Order Statistics and Probabilistic Robust Control

TL;DR

This paper integrates order statistics with probabilistic robust control to determine minimal sample sizes for reliable estimates, extends theory to non-continuous distributions, and offers practical guidelines for performance-risk tradeoffs.

Contribution

It develops a distribution-free approach for estimating bounds of uncertain quantities, relaxing the continuity assumption in order statistics theory, and provides a practical tradeoff guideline.

Findings

Derived a CDF for order statistics without continuity assumptions.

Established an upper bound inequality for the distribution of order statistics.

Provided conditions for minimal sample size under weaker assumptions.

Abstract

Order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution. Also, the concept of distribution-free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution. To overcome the limitations imposed by the continuity assumption in the existing order statistics theory, we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied. By applying this inequality, we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound (or lower bound) and the range of a quantity. We also give conditions, which are much weaker than the absolute continuity assumption, for the existence of such minimum sample size. Furthermore, the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established. This guideline can be applied in general without continuity assumption.