I'm developing a method which estimates a value $t$ for a very specific problem.

Assume that we have an individual $s$ with its associated data but where $t$ is not precisely known and a reference series $S$, with known associated values.

By using simple correlation between $s$ and the individuals of $S$, I get a correlation profile, which (because of the nature of the data) may be oscillatory. We know that the estimate of $t$ we want is one of the peaks in this profile, which may not necessarily be the highest one. Since we have a rough estimate of $t$, the approach we have opted for consists of building a gaussian around it, and scoring the peaks accounting for their density on the "guess gaussian" as well as their correlation score. This image may clear up the situation a bit (the gaussian is not to scale) :


When the peaks are rather clean like above, the only error estimate I can give would be the resolution of the reference series. The issue comes with data that produces correlation profiles with "plateaus" rather than peaks. In such cases, the "guess gaussian" will have a rather significant impact on which local maximum the method will pick and to account for this problem, we have implemented a kind of bootstrap. This bootstrap shifts the initial guesstimate within a given window and re-estimates $t$ (note that the correlation profile does not change). By doing this about a hundred times, we get an average estimate which is at the plateau's center and here is where I'm split on the error estimation, what do I do? Can I give the $2.5$ and $97.5$ quantiles of the different estimates and call it a confidence interval ? Is there an specific error I can compute here ?


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