Nonparametric Inference For Density Modes
This work addresses the need for reliable statistical inference in density estimation, particularly for mode analysis, offering practical tools for researchers in statistics and data science, though it is incremental in building on existing nonparametric methods.
The paper tackles the problem of inferring the strength and shape of density modes by deriving nonparametric confidence intervals for Hessian eigenvalues at modes, using a data-splitting approach and a bootstrap method to handle eigenvalue multiplicities. It also introduces a bandwidth selection method that maximizes significant modes, showing effectiveness even with singular distributions where cross-validation fails.
We derive nonparametric 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 using the first half of the data and inference is done with the second half of the data. To get valid confidence sets for the eigenvalues, we use a bootstrap based on an elementary-symmetric-polynomial (ESP) 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.