I would like to estimate extreme right tail quantiles (e.g. 99% or 99.5% tail) of a random variable $X$. The explicit representation of $X$ is unknown and there is no a-priori reason to assume it belongs to a well known parametric family. I do have access to simulated values of $X$ and I estimate my tail quantile by simulating a sample of a few thousand (maybe more) iid instances $(X_i)$ of $X$. Then I use the $\alpha N$ order statistic $T_{\alpha}[X] = X_{(\alpha N)}$ as an estimate of the tail quantile.

The problem is that I cannot simulate $X$ directly. I can only obtain noisy observations $(Y_i)$ of a distribution $Y= X + \epsilon$. The error $\epsilon$ has a normal distribution $\epsilon\sim \mathcal{N}(0, \sigma(x))$. While the size of $\sigma$ may be a function of $X$ the $\epsilon$ pertaining to different $X$ are independent. Since $X$ and thus also $Y$ depend on other risk factors in my simulation, I am able to obtain a few (say 3 to 5) repeated measurements of $Y(\omega)$ for the same underlying value $X(\omega)$ .

Sadly, the noise is very large in comparison to the size and differences of adjacent order statistics of $(Y_i)$. This leads to something very similar to the "regression to the mean" effect (see also here): My tail quantile estimate for $X$ obtained by the order statistic from the sample $(Y_i)$ is terribly biased upwards, i.e. much too large, due to the noise. This is the case even though I currently average over the repeated measurements, hence reducing the noise somewhat.

The next thing to try is to regress $X$ on $Y$, i.e. calculate the regression function $f(y)=E[X|Y=y]$. I am slightly reluctant, since this is a lot of work and hard to get right. I would like to make sure that there is no better alternative and I will obtain a sufficiently strong improvement of the estimate.


  1. What can be said about the size of the bias in the noisy estimate $T_{\alpha}[Y]$ versus the "clean" estimate value $T_{\alpha}[X]$?
  2. Are there general properties of $X$ (or ideally observable ones from $Y$) which drive or limit this error?
  3. Assume a good parameterisation of the regression function is possible and I find it. Does the regression approach lead to improved estimates of tail quantiles? Can this improvement be quantified? Practically speaking, at what value $y$ would I evaluate the regression function $f(y)$ to obtain an estimate for the $\alpha$ quantile of $X$?
  4. Are there other approaches to this estimation problem? Especially ones which do not require an explicit parameterisation of the regression function.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.