convergence then
follows via continuous mapping. While I do not go into the details of the rep-
resentation of the MLEs in terms of these operators, their relevance is readily
seen by examining the displays characterizing fv
i
g and fv
i
g above. Note, in
particular, the dichotomous representation of fv
i
g depending on whether the
index is less than or greater than m and the constraints imposed in each seg-
ment via the max and min operations, which is structurally similar to the
dichotomous representation of slogcm
0
depending on whether one is to the
left or the right of 0. I do not provide a detailed derivation of LRS(
0
) in this
review. Detailed proofs are available both in Banerjee (2000) and Banerjee
and Wellner (2001), where it is shown that
Z
f(g
1;1
(h))
2
(g
1;1
(h))
2
gdh:
LRS(
0
) !
d
D
However, I do illustrate whyDhas the particular form above by resorting to a
residual sum of squares statistic (RSS), which leads naturally to this form. So,
for the moment, let's forget LRS(
0
) and view the current status model as a
binary regression model, indeed, the conditional distribution of
i
given Ui
i
is
Bernoulli (F(U
i
)) and given the (Ui))
i
's (which we now think of as covariates), the
Search WWH ::
Custom Search