curve fittingdistributions
I want to fit an ex-Gaussian distribution (which has three parameters: $\mu$, $\sigma$, and $\nu$) to reaction time data.
How many data points are needed as a minimum to reasonably fit this distribution?
I am aware that an answer to this question depends on many factors (e.g., how much data actually deviate from the theoretical distribution) and that "more is better" in general. Anyway, are there any rules-of-thumb, or publications which could point me to the needed minimum of data points?
The first thing to do, if possible, is to take care of the heteroscedasticity. Notice how the spread of the residuals consistently increases with the fit: in fact, the spread seems to increase almost quadratically with larger fit.
A standard cure is to return to the original response ($log(xy)$) and apply a strong transformation, such as a logarithm or even a reciprocal: something in that range is suggested by this pattern of heteroscedasticity. Then redo the fitting and recheck the residuals.
It's a good idea to fit lines by eye, using graphs of transformed $xy$ against $z$ (or $\log(z)$. This usually reveals more than any amount of manipulating a regression routine. Once you have a suitable model, you can finally use least squares (or robust regression) to produce a final fit.
In this instance you might also want to explore the relationships among $x$ and $z$ and $y$ and $z$ separately to see whether just one of $x$, $y$ is causing the sudden change in slope between 2.9 and 3.6. The change clearly is not quadratic: both "limbs" of the residual plot are linear. One way to model this change--if it persists after you have dealt with the heteroscedasticity--is with a changepoint model that posits one value of the slope $\beta_1$ for, say, $z \le 3.2$, and a different value for $z \gt 3.2$.
Finally, in Monte-Carlo simulations you have full control and you know exactly the mechanism that produces the responses. It can be useful to subject this to some analysis to find out just what the correct relationship among the triples ought to be.
Well, in the special case of loss data I think most appropriate is the second source, but I give you some others too:
http://imash.leeds.ac.uk/dicode/wp4/Stats2-Forum/view_topic.php?id=10079
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.
Best Answer
If I recall correctly, as sample size decreases, mu will manifest positive bias while tau will manifest negative bias. Sigma will manifest either positive or negative bias depending on true values of the other parameters. This paper compares two methods of fitting ex-Gaussians, and from the figures I'd probably avoid going below 40 observations.