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Non‐parametric inference for density modes
Author(s) -
Genovese Christopher R.,
PeronePacifico Marco,
Verdinelli Isabella,
Wasserman Larry
Publication year - 2016
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12111
Subject(s) - hessian matrix , eigenvalues and eigenvectors , inference , mathematics , parametric statistics , bandwidth (computing) , confidence interval , algorithm , statistics , computer science , artificial intelligence , physics , computer network , quantum mechanics
Summary We derive non‐parametric confidence intervals for the eigenvalues of the Hessian at modes of a density estimate. This provides information about the strength and shape of modes and can also be used as a significance test. We use a data‐splitting approach in which potential modes are identified by using the first half of the data and inference is done with the second half of the data. To obtain valid confidence sets for the eigenvalues, we use a bootstrap based on an elementary symmetric polynomial transformation. This leads to valid bootstrap confidence sets regardless of any multiplicities in the eigenvalues. We also suggest a new method for bandwidth selection, namely choosing the bandwidth to maximize the number of significant modes. We show by example that this method works well. Even when the true distribution is singular, and hence does not have a density (in which case cross‐validation chooses a zero bandwidth), our method chooses a reasonable bandwidth.