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Inference for the mode of a log-concave density: a likelihood ratio test and confidence intervals

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STS - Statistical scalability

I will discuss a likelihood ratio test for the mode 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 maximum likelihood estimator, where the constraint is that the mode of the density is fixed, say at m. The constrained estimators have many properties in common with the unconstrained estimators discussed by Walther (2001), Pal, Woodroofe, and Meyer (2007), Dümbgen and Rufibach (2009), and Balabdaoui, Rufibach and Wellner (2010), but they differ from the unconstrained estimator under the null hypothesis on n^{−1/5} neighborhoods of the mode m. Using joint limiting properties of the unconstrained and constrained estimators we show that under the null hypothesis (and strict curvature of – log f at the mode), 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 distribution in classical parametric statistical problems. By inverting this family of 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 confidence intervals seem to have several advantages over existing procedures.   This talk is based on joint work with Charles Doss.



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