Generating correlated random variables for quality control applications

Abstract

A great deal of data in business, economics, engineering, and the natural sciences occur in the form of time series, where observations are dependent. In such cases, identification techniques may be used to identify a specific model and the likelihood function can be used to provide maximum likelihood estimates of the model parameters. It may also be desirable to generate a sequence of numbers x1, x2,...,xn so that the individual xi's have specified univariate distributions and the correlations ρij between xi and xj are specified by the modeler. For quality control applications, when the charting variable is the sample standard deviation s, the control limits are functions of E (s) and std (s). In this paper the density function of s is derived for n=2 and n=3, and a sequence of random numbers is generated which possesses that density function. For n ≥ 4 the density function f(s) becomes extremely complicated and other methods have to be used.