Shrinkage estimation of the slope parameters of two parallel regression lines under uncertain prior information

Abstract

The estimation of the slope parameter of two linear regression models with normal errors are considered, when it is suspected that the two lines are parallel. The uncertain prior information about the equality of slopes is presented by a null hypothesis and a coefficient of distrust on the null hypothesis is introduced. The unrestricted estimator (UE) based on the sample responses and shrinkage restricted estimator (SRE) as well as shrinkage preliminary test estimator (SPTE) based on the sample responses and prior information are defined. The relative performances of the UE, SRE and SPTE are investigated based on the analysis of the bias, quadratic bias and quadratic risk functions. An example based on a health study data is used to illustrate the method. The SPTE dominates other two estimators if the coefficient of distrust is not far from 0 and the difference between the population slopes is small.