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I often have data, sample sizes typically around $N = 10000$, where I know the data follows a specific distribution -- either because it is simulated or I understand the physics processes that drive it.

What are the best methods of determining the distribution parameters?

The approach I used to take is to bin the data by some rule e.g. Freedman–Diaconis rule or the square-root-$N$ rule, and get the count number for each bin centre and then fit the function to these coordinates. This is very robust, and gives reasonable numbers but as I understand it, this isn't a good approach as this can result in information being "hidden" or lost -- and of course is sensitive to the number of bins and indeed the bin width. However this seems to be the most robust and reliable approach.

Other approaches I have tried include likelihood evaluations, often inbuilt into routines and functions in software such as Mathematica, give results but the more free parameters in a given distribution seems to make finding reasonable values more challenging.

What is the best approach to extract distribution parameters from a data set, when the distribution is known?

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  • $\begingroup$ 1. Usually in large samples, choosing the parameters that maximize the likelihood will do about as well as you can do for parameter estimation (at least if your loss function is of the most typical form). 2. Why do you need to bin the data? What makes it in any way more robust or reliable than simply using all the data unbinned? $\endgroup$
    – Glen_b
    Commented Oct 30, 2019 at 22:47
  • $\begingroup$ @Glen_b This is a good question -- it require a little more detail so I'll add it to my question. $\endgroup$
    – user27119
    Commented Nov 1, 2019 at 11:55
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    $\begingroup$ You seem simply to be stating it's "robust and reliable" a couple of times which doesn't really seem to explain anything. Compared to using unbinned data efficiently, in what sense is it more reliable? More robust against what? $\endgroup$
    – Glen_b
    Commented Nov 1, 2019 at 15:37

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When the distribution (family) is known, and the sample size is large, it is generally difficult to find something better than maximum likelihood. Anyhow, to give more specific advice than that we would need to know the specifics.

It is unclear why you think binning is more robust, generally it is just information loss. But you can still use maximum likelihood with binned data, see Estimate of parameter of exponential distribution with binned data. See also the useful info in comments.

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  • $\begingroup$ I have since learnt the usefulness of Maximum-Likelihood evaluation, thanks! $\endgroup$
    – user27119
    Commented Oct 17, 2020 at 21:09
  • $\begingroup$ Well, your question was very general and did not mention Gaussian mixtures ... $\endgroup$
    – kjetil b halvorsen ♦
    Commented Oct 17, 2020 at 21:10
  • $\begingroup$ I accidentally commented on the wrong post previously! $\endgroup$
    – user27119
    Commented Oct 17, 2020 at 21:11

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