Estimation of marginal and spectral modes

We investigate the asymptotic behaviour of the empirical mode of an i.i.d. sample. We also study the asymptotic properties of the empirical spectral mode of a sample generated by a Gaussian stationary process. The empirical estimators are built using the empirical histogram or the empirical spectrogram, respectively. Under various regularity assumptions, including the case of a singular mode, we compute the optimal window width and study the rate of convergence of our estimates.