Minimax lower bounds for nonparametric estimation of the instantaneous frequency- and time-varying amplitude of a harmonic signal. (Q1566664)

Estimation of the instantaneous frequency- and time-varying amplitude along with their derivatives is considered for a harmonic complex-valued signal given with an additive noise. Asymptotic minimax lower bounds are derived for the mean-squared errors of estimation provided that the phase and amplitude are arbitrary piecewise differentiable functions of time. It is shown that these lower bounds are different only in constant factors from the optimal upper bounds of mean-squared errors of estimates given by the generalized local polynomial periodogram. The time-varying phase and amplitude are derived which are ``worst'', respectively, for estimation of the instantaneous frequency, amplitude and their derivative. These ``worst'' functions can be applied in order to test the accuracy of algorithms used for estimation of the instantaneous frequency and amplitude.