Optimality

168 D. M. Dabrowska
We have|x(t)| ≤ max{�y�∞,�y −�∞} exp � t
0 d�b�v and� exp[− �
· d�b�v]|x|�∞ ≤
max{�y�∞,�y −�∞}. If yθ(t), and bθ(t) = � t
0 kθ(u)n(du) are functions dependent
on a Euclidean parameter θ∈Θ⊂R d , and|kθ|(t)≤k(t), then these bounds hold
pointwise in θ and
sup{exp[−
t≤τ
θ∈Θ
� t
Acknowledgement
0
k(u)n(du)]|xθ(t)|}≤max{sup
u≤τ
θ∈Θ
|yθ|(u), sup|yθ(u−)|}.
u≤τ
θ∈Θ
The paper was presented at the First Erich Leh–mann Symposium, Guanajuato,
May 2002. I thank Victor Perez Abreu and Javier Rojo for motivating me to write it.
I also thank Kjell Doksum, Misha Nikulin and Chris Klaassen for some discussions.
The paper benefited also from comments of an anonymous reviewer and the Editor
Javier Rojo.
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