  Quasi-periodic oscillations (QPOs) are an important key to understand the
dynamic behavior of astrophysical objects during transient events like
gamma-ray bursts, solar flares, and magnetar flares. Searches for QPOs often
use the periodogram of the time series and perform spectral density estimation
using a Whittle likelihood function. However, the Whittle likelihood is only
valid if the time series is stationary since the frequency bins are otherwise
not statistically independent. We show that if time series are non-stationary,
the significance of QPOs can be highly overestimated and estimates of the
central frequencies and QPO widths can be overconstrained. The effect occurs if
the QPO is only present for a fraction of the time series and the noise level
is varying throughout the time series. This can occur for example if background
noise from before or after the transient is included in the time series or if
the low-frequency noise profile varies strongly over the time series. We
confirm the presence of this bias in previously reported results from solar
flare data and show that significance can be highly overstated. Finally, we
provide some suggestions that help identify if an analysis is affected by this
bias.
