Until the recent advent of extended covariance analysis utilizing quasi-linearization techniques, the only approach for assessing the performance of a nonlinear system with random inputs and initial conditions has been the monte carlo method. This method involves direct simulation, i.e., determining the system response to a finite number of "typical" initial conditions and noise input functions which are generated according to their specified statistics, and averaging over the resulting ensemble of responses ("trials") to obtain estimated or sample statistics.
While the monte carlo method remains the most general trustworthy technique available for the estimation of nonlinear system performance statistics, sample statistics may be unreliable unless hundreds or perhaps thousands of trials are performed. Since computer budget constraints may not permit such extensive simulation of high-order systems, there is often a temptation to "make do" with sample statistics based on 20 to 25 trials. While this procedure might provide meaningful results in cases that are "nearly gaussian", it is dangerous to rely on limited sample statistics if deviations from normality are significant.
This point is demonstrated in detail in this paper using a generalized confidence band concept (not Chisquare) as a measure of the reliability of monte carlo sample statistics for nongaussian random variables. Some compensatory approaches are discussed, including estimating higher-order moments and generating histograms (approximate cumulative distributions).
' E[ ] denotes the expected value of the bracketed variable.