Regular and Inverse interval Estimation of Individual Observations Using Uniform Confidence Bands

Abstract

Bowden and Graybill construct uniform confidence bands under the familiar simple linear regression model, and use those bands to write joint confidence intervals for expected Y given X = X_o for all X_o in a specified finite range X_a ≤ X_o ≤ X_b . Their approach is extended here to provide joint confidence intervals for individual observations on Y given X, and also for individual X given Y (the inverse interval estimation problem), in both cases using uniform confidence bands constructed for X_a ≤ X ≤ X_b . We show that Bowden and Graybill's tables can be used in these applications, and give supplementary tables to cover additional cases which more commonly arise in these new contexts. Asymptotic results are also given, allowing the user to generate additional “table” values.

The relative efficiencies of the uniform and Scheffé procedures for inverse interval estimation of individual X given Y are extensively compared. It seems that one need not pay a price in efficiency for the convenience of the uniform procedure.

KEY WORDS:

• Interval estimation
• Uniform confidence bands
• Inverse estimation
• Calibration
• Scheffé confidence bands