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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Inference for the mode of a log-concave density: a
likelihood ratio test and confidence intervals -
Jon August Wellner (University of Washington)
DTSTART;TZID=Europe/London:20180403T110000
DTEND;TZID=Europe/London:20180403T120000
UID:TALK103657AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/103657
DESCRIPTION:I will discuss a likelihood ratio test for the mod
e of a log-concave density. The new test is based
on comparison of the log-likelihoods corresponding
to the unconstrained maximum likelihood estimator
of a log-concave density and the constrained maxi
mum likelihood estimator\, where the constraint is
that the mode of the density is fixed\, say at m.
The constrained estimators have many properties i
n common with the unconstrained estimators discuss
ed by Walther (2001)\, Pal\, Woodroofe\, and Meyer
(2007)\, Dü\;mbgen and Rufibach (2009)\, and
Balabdaoui\, Rufibach and Wellner (2010)\, but the
y differ from the unconstrained estimator under th
e null hypothesis on n^{&minus\;1/5} neighborhoods
of the mode m. Using joint limiting properties of
the unconstrained and constrained estimators we s
how that under the null hypothesis (and strict cur
vature of - log f at the mode)\, the likelihood ra
tio statistic is asymptotically pivotal: that is\,
it converges in distribution to a limiting distri
bution which is free of nuisance parameters\, thus
playing the role of the chi-squared distribution
in classical parametric statistical problems. By i
nverting this family of tests\, we obtain new (lik
elihood ratio based) confidence intervals for the
mode of a log-concave density f. These new interva
ls do not depend on any smoothing parameters. We s
tudy the new confidence intervals via Monte Carlo
studies and illustrate them with several real data
sets. The new confidence intervals seem to have s
everal advantages over existing procedures.
\; This talk is based on joint work with Charles
Doss.

LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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