# Why Loss function has to be bounded from below (statistical decision theory)?

In statistical decision theory the loss function $L(\theta, a)\ge-K > -\infty$ is often chosen for technical convenience (e.g. See  p.3 ). Can anyone explain why the above condition is convenient, and if it is feasible, provide a simple example.

[1.] James O. Berger, "Statistical Decision Theory and Bayesian Analysis"

• One problem leaps out immediately: you might have a hard time using any procedure that is based on expected loss. You could end up comparing a lot of values of $-\infty$ to each other. More pragmatically, since an infinite loss is unrealistic, why would one want to tackle the technical complications of allowing for it in the first place? – whuber Jan 16 '18 at 17:02

## 1 Answer

If loss is not bounded below, there is no admissible estimator. An admissible estimator is desirable because it precludes stupid estimators like the Hodge's superefficient estimator. Hodge's estimator is "better on paper" than the MLE (unbiased yet MORE efficient) but in fact it isn't. The risk behaves erratically and can explode to infinity for probability models where the truth approaches the null.

https://en.wikipedia.org/wiki/Admissible_decision_rule

https://en.wikipedia.org/wiki/Hodges%27_estimator