The bias-variance trade-off in Thomson's multitaper estimator

At the heart of non-parametric spectral estimation, lies the dilemma known as the bias-variance trade-off: low biased estimators tend to have high variance and low variance estimators tend to have high bias. In 1982, Thomson introduced a multitaper method where this trade-off is made explicit by choosing a target bias resolution and obtaining a corresponding variance reduction. The method became the standard in many applications. Its favorable bias-variance trade-off is due to an empirical fact, conjectured by Thomson based on numerical evidence: assuming bandwidth W and N time domain observations, the average of the square of the first $K=\left\lfloor 2NW\right\rfloor$ Slepian functions approaches, as K grows, an ideal band-pass kernel for the interval [-W,W]. We provide an analytic proof of this fact and quantify the approximation error in the L1 norm; the approximation error is then used to control the bias of the multitaper estimator resulting from spectral leakage. This leads to new performance bounds for the method, explicit in terms of the bandwidth W and the number N of time domain observations. Our method is flexible and can be extended to higher dimensions and different geometries.