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I am looking for references about calculating confidence intervals for mode (in general). Bootstrap may seem to be natural first choice, but as discussed by Romano (1988), standard bootstrap fails for mode and it does not provide any simple solution. Did anything change since this paper? What is the best way to calculate confidence intervals for mode? What is the best bootstrap-based approach? Can you provide any relevant references?

Romano, J.P. (1988). Bootstrapping the mode. Annals of the Institute of Statistical Mathematics, 40(3), 565-586.

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  • $\begingroup$ For "in general" you mean a multivariate possibly multimodal joint density with unbounded domain and no pre-specified parametric form? Or are there some constraints? $\endgroup$
    – GeoMatt22
    Commented Oct 16, 2016 at 21:05
  • $\begingroup$ @GeoMatt22 say that we are dealing with unimodal distribution, with or without pre-specified parametric form. As calculating mode in multidimensional case gets complicated, it'd be interesting enough to start with unidimensional case. $\endgroup$
    – Tim
    Commented Oct 16, 2016 at 21:08
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    $\begingroup$ OK, and also unbounded then? (e.g. not Beta w/a mode at 0 or 1.) The parametric case seems easiest, as the mode would be well defined in terms of the parameters. $\endgroup$
    – GeoMatt22
    Commented Oct 16, 2016 at 21:15
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    $\begingroup$ How are you estimating the location of the mode? $\endgroup$
    – Glen_b
    Commented Oct 16, 2016 at 22:32
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    $\begingroup$ FYI for KDE modes, the "mean shift" algorithm of computer vision may be relevant. (Not an answer, but perhaps a pointer to another relevant branch of the literature.) $\endgroup$
    – GeoMatt22
    Commented Oct 17, 2016 at 15:45

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While it appears there hasn't been too much research into this specifically, there is a paper that did delve into this on some level. The paper On bootstrapping the mode in the nonparametric regression model with random design (Ziegler, 2001) suggests the use of a smoothed paired bootstrap (SPB). In this method, to quote the abstract, "bootstrap variables are generated from a smooth bivariate density based on the pairs of observations."

The author claims that SPB "is able to capture the correct amount of bias if the pilot estimator for m is over smoothed." Here, m is the regression function for two i.i.d. variables.

Good luck, and hope this gives you a start!

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  • $\begingroup$ Smoothed bootstrap would be something that I'd actually consider, but haven't seen it suggested anywhere yet. Thanks! There is no other answers so I'm awarding the bounty to this answer. I'm not accepting it since I'm still hoping to possibly get other answers and suggestions. $\endgroup$
    – Tim
    Commented Mar 6, 2017 at 21:59

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