I have one conceptual (but sort of vague) question regarding distribution fitting.

How many observations one would need to best fit any statistical distribution to given data. Like I am dealing with loss data and in one situation, I had only three observations still I could fit Normal distribution, which in reality I think is absurd. One more data and the scenario may change.

Hence, is there any thumb rule for minimum number of observations in order to have reasonably stable distribution fitting?

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    $\begingroup$ Such rules will only really be feasible if you fairly precisely define 'reasonably stable' - and even so it will depend on the distributions. $\endgroup$ – Glen_b Apr 10 '13 at 0:16

Well, in the special case of loss data I think most appropriate is the second source, but I give you some others too:

t1 http://imash.leeds.ac.uk/dicode/wp4/Stats2-Forum/view_topic.php?id=10079

t2 http://www.mathwave.com/articles/distribution-fitting-preliminary.html

a pdf: http://www.vanbelle.org/chapters/webchapter2.pdf

and see this related question: How many data points to fit an ex-Gaussian distribution?

I can tell you from my research, that in case of loss returns from stocks you need quite a lot of values to estimate an appropriate distribution correctly. Loss distributions are often leptokurtic and skewed, therefore the assumption of normality can be rejected. Especially if you are interested in Risk Management you focus on the tails. Therefore you will need at least 200 observations from my knowledge to get a basic idea of the distributions of extreme values. If you really want to do extreme value theory for risk management you will have to consider way more observations, since these are values which occur mostly with a small probability, so in case of, let's say 5% you have just 0,05*200=10 extreme values.

Please note, you say "stable distribution". This is not really convenient in this context, since stable distributions are a different topic, e.g. the Cauchy distribution is stable. I think it is more appropriate to say, that the estimators should have a small variance, so they do not vary to much, if you do a reestimation with another sample. This will lead to another statistical topic.


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