Modelling the tail only

Question

I'm trying to model a real-world random variable that behaves approximately as a Gaussian, so a Normal distribution fit is reasonable but far from perfect.

However, I only care about its tail, that is, I only care about finding $x$ such as $P(X > x) < p$, with $p$ anywhere between 1% and 5%. I'm modelling the random variable as $X\sim\mathcal{N}(\mu, \sigma^2)$, with parameters estimated from all the samples.

I tried to search for specific ways to handle this and I got a lot of results regarding Extreme Value Theory, but it seems to apply mostly to even smaller probabilities than I care or to a heavy-tailed distributions, which is not my case.

Any pointers?

Answer 1

The devil, as they say, is in the tails. Many have been led to misleading and wrong conclusions by using an easy-to-handle modeling tactic (e.g. the Normal!), which failed them spectacularly when the time came to do statistical inference -which mostly depends on what happens in the tails -and the empirical distribution of the data may have looked adequately normal in its "main body", but a different story in the tails. And you are exclusively concerned with the tails.

In a comment, you mention that you have some skewness, but most importantly, negative excess kurtosis. Not many distributions supported on the real line exhibit negative excess kurtosis. In fact the first that comes to mind is the "Generalized Normal Distribution". It can exhibit negative excess kurtosis if the new parameter (instead of the square) is sufficiently large. Its second version can also handle skewness. But you should find a way to calculate the relevant probabilities - since they will depend on the value of this new parameter, I doubt that any tables already calculated by others will match your case.