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SUMMARY:Inference for the mode of a log-concave density: a likelihood rati
 o test and confidence intervals - Jon August Wellner (University of Washin
 gton)
DTSTART:20180403T100000Z
DTEND:20180403T110000Z
UID:TALK103657@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:I will discuss a likelihood ratio test for the mode of a log-c
 oncave 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 maximum likelihood estimator\, where t
 he constraint is that the mode of the density is fixed\, say at m. The con
 strained estimators have many properties in common with the unconstrained 
 estimators discussed by Walther (2001)\, Pal\, Woodroofe\, and Meyer (2007
 )\, D&uuml\;mbgen and Rufibach (2009)\, and Balabdaoui\, Rufibach and Well
 ner (2010)\, but they differ from the unconstrained estimator under the nu
 ll hypothesis on n^{&minus\;1/5} neighborhoods of the mode m. Using joint 
 limiting properties of the unconstrained and constrained estimators we sho
 w that under the null hypothesis (and strict curvature of - log f at the m
 ode)\, the likelihood ratio statistic is asymptotically pivotal: that is\,
  it converges in distribution to a limiting distribution which is free of 
 nuisance parameters\, thus playing the role of the chi-squared distributio
 n in classical parametric statistical problems. By inverting this family o
 f tests\, we obtain new (likelihood ratio based) confidence intervals for 
 the mode of a log-concave density f. These new intervals do not depend on 
 any smoothing parameters. We study the new confidence intervals via Monte 
 Carlo studies and illustrate them with several real data sets. The new con
 fidence intervals seem to have several advantages over existing procedures
 .  &nbsp\;  This talk is based on joint work with Charles Doss.  <br><br><
 br><br>
LOCATION:Seminar Room 2\, Newton Institute
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