38 research outputs found
A Proof of the Bomber Problem's Spend-It-All Conjecture
The Bomber Problem concerns optimal sequential allocation of partially
effective ammunition while under attack from enemies arriving according to
a Poisson process over a time interval of length . In the doubly-continuous
setting, in certain regions of -space we are able to solve the integral
equation defining the optimal survival probability and find the optimal
allocation function exactly in these regions. As a consequence, we
complete the proof of the "spend-it-all" conjecture of Bartroff et al. (2010b)
which gives the boundary of the region where
Asymptotically optimal multistage tests of simple hypotheses
A family of variable stage size multistage tests of simple hypotheses is
described, based on efficient multistage sampling procedures. Using a loss
function that is a linear combination of sampling costs and error
probabilities, these tests are shown to minimize the integrated risk to second
order as the costs per stage and per observation approach zero. A numerical
study shows significant improvement over group sequential tests in a binomial
testing problem.Comment: Published in at http://dx.doi.org/10.1214/009053607000000235 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficient three-stage -tests
Three-stage -tests of separated one-sided hypotheses are derived,
extending Lorden's optimal three-stage tests for the one-dimensional
exponential family by using Lai and Zhang's generalization of Schwarz's optimal
fully-sequential tests to the multiparameter exponential family. The resulting
three-stage -tests are shown to be asymptotically optimal, achieving the
same average sample size as optimal fully-sequential tests.Comment: Published at http://dx.doi.org/10.1214/074921706000000626 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Berry-Esseen bound for the uniform multinomial occupancy model
The inductive size bias coupling technique and Stein's method yield a
Berry-Esseen theorem for the number of urns having occupancy when
balls are uniformly distributed over urns. In particular, there exists a
constant depending only on such that \sup_{z \in
\mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left(
\frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$
and $m \ge 2$,} where and are the standardized
count and variance, respectively, of the number of urns with balls, and
is a standard normal random variable. Asymptotically, the bound is optimal up
to constants if and tend to infinity together in a way such that
stays bounded.Comment: Typo corrected in abstrac
A Rejection Principle for Sequential Tests of Multiple Hypotheses Controlling Familywise Error Rates
We present a unifying approach to multiple testing procedures for sequential
(or streaming) data by giving sufficient conditions for a sequential multiple
testing procedure to control the familywise error rate (FWER), extending to the
sequential domain the work of Goeman and Solari (2010) who accomplished this
for fixed sample size procedures. Together we call these conditions the
"rejection principle for sequential tests," which we then apply to some
existing sequential multiple testing procedures to give simplified
understanding of their FWER control. Next the principle is applied to derive
two new sequential multiple testing procedures with provable FWER control, one
for testing hypotheses in order and another for closed testing. Examples of
these new procedures are given by applying them to a chromosome aberration data
set and to finding the maximum safe dose of a treatment