The Living Thing / Notebooks :

Rare-event-conditional estimation

As seen in tail risk estimation.

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

\begin{equation*} f^*(L) = \frac{1}{\ell_\gamma}\mathbb{I}\{L\geq \gamma\} \end{equation*}


\begin{equation*} \ell(\gamma):=\mathbb{I}\{L\geq \gamma\} \end{equation*}

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:

\begin{equation*} \hat{\ell}(\gamma) = \frac{1}{N}\sum_{i=1}^N \mathbb{I}\{L_i\geq \gamma\} \end{equation*}

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 what we regard as a “good” estimate.

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

Importance sampling

TODO: explain explicitly the variance of this estimator.

I simulate using a different variable \(L'=S(L)\). I am interested in the probability of a random portfolio loss \(L\) exceeding a threshold, \(\mathbb{P}(L\geq\gamma)\).

\begin{equation*} \ell(\gamma) := \mathbb{P}\left(S\right) \end{equation*}


Dynamic splitting



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Botev, Z. I.(2016) The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), n/a-n/a. DOI.
Botev, Z. I., & Kroese, D. P.(2008) An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting. Methodology and Computing in Applied Probability, 10(4), 471–505. DOI.
Botev, Z. I., & Kroese, D. P.(2012) Efficient Monte Carlo simulation via the generalized splitting method. Statistics and Computing, 22(1), 1–16. DOI.
Lam, K., & Botev, Z. (2015, November 1) The Dynamic Splitting Method with an application to portfolio credit risk.
Rubinstein, R. Y., & Kroese, D. P.(2016) Simulation and the Monte Carlo Method. (3 edition.). Hoboken, New Jersey: Wiley