Design Aspects of Regression-Based Ratio Estimation

Abstract

This article is concerned with statistical design for the estimation of a ratio of mean values in a regression model. This work is motivated, in part, by an important regulatory issue whereby automotive emissions (hydrocarbon, carbon monoxide, and oxides of nitrogen) are related linearly to accumulated vehicle mileage. The statistical design problem is the specification of the frequency and location of points in the design space where observations are to be taken. The optimal design is determined by minimizing the asymptotic variance of the ratio of least squares point estimators. We derive such designs for the ordinary linear-regression model and for the linear-spline-regression model. In both cases, we allow some of the observations to be fixed so that they are not available for allocation in the variance minimization. The designs depend explicitly on the unknown regression parameters. Thus an assessment of sensitivity of these designs to reasonable specifications of model parameters is recommended. The automotive-emissions example is used to illustrate the derivation of these new designs.

KEY WORDS:

• Asymptotics
• Linear model
• Linear-spline regression
• Optimal designs
• Sensitivity analysis