On finding confidence intervals for the inverse regression method of linear calibration

Abstract

In the linear calibration problem one estimates the independent variable x in a regression situation for a measured value of the dependent variable y. The Classical estimator is obtained by expressing the linear model as y_i = α + βx_i + ε_i, obtaining the least squares estimator for y for a given value of x and solving for x. The Inverse estimator is gotten by expressing the linear model as x_i = γ + δy_i + ε_i̇ and using the resulting least squares estimator to estimate x. The purpose of this dissertation is to find confidence intervals for the Inverse Method of linear calibration, then make a comparison of the Inverse and Classical estimators using the criterion of confidence intervals.

First, Monte Carlo techniques are used to determine the worth of a Bayes confidence interval for the Inverse Method. Next, a confidence interval for the Inverse Method is derived which has the same length, confidence, and restriction possessed by the known confidence interval for the Classical Method. Finally, an unrestricted confidence interval for the Inverse Method is sought by making use of the density of the Inverse estimator. In an effort to accomplish this, attempts are made at both deriving a workable expression for the density and describing the density with Pearson's system of frequency curves.