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What is the equivalent of $\sigma$ confidence limits for a normal distribution (68–95–99.7 %) on a gamma distribution ?

And is there a way relate it to the standard deviation and mean (or median) of the distribution, or we have to make a fit on the distribution (since I suppose errorbars will be asymetric) ?

Thank you very much in advance for your help.

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  • $\begingroup$ stats.stackexchange.com/questions/89230/… $\endgroup$
    – Deep North
    Commented Sep 22, 2017 at 14:33
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    $\begingroup$ Since the gamma is asymmetric it's not quite clear what you intend by "equivalent". What is it you need this "equivalent" to tell you? $\endgroup$
    – Glen_b
    Commented Sep 22, 2017 at 15:34
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    $\begingroup$ You seem to be confusing a rule of thumb about probability distributions with some kind of statement about confidence limits. Could you clarify what you're referring to? If you really are referring to the standard 68-95-99.7 rule (which holds for much more than Normal distributions, btw), then could you specify which Gamma distributions you are concerned about? This rule applies nicely to Gamma distributions with shape parameters greater than 20 but not so well to the other Gamma distributions, especially with parameters less than 2. $\endgroup$
    – whuber ♦
    Commented Sep 22, 2017 at 17:39
  • $\begingroup$ Well, approximately holds under some suitable conditions. I don't mind using the 95 rule under unimodality (IMO it performs somewhat better than the other two), but even then it's sometimes a bit rough; IIRC the bounds are between 88.9 and 100% $\endgroup$
    – Glen_b
    Commented Sep 23, 2017 at 1:24

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There will not be a single simple rule for the gamma distribution because one of the parameters affects the shape and the rule will change with the shape. Some gamma distributions can be approximated by a normal distribution fairly well, so the normal 68-95-99.7 rule will be a decent approximation. But other gamma's are extremely skewed (the exponential is a special case of the gamma) and the normal rule will not work in those cases.

One rule that you may be interested in is Chebyshev's Inequality. This tells us that for any distribution with a finite mean and standard deviation that at least 0% of the values will be within 1 standard deviation, 75% within 2 standard deviations, and 89% within 3 standard deviations. See the Wikipedia article linked above for the general rule, a table of other values of interest, and related inequalities that may give you better estimates for specific gamma distributions.

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  • $\begingroup$ Could you clarify why you expect the rule to change with the scale? Because it expresses differences as multiples of $\sigma$ (presumably the SD), it is explicitly scale-independent. Are you perhaps implicitly remarking on the confusion in the question between confidence limits and standard deviation limits? $\endgroup$
    – whuber ♦
    Commented Sep 22, 2017 at 16:09
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    $\begingroup$ @whuber, I was thinking about distributions in general where the mean and sd are not independent (which the 2nd paragraph is about), but you are correct that for the gamma the scale should not matter since the question was in terms of the standard deviation. I have edited my answer accordingly. $\endgroup$
    – Greg Snow
    Commented Sep 22, 2017 at 17:48

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