On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

TL;DR

This paper develops efficient variance reduction techniques for accurately estimating the cumulative distribution function of the sum of order statistics, crucial for analyzing wireless communication system performance, especially in low outage probability scenarios.

Contribution

It introduces two novel variance reduction methods, including importance sampling and conditional Monte Carlo estimators, with proven efficiency for challenging distributions.

Findings

The importance sampling estimator has bounded relative error for most distributions.

The conditional Monte Carlo estimator is efficient for Generalized Gamma and Log-normal distributions.

Numerical experiments demonstrate the effectiveness of the proposed methods.

Abstract

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.