  An optimal estimate for Stokes parameters is derived for the situation in
X-ray astronomy where the instrument has a modulation factor that varies
significantly with energy but the signals are very weak or mildly polarized.
For such sources, the band of analysis may be broadened in order to obtain a
significant polarization measurement. Optimal estimators are provided for the
cases of binned and unbinned data and applied to data such as might be obtained
for faint or weakly polarized sources observed using the Imaging X-ray
Polarimetry Explorer (IXPE). For a sample situation, the improvement in the
minimum detectable polarization is 6-7% using a count weighted root-mean-square
of the modulation factor, when compared to a count weighted average. Improving
the modulation factor, such as when using a neural network approach to IXPE
event tracks, can provide additional improvement up to 10-15%. The actual
improvement depends on the spectral shape and the details of the instrument
response functions.
