  In stochastic simulation, input uncertainty refers to the propagation of the
statistical noise in calibrating input models to impact output accuracy, in
addition to the Monte Carlo simulation noise. The vast majority of the input
uncertainty literature focuses on estimating target output quantities that are
real-valued. However, outputs of simulation models are random and real-valued
targets essentially serve only as summary statistics. To provide a more
holistic assessment, we study the input uncertainty problem from a
distributional view, namely we construct confidence bands for the entire output
distribution function. Our approach utilizes a novel test statistic whose
asymptotic consists of the supremum of the sum of a Brownian bridge and a
suitable mean-zero Gaussian process, which generalizes the Kolmogorov-Smirnov
statistic to account for input uncertainty. Regarding implementation, we also
demonstrate how to use subsampling to efficiently estimate the covariance
function of the Gaussian process, thereby leading to an implementable
estimation of the quantile of the test statistic and a statistically valid
confidence band. Numerical results demonstrate how our new confidence bands
provide valid coverage for output distributions under input uncertainty that is
not achievable by conventional approaches.
