In this article, the preliminary test estimator is considered under the BLINEX loss function.
The problem under consideration is the estimation of the location parameter from
a normal distribution. The risk under the null hypothesis for the preliminary test estimator,
the exact risk function for restricted maximum likelihood and approximated risk
function for the unrestricted maximum likelihood estimator, are derived under BLINEX
loss and the different risk structures are compared to one another both analytically and
computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for
the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary
test estimator under LINEX loss and it is shown that the LINEX expected loss
is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are
derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined
and compared to the classical preliminary test estimator.