I'm working with measurements for a large sample set and need to quantify the expected extremes. The measurements obviously have some degree of random noise simply in the measurement itself (for the single measurement) but also noise associated with repeated measurement.

The most naive approach is to take the minimum and maximum, but this obviously doesn't account for measurement noise.

Instead, since I have repeated measurements on a small subset of samples, I can calculate a coefficient of variation (CV). If I start with the predetermined CV value, for any given measurement from the bigger sample set, I can determine a measurement-dependent standard deviation:
\begin{align} CV &= \frac{\sigma}{\mu} \\[7pt] \sigma &= CV * \mu \end{align} Then for any new measurement, I can come up with a $\sigma^\prime$ for that measurement level. Using this approach I can simulate measurement variation by drawing from a Gaussian distribution with $\mu = 0$ and $\sigma = \sigma^\prime$, and then adding this value to the actual measurement. Repeating this simulation many times, I can find an empirical confidence interval on the value of any given measurement.

For the smaller values, in this case I'm interested in the minimum, this occasionally results in a sampled noise value that is larger than the measurement itself, and so the adjusted measurement goes below zero. This often happens when there is a large CV. There is clearly a distribution of CV values for the entire distribution of measurement values, but I can't measure that due to costs.

I've thought about using order statistics to approach this, but I don't see how that can account for the variation in repeated measurement described by CV. What other ways might I simulate the expected distribution of the maximum and minimum values taking into account the known variability in the measurement itself?


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