  We analytically and numerically investigate the performance of weak-value
amplification (WVA) and related parameter estimation methods in the presence of
temporally correlated noise. WVA is a special instance of a general measurement
strategy that involves sorting data into separate subsets based on the outcome
of a second "partitioning" measurement. Using a simplified noise model that can
be analyzed exactly together with optimal statistical estimators, we compare
WVA to a conventional measurement method. We find that introducing WVA indeed
yields a much lower variance of the parameter of interest than does the
conventional technique, optimized in the absence of any partitioning
measurements. In contrast, a statistically optimal analysis that employs
partitioning measurements, incorporating all partitioned results and their
known correlations, is found to yield an improvement -- typically slight --
over the noise reduction achieved by WVA. This is because the simple WVA
technique is not tailored to a given noise environment and therefore does not
make use of correlations between the different partitions. We also compare WVA
to traditional background subtraction, a familiar technique where measurement
outcomes are partitioned to eliminate unknown offsets or errors in calibration.
Surprisingly, in our model background subtraction turns out to be a special
case of the optimal partitioning approach in the balanced case, possessing a
similar typically slight advantage over WVA. These results give deeper insight
into the role of partitioning measurements, with or without post-selection, in
enhancing measurement precision, which some have found puzzling. We finish by
presenting numerical results to model a more realistic laboratory situation of
time-decaying correlations, showing our conclusions hold for a wide range of
statistical models.
