An admissible decision rule is a rule for making a statistical decision; There isn’t any other rule which is, generally speaking, better.
If it’s not admissible, then it’s inadmissible. An inadmissible decision rule is never worth using, since by definition there will always be another rule that is better than it. But just because a rule is admissible doesn’t mean it’s the best or even the most sensible rule. Although there is no rule that is generally better than it, there may be decision rules that are better over specific ranges—and those ranges might be the ranges that are most interesting to our research.
Technical Definition of an Admissible Decision Rule
Let δ be our decision rule, and R(θ, x) the risk function. The risk function quantifies the risk we take in using δ, or the likelihood we make a wrong decision with our rule. θ here is a variable in Θ, which includes all the states of nature, in a way of speaking, or every point in an unlimited range.
Then we can define an admissible decision rule by saying that:
If there is no δ′ such that both
R(θ δ′) ≤ R(θ δ) for all θ, and
R(θ δ′) < R(θ δ) for at least one θ
then the decision rule δ is admissible.
Another way of saying this is that if our decision rule δ isn’t dominated by another decision rule δ′ everywhere, δ is admissible.
References
Shao, J. (2013). Mathematical Statistics. Springer.
Lecture Notes on Decision Theory. Retrieved from http://www.maths.lu.se/fileadmin/maths/forskning_research/Undervisningsmaterial/Inferensteori_MASC02/decision.theory_02.pdf on February 18, 2018
Wellner, Jon. Stat 580 Lecture Notes, Chapter 5: Bayes Method and Elementary Decision Theory.
Retrieved from https://www.stat.washington.edu/jaw/COURSES/580s/581/LECTNOTES/ch5.pdf on February 18, 2018