Characterising non-parametric estimators in closed-loop: The finite data case

We consider indirect frequency domain non-parametric transfer function plant estimators in closed-loop. We make the assumption that the real and imaginary parts of the corresponding closed-loop system transfer function estimate at each frequency have Gaussian distribution, but do not necessarily have equal variance, nor are they necessarily independent. We characterise the probability density function of the plant transfer function estimate, and show it to have a unique minimum, and at most two maxima.