  While the Lomb-Scargle periodogram is foundational to astronomy, it has a
significant shortcoming: the variance in the estimated power spectrum does not
decrease as more data are acquired. Statisticians have a 60-year history of
developing variance-suppressing power spectrum estimators, but most are not
used in astronomy because they are formulated for time series with uniform
observing cadence and without seasonal or daily gaps. Here we demonstrate how
to apply the missing-data multitaper power spectrum estimator to spacecraft
data with uniform time intervals between observations but missing data during
thruster fires or momentum dumps. The F-test for harmonic components may be
applied to multitaper power spectrum estimates to identify statistically
significant oscillations that would not rise above a white noise-based false
alarm probability. Multitapering improves the dynamic range of the power
spectrum estimate and suppresses spectral window artifacts. We show that the
multitaper - F-test combination applied to Kepler observations of KIC 6102338
detects differential rotation without requiring iterative sinusoid fitting and
subtraction. Significant signals reside at harmonics of both fundamental
rotation frequencies and suggest an antisolar rotation profile. Next we use the
missing-data multitaper power spectrum estimator to identify the oscillation
modes responsible for the complex "scallop shell" shape of the K2 light curve
of EPIC 203354381. We argue that multitaper power spectrum estimators should be
used for all time series with regular observing cadence.
