# Assessing and Visualizing Simultaneous Simulation Error

## TL;DR

This paper develops a multivariate CLT for simultaneous estimation of means and quantiles in Monte Carlo experiments, providing a practical visualization tool for assessing the reliability of these estimates.

## Contribution

It introduces a multivariate CLT for combined means and quantiles under mixing conditions and a fast algorithm for hyperrectangular confidence regions with simultaneous coverage.

## Key findings

- The method applies to i.i.d. and MCMC samples.
- It enables visual assessment of Monte Carlo estimate reliability.
- The approach improves interpretation of simulation results.

## Abstract

Monte Carlo experiments produce samples in order to estimate features of a given distribution. However, simultaneous estimation of means and quantiles has received little attention, despite being common practice. In this setting we establish a multivariate central limit theorem for any finite combination of sample means and quantiles under the assumption of a strongly mixing process, which includes the standard Monte Carlo and Markov chain Monte Carlo settings. We build on this to provide a fast algorithm for constructing hyperrectangular confidence regions having the desired simultaneous coverage probability and a convenient marginal interpretation. The methods are incorporated into standard ways of visualizing the results of Monte Carlo experiments enabling the practitioner to more easily assess the reliability of the results. We demonstrate the utility of this approach in various Monte Carlo settings including simulation studies based on independent and identically distributed samples and Bayesian analyses using Markov chain Monte Carlo sampling.