Signals in coloured noise: joint non-parametric estimation of signal and of noise spectrum

There is a considerable literature on non-parametric estimation of signals buried in coloured noise of known or finitely parameterised spectrum. And a considerable literature on non-parametric estimation of the spectrum of an observed coloured noise. Here we consider the bivariate problem of jointly, non-parametrically estimating a signal in coloured noise as well as the coloured noise spectrum. It is straightforward enough to develop estimators for each infinite dimensional parameter but what is much less clear is how to choose jointly the two required tuning parameters. We develop, apparently for the first time, a criterion that achieves this.