  With the growing interest in deploying robots in unstructured and uncertain
environments, there has been increasing interest in factoring risk into
safety-critical control development. Similarly, the authors believe risk should
also be accounted in the verification of these controllers. In pursuit of
sample-efficient methods for uncertain black-box verification then, we first
detail a method to estimate the Value-at-Risk of arbitrary scalar random
variables without requiring \textit{apriori} knowledge of its distribution.
Then, we reformulate the uncertain verification problem as a Value-at-Risk
estimation problem making use of our prior results. In doing so, we provide
fundamental sampling requirements to bound with high confidence the volume of
states and parameters for a black-box system that could potentially yield
unsafe phenomena. We also show that this procedure works independent of system
complexity through simulated examples of the Robotarium.

Systems and Control;Systems and Control