Rare-event-conditional estimation

As seen in tail risk estimation.

At the moment I mostly care about splitting simulation, but the set-up for that problem is here.

I consider the problem of simulating some quantity of interest conditional on a tail-event defined on a \(d\)-dimensional continuous random variable \(X\) and importance function \(S: \mathbb{R}^d\rightarrow \mathbb{R}\). We write the conditional density \(f^*\) in terms of the density of \(X\) as

\[ f^*(L) = \frac{1}{\ell_\gamma}\mathbb{I}\{L\geq \gamma\} \]

where

\[ \ell(\gamma):=\mathbb{I}\{L\geq \gamma\} \]

is a normalising constant; specifically, \(\ell(\gamma)\) is the cumulative distribution of the random variable \(L.\)

So, using naïve Monte Carlo, you can estimate this by taking the empirical cdf of \(N\) independent simulations of variable \(L_i\) as an estimate of the true cdf:

\[ \hat{\ell}(\gamma) = \frac{1}{N}\sum_{i=1}^N \mathbb{I}\{L_i\geq \gamma\} \]

Note that if the quantity of interest is precisely this cdf for values of \(\gamma\) close to the expectation we may already be done, depending on what we regard as a “good” estimate.

But if we care about rare tail events specifically, we probably need to work harder. Suppose we hold \(\gamma\) fixed and \(\ell (\gamma) \ll 10^{-2}\), we have a bad convergence rate for this estimator. 🚧TODO🚧 clarify convergence rates, number of samples.