  This article describes two Monte Carlo methods for calculating confidence
intervals on cumulative density function (CDF) based multivariate normal
quantiles that allows for controlling the tail regions of a multivariate
distribution where one is most concerned about extreme responses. The CDF based
multivariate normal quantiles associated with bivariate distributions are
represented as contours and for trivariate distributions represented as
iso-surfaces. We first provide a novel methodology for an inverse problem,
characterizing the uncertainty on the $\tau^{\mathrm{th}}$ multivariate
quantile probability, when using concurrent univariate quantile probabilities.
The uncertainty on the $\tau^{\mathrm{th}}$ multivariate quantile probability
demonstrates inadequacy in univariate methods which neglect correlation between
multiple variates. Limitations of traditional multivariate normal tolerance
regions and simultaneous univariate tolerance methods are discussed thereby
necessitating the need for confidence intervals on CDF based multivariate
normal quantiles. Two Monte Carlo methods are discussed; the first calculates
the CDF over a tessellated domain followed by taking a bootstrap confidence
interval over the tessellated CDF. The CDF based multivariate quantiles are
then estimated from the CDF confidence intervals. For the second method, only
the point associated with highest probability density along the CDF based
quantile is calculated, which greatly improves the computational speed compared
to the first method. Monte Carlo simulation studies are used to assess the
performance of the various methods. Finally, real data analysis is performed to
illustrate a workflow for CDF based multivariate normal quantiles in the domain
of mechanical shock and vibration to specify a minimum conservative test level
for environmental specification.
