Confidence Regions in Linear Functional Relationships

Abstract

A unified approach to deriving confidence regions in linear functional relationship models is presented, based on the conditional likelihood ratio method of Knowles, Siegmund and Zhang. In the case of a single latent predictor, the confidence region for the slope produced by this approach is the familiar one of Fieller and Creasy. However, here it is shown how to derive a confidence region for the slope, when it is known that the slope is positive, that improves on merely intersecting the region for an unrestricted slope with $(0,\infty)$. A geometric interpretation is given for Fieller-Creasy confidence region for the ratio of population means (Fieller-Creasy problem). Regions are also derived for simultaneous estimation of the slope and intercept in the model with a single latent predictor, and for the slopes in a model with two latent predictors.