Confidence regions for non-parametric estimators with periodic excitation and coloured noise

We construct confidence regions for non-parametric transfer function estimates when the system noise is coloured Gaussian and the excitation is periodic. Our construction stems from the observation that the real and imaginary parts of the corresponding probability density function are independent after a rotation whose angle is independent of the measurement data. However the construction also requires the errors in each periodogram to be independent, so the regions are not suitable for data where the periodograms are recorded consecutively. We illustrate our results with a simulation.