1,021 research outputs found
On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances
We consider the problem of evaluating the cumulative distribution function
(CDF) of the sum of order statistics, which serves to compute outage
probability (OP) values at the output of generalized selection combining
receivers. Generally, closed-form expressions of the CDF of the sum of order
statistics are unavailable for many practical distributions. Moreover, the
naive Monte Carlo (MC) method requires a substantial computational effort when
the probability of interest is sufficiently small. In the region of small OP
values, we propose instead two effective variance reduction techniques that
yield a reliable estimate of the CDF with small computing cost. The first
estimator, which can be viewed as an importance sampling estimator, has bounded
relative error under a certain assumption that is shown to hold for most of the
challenging distributions. An improvement of this estimator is then proposed
for the Pareto and the Weibull cases. The second is a conditional MC estimator
that achieves the bounded relative error property for the Generalized Gamma
case and the logarithmic efficiency in the Log-normal case. Finally, the
efficiency of these estimators is compared via various numerical experiments
The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables
We propose a new algorithm to compute numerically the distribution function
of the sum of dependent, non-negative random variables with given joint
distribution.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ284 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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Numerical Methods for PDE Constrained Optimization with Uncertain Data
Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization.
The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods
Importance Sampling and Stratification for Copula Models
An importance sampling approach for sampling from copula models is introduced. The proposed algorithm improves Monte Carlo estimators when the functional of interest depends mainly on the behaviour of the underlying random vector when at least one of its components is large. Such problems often arise from dependence models in finance and insurance. The importance sampling framework we propose is particularly easy to implement for Archimedean copulas. We also show how the proposal distribution of our algorithm can be optimized by making a connection with stratified sampling. In a case study inspired by a typical insurance application, we obtain variance reduction factors sometimes larger than 1000 in comparison to standard Monte Carlo estimators when both importance sampling and quasi-Monte Carlo methods are used.NSERC, Grant 238959
NSERC, Grant 501
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