  The direct evaluation of manifestly optimal, cut-sky CMB power spectrum and
bispectrum estimators is numerically very costly, due to the presence of
inverse-covariance filtering operations. This justifies the investigation of
alternative approaches. In this work, we mostly focus on an inpainting
algorithm that was introduced in recent CMB analyses to cure cut-sky
suboptimalities of bispectrum estimators. First, we show that inpainting can
equally be applied to the problem of unbiased estimation of power spectra. We
then compare the performance of a novel inpainted CMB temperature power
spectrum estimator to the popular apodised pseudo-$C_l$ (PCL) method and
demonstrate, both numerically and with analytic arguments, that inpainted power
spectrum estimates significantly outperform PCL estimates. Finally, we study
the case of cut-sky bispectrum estimators, comparing the performance of three
different approaches: inpainting, apodisation and a novel low-l leaning scheme.
Providing an analytic argument why the local shape is typically most affected
we mainly focus on local type non-Gaussianity. Our results show that inpainting
allows to achieve optimality also for bispectrum estimation, but interestingly
also demonstrate that appropriate apodisation, in conjunction with low-l
cleaning, can lead to comparable accuracy.
