Biomedical Engineering Reference
In-Depth Information
Moreover, suppose that (L
i
;R
i
) are i.i.d. observations from (L;R). Verify
that (L
i
;R
i
)'s are of the three aforementioned forms. Then the MLE of p
under L is
(n
1
+ n
2
)
!
P(T = 3;U = 2)
n
2
p=2
p =
=
2
p
6= p; unless p = 0 or 1.
1
1
P(U = 2)
4
(1 p) +
In this example, the full likelihood is
= (pf
UjT
(2j3))
n
2
((1 p)f
UjT
(2j2))
n
1
(1 pf
UjT
(2j3) (1 p)f
UjT
(2j2))
n
3
:
The MLE under satises
p f
UjT
(2j3) = n
2
=n and (1 p) f
UjT
(2j2) = n
1
=n:
It is easy to see that if if
UjT
is also unknown, then p is not identifiable. How-
ever, if either if
UjT
(2j2) or f
UjT
(2j3) is known, then the MLE of p under
the full likelihood is consistent. Moreover, if one assumes that U and V are
independent (U ? T), instead of making assumption (2.2), then the MLE
p = n
2
=(n
1
+ n
2
) under L is also consistent.
Example 1 indicates that additional assumptions on the relation between
the underlying censoring vector, say C (= (U;V ) in the example), survival
time T, and the observable random vector (L;R) are needed in order to ensure
the valid statistical inferences under the likelihood function given in Equation
(2.1). Such additional assumptions are discussed in the next two sections.
2.2
Independent Interval Censorship Models
In Section 2.2.1, we state the necessary and sucient condition that the full
likelihood can be simplied to Peto's likelihood . In Section 2.2.2, we introduce
various models for C1 data or C2 data. In Section 2.2.3, we introduce various
models for MIC data. The models in Section 2.2.2 and Section 2.2.3 all make
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