Confidence Intervals on Multivariate Normal Quantiles for Environmental Specification Development in Multi-axis Shock and Vibration Testing

Abstract: 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.