# Variance Reduction Techniques for Rare Events Simulations

PGR-P-1509

## Key facts

Type of research degree
PhD
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Schools
School of Mathematics
Research groups/institutes
Probability and Financial Mathematics, Statistical methodology and probability, Statistics

In many real-life applications, one often encounters the problem of estimating events with very low probabilities, but their occurrences are critical and can result in severe consequences (major earthquakes, floods, etc.). In the field of wireless communication systems, the rare event can, for instance, be the event that the system is in an outage, and hence it does not operate properly. For sake of illustration, error probabilities of the order of 10-9 need to be estimated with high precision for ultra-reliable 5G and 6G wireless systems. Naive Monte Carlo methods can be used to estimate these probabilities. However, it is well-acknowledged that these methods are computationally expensive when dealing with rare events, that is a substantial amount of computational effort is required in order to achieve a good accuracy requirement. Variance reduction techniques are good alternatives that can be used to overcome the failure of naive Monte Carlo methods. Importance sampling, conditional Monte Carlo, and splitting are among the most common variance reduction techniques that were extensively used to develop efficient estimators for rare event probabilities [1]. The main objective of the Ph.D. project is to develop efficient importance sampling techniques to estimate critical rare event quantities. The focus will be on estimating tail probabilities of the form P(S(X) &gt; b), where X is a random vector, S is a given real-valued function, and b is a threshold value. The tail probabilities of this form are motivated by several engineering applications. In financial engineering, the problem of estimating the value-at-risk can be related to that computing the left tail of sums of random variables [3]. The right tail of sums of random variables can model the ruin probability of an insurance company receiving a large number of claims [2]. Finally, tail probabilities of the form P(S(X) &gt; b) can serve to compute error probabilities in the field of wireless communication systems [4,5]. Technically, the focus of the project will be to develop state-dependent importance sampling techniques to efficiently estimate the quantity of interest. By state-dependent schemes, we mean that the importance sampling parameters are dynamically chosen as a function of the system&rsquo;s current time and space. Techniques from dynamic programming will be employed to obtain the optimal importance sampling parameters. The efficiency of the proposed importance sampling estimator will be then compared to state-of-the-art approaches in terms of the amount of variance reduction as well as the total computational effort.

<p style="margin-bottom:11px">References</p> <p>[1] D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo methods, <a href="https://onlinelibrary.wiley.com/series/1345">Wiley Series in Probability and Statistics</a>, 2011.</p> <p>[2] S. Asmussen and P. W. Glynn, Stochastic simulation: algorithms and analysis, Springer Science &amp; Business Media,&nbsp;2007.</p> <p>[3] S. Asmussen, J. L. Jensen, and L. Rojas-Nandayapa, Exponential family techniques for the Lognormal left tail, Scandinavian Journal of Statistics, Vol. 43, No. 3, 2016.</p> <p>[4] N. Ben Rached, A. Kammoun, M.-S. Alouini, and R. Tempone, Unified importance sampling Schemes for efficient simulation of outage capacity over generalized fading channels, IEEE Journal of Selected Topics in Signal Processing (Special Issue on: Stochastic Simulation and Optimization in Signal Processing), Vol. 10, issue 2, Mar. 2016.</p> <p>[5] E. Ben Amar, N. Ben Rached, A.-L. Haji-Ali, and R. Tempone, State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables, arXiv preprint arXiv:2201.01340, 2022.</p>

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