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My goal is to find a way to report a mean $\pm$ error for different estimators and quantiles of the same distribution (same measurement).

I am measuring the width of a distribution (Gaussian core and long tails) with different estimators (to check the stability) and for different quantiles (restricting more and more towards the inner core). I found out two stable estimators:

  • the sigma of a Gaussian fit for various quantiles: from 100% (considering the full distribution for the fit) to 80% inner core -- basically cutting more and more of the tails.
  • the median absolute deviation (of the median), again for the same quantiles.

For completeness, below is a picture of the distribution. In there, I fit a Gaussian to the blue part (98% quantile, by removing 1% from both sides of the distribution). For this 98% core, alongside the Gauss sigma reported in red, I get estimators such as AAD, RMS and MAD.

For the different estimators and quantiles I get a list of values and their uncertainty, which I report below:

Estimator, quantile, value, error
MAD, 100, 0.371, 0.018
MAD, 99, 0.364, 0.017
MAD, 98, 0.367, 0.017
MAD, 97, 0.347, 0.017
MAD, 95, 0.363, 0.017
MAD, 93, 0.345, 0.017
MAD, 90, 0.378, 0.018
MAD, 85, 0.363, 0.017
MAD, 80, 0.383, 0.018
Gauss_sigma, 100, 0.335, 0.017
Gauss_sigma, 99, 0.335, 0.017
Gauss_sigma, 98, 0.334, 0.017
Gauss_sigma, 97, 0.339, 0.017
Gauss_sigma, 95, 0.357, 0.017
Gauss_sigma, 93, 0.351, 0.017
Gauss_sigma, 90, 0.337, 0.017
Gauss_sigma, 85, 0.332, 0.017
Gauss_sigma, 80, 0.314, 0.016

Now, these measurements all belong to the same distribution. So, the errors are fully correlated. I am trying to figure out a way to report a mean and the error from these measurements. I think for the error I can do error propagation, considering the correlation coefficient of 1. I would get: $\sigma = \sqrt{\sigma_A^2 + \sigma_B^2 + 2\sigma_A\sigma_B}$ For the mean, I am not sure if calculating a mean from the values makes sense, considering that the values I reported above are not different measurements, but different estimators and quantiles of the same measurement.

Any hints are appreciated :) Thank you!

enter image description here

LATER EDIT:

A bit of clarification on the nature of distribution and why the choice of estimators/quantiles.

The distribution reflects the scattering angle, given by the Moliere theory. When elementary charged particles pass through some material, they suffer thousands of elastic collisions with the electric field of atomic nuclei. The sum of these myriad of interactions is called multiple scattering.

The large number of individual single scatterings lead to a Gaussian scattering angle distribution (according to the central limit theorem) with a mean at 0. There are also single scattering events (hard scattering) which result in a large angle. Such events are more rare and are responsible for the tails you see in the distribution.

The Moliere theory is able to explain the distribution in its entirety. However, I am working with a widely-used (in our community) parametrization (given by Highland-Lynch-Dahl) of the probability density function for the effective scattering angle projected onto one of the lateral dimensions.

In this parametrization, the width of the central part of the distribution (usually the inner 98% quantile) is described by an equation which relates the width of the distribution to a parameter called radiation length, which is material specific. Therefore, if one is to get a measure of the width of the distribution upon irradiating a material sample, one can infer the radiation length of that specific sample.

Moliere theory does not have such a relation to the parameter I am interested. As such, I have to use the parameterization. Moreover, it has been observed that this might not hold in all cases, which is what I am trying to understand.

Therefore, my goal is not to fit such a distribution, but extract some information from it.

For completeness, I have ways of better fitting the distribution (for example a convolution of a Gauss + Student's t distribution). From which I also extract an estimator.

In the end I have a number of estimators, each for a number of quantiles. My goal is to see which ones are robust, what precision they give me, etc ...

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    $\begingroup$ Please edit the question to say more about the nature of the data and what you are trying to accomplish with these measures of the distribution. At first it doesn't seem to make much sense to fit a Gaussian distribution to clearly non-Gaussian data. There might be a better way to accomplish your ultimate goal (perhaps by bootstrapping?), but it's hard to tell without more information about the underlying subject matter. $\endgroup$
    – EdM
    Commented Jun 22 at 14:18
  • $\begingroup$ Is your objective to study these various estimators or would it be to develop a single good estimator of some property of the underlying distribution? In either case, what property would it be? (A "Gauss sigma" is not a recognizable property of a non-Gaussian distribution!) Maybe something like a FWHM? $\endgroup$
    – whuber ♦
    Commented Jun 22 at 15:33
  • $\begingroup$ Hello and thank you for answering. I tried avoiding giving more information in the original post, in order to keep it short. I added a few more details explaining why I made these choices. In the end, what I showed above is an example of an estimator. It might be good, it might be bad. If I find a way of combining the values there into a mean+- error, I can further compare different estimators, their robustness and see which ones might pertain to my distribution (keeping in mind the physics behind) $\endgroup$
    – nyw
    Commented Jun 22 at 16:28

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