12 research outputs found
Unstructured sequential testing in sensor networks
We consider the problem of quickly detecting a signal in a sensor network
when the subset of sensors in which signal may be present is completely
unknown. We formulate this problem as a sequential hypothesis testing problem
with a simple null (signal is absent everywhere) and a composite alternative
(signal is present somewhere). We introduce a novel class of scalable
sequential tests which, for any subset of affected sensors, minimize the
expected sample size for a decision asymptotically, that is as the error
probabilities go to 0. Moreover, we propose sequential tests that require
minimal transmission activity from the sensors to the fusion center, while
preserving this asymptotic optimality property.Comment: 6 two-column pages, To appear in the Proceedings 2013 IEEE Conference
on Decision and Control, Firenze, Italy, December 201
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
A Linear Programming Approach to Sequential Hypothesis Testing
Under some mild Markov assumptions it is shown that the problem of designing
optimal sequential tests for two simple hypotheses can be formulated as a
linear program. The result is derived by investigating the Lagrangian dual of
the sequential testing problem, which is an unconstrained optimal stopping
problem, depending on two unknown Lagrangian multipliers. It is shown that the
derivative of the optimal cost function with respect to these multipliers
coincides with the error probabilities of the corresponding sequential test.
This property is used to formulate an optimization problem that is jointly
linear in the cost function and the Lagrangian multipliers and an be solved for
both with off-the-shelf algorithms. To illustrate the procedure, optimal
sequential tests for Gaussian random sequences with different dependency
structures are derived, including the Gaussian AR(1) process.Comment: 25 pages, 4 figures, accepted for publication in Sequential Analysi
Sequential Tests of Multiple Hypotheses Controlling Type I and II Familywise Error Rates
This paper addresses the following general scenario: A scientist wishes to
perform a battery of experiments, each generating a sequential stream of data,
to investigate some phenomenon. The scientist would like to control the overall
error rate in order to draw statistically-valid conclusions from each
experiment, while being as efficient as possible. The between-stream data may
differ in distribution and dimension but also may be highly correlated, even
duplicated exactly in some cases. Treating each experiment as a hypothesis test
and adopting the familywise error rate (FWER) metric, we give a procedure that
sequentially tests each hypothesis while controlling both the type I and II
FWERs regardless of the between-stream correlation, and only requires arbitrary
sequential test statistics that control the error rates for a given stream in
isolation. The proposed procedure, which we call the sequential Holm procedure
because of its inspiration from Holm's (1979) seminal fixed-sample procedure,
shows simultaneous savings in expected sample size and less conservative error
control relative to fixed sample, sequential Bonferroni, and other recently
proposed sequential procedures in a simulation study