Variance Reduction Techniques for Rare Events Simulations

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) > 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) > 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’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 & Business Media, 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>

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>.Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PhD Statistics FT,</strong></em> in the research information section that the research degree you wish to be considered for is <em><strong><span style="font-size:11.0pt"><span style="line-height:107%"><span style="font-family:"Calibri",sans-serif">Variance Reduction Technique for Rare Events Simulations</span></span></span></strong></em> as well as <a href="https://eps.leeds.ac.uk/faculty-engineering-physical-sciences/staff/12206/dr-nadhir-ben-rached">Dr Nadhir Ben Rached</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong></em>.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p class="MsoNoSpacing"><strong>Please note that you must provide the following documents in support of your application by the closing date of Monday 6 January 2025 if applying for the China Scholarship Council-University of Leeds Scholarship, Monday 3 February 2025 if applying for Leeds Doctoral Scholarship or Tuesday 1 April 2025 for Leeds Opportunity Research Scholarship.</strong></p> <p><strong>If you are applying for the School of Mathematics Scholarship 2025/26, or with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

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