THEORETICAL OPTIMALITY OF IMS 71
contain (0,t
0
); these intervals are shown in black. In such cases, the integral (4.28)
can be reduced by breaking B
?
t
into two parts: one part takes more of (0,t
0
), where
V (t) is smallest, and the other part is chosen to satisfy the score-balance condition
(4.27). But when t ≥ t
0
, no improvement can be made by changing B
?
t
; these cases
are shown in gray. So, in this sense, the intervals B
?
t
in (4.29) are not too bad even if
(4.30) fails.
On the other hand, violations of (4.30) are due to the choice of the parametriza-
tion. Indeed, under mild assumptions, there exists a transformation η = η(θ ) such
that the corresponding V (t) function for η satisfies (4.30). Then the predictive ran-
dom set S
?
in Proposition 4.3 is the ...