Estimation of the mean and variance response surfaces when the means and variances of the noise variables are unknown

Abstract

The means and variances of noise variables are typically assumed known in the design and analysis of robust design experiments. However, these parameters are often not known with certainty and estimated with field data. Standard experimentation and optimization conducted with the estimated parameters can lead to results that are far from optimal due to variability in the data. In this paper, the estimation of the mean and variance response surfaces are considered using a combined array experiment in which estimates of the means and variances of the noise variables are obtained from random samples. The effects of random sampling error on the estimated mean and variance models are studied and a method to guide the design of the sampling effort and experiment to improve the estimation of the models is proposed. Mathematical programs are formulated to find the sample sizes for the noise variables and number of factorial, axial and center point replicates for a mixed resolution design that minimize the average variances of the estimators for the mean and variance models. Furthermore, an algorithm is proposed to find the optimal design and sample sizes given a candidate set of design points.