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

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


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

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 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. 🏗 convergence rates, number of samples.

Importance sampling

🏗 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, \(\mathbb{P}(L\geq\gamma)\).

\[ \ell(\gamma) := \mathbb{P}\left(S\right) \]


Dynamic splitting


Asmussen, Søren, and Peter W. Glynn. 2007. Stochastic Simulation: Algorithms and Analysis. 2007 edition. New York: Springer.

Ben Rached, Nadhir, Zdravko Botev, Abla Kammoun, Mohamed-Slim Alouini, and Raul Tempone. 2018. “On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances.” IEEE Transactions on Wireless Communications 17 (11): 7801–13. https://doi.org/10.1109/TWC.2018.2871201.

Botev, Zdravko I., and Dirk P. Kroese. 2008. “An Efficient Algorithm for Rare-Event Probability Estimation, Combinatorial Optimization, and Counting.” Methodology and Computing in Applied Probability 10 (4): 471–505. https://doi.org/10.1007/s11009-008-9073-7.

———. 2012. “Efficient Monte Carlo Simulation via the Generalized Splitting Method.” Statistics and Computing 22 (1): 1–16. https://doi.org/10.1007/s11222-010-9201-4.

Botev, Zdravko I., Robert Salomone, and Daniel Mackinlay. 2019. “Fast and Accurate Computation of the Distribution of Sums of Dependent Log-Normals.” Annals of Operations Research, February. https://doi.org/10.1007/s10479-019-03161-x.

Botev, Zdravko, and Pierre L’Ecuyer. 2017. “Simulation from the Normal Distribution Truncated to an Interval in the Tail.” In Proceedings of the 10th EAI International Conference on Performance Evaluation Methodologies and Tools, 23–29. VALUETOOLS’16. ICST, Brussels, Belgium, Belgium: ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering). https://doi.org/10.4108/eai.25-10-2016.2266879.

Botev, Z. I. 2017. “The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79 (1): 125–48. https://doi.org/10.1111/rssb.12162.

Cérou, F., P. Del Moral, T. Furon, and A. Guyader. 2011. “Sequential Monte Carlo for Rare Event Estimation.” Statistics and Computing 22 (3): 795–808. https://doi.org/10.1007/s11222-011-9231-6.

Johansen, Adam M., Pierre Del Moral, and Arnaud Doucet. 2006. “Sequential Monte Carlo Samplers for Rare Events.” In Proceedings of the 6th International Workshop on Rare Event Simulation, 256–67. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. Simulation and the Monte Carlo Method. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.