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whuber
whuber
3:20 PM
@Szabolcs Actually there are many "obvious straightforward methods" that are routinely brought to bear on questions like this. Coincidentally I was answering a similar question when you and @Glen_b were carrying out this conversation:
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A: Expected standard deviation for a sample from a uniform distribution?

whuberThe integration is difficult even with as few as $3$ values. Why not estimate the bias in the sample SD by using a surrogate measure of spread? One set of choices is afforded by differences in the order statistics. Consider, for instance, Tukey's H-spread. For a data set of $n$ values, let $m...

Two ideas were exploited in that analysis: (1) transforming the independent variable in a way motivated by statistical reasoning, intuition, or theory; and (2) using a theoretical suggestion of the nature of the nonlinearity to fit a series of successive approximations (which often is a Taylor series or an asymptotic expansion).
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Regression techniques enable you to model the deviations between the data and the expected asymptotic linear behavior (or whatever the asymptotic behavior is expected to be). When the regression is successful at getting strong fits, you just read the relevant slope (or other information) from its estimates.
In your case, where physical law is involved, you almost always can make reasonable guesses about an appropriate way to model the data.
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