Nonparametric Mode Hunting

We propose a procedure for detecting the modes of a density estimate and test their significance. We use a data-splitting approach: potential modes are identified using the first half of the data and their significance is tested with the second half of the data. The mode test is based on nonparametric confidence intervals for the eigenvalues of the Hessian. In order to get valid bootstrap confidence sets even in presence of multiplicity of the eigenvalues, we use a bootstrap based on an elementary-symmetric-polynomial transformation.