Risk of a statistical procedure

A characteristic giving the mean loss of an experimenter in a problem of statistical decision making and thus defining the quality of the statistical procedure under consideration.

Suppose that one has to make a decision $ d $ in a measurable decision space $ ( D, {\mathcal A}) $ with respect to a parameter $ \theta $ on the basis of a realization of a random variable $ X $ with values in a sampling space $ ( \mathfrak X, \mathfrak B, {\mathsf P} _ \theta ) $, $ \theta \in \Theta $. Further, let the loss of a statistician caused by making the decision $ d $ when the random variable $ X $ follows the law $ {\mathsf P} _ \theta $ be $ L( \theta , d) $, where $ L $ is some loss function given on $ \Theta \times D $. In this case, if the statistician uses a non-randomized decision function $ \delta : \mathfrak X \rightarrow D $ in the problem of decision making, then as a characteristic of this function $ \delta $ the function

$$ R( \theta , \delta ) = {\mathsf E} _ \theta L( \theta , \delta ( X)) = \ \int\limits _ { \mathfrak X } L( \theta , \delta ( X)) d {\mathsf P} _ \theta ( x) $$

is used. It is called the risk function or, simply, the risk, of the statistical procedure based on the decision function $ \delta $ with respect to the loss $ L $.

The concept of risk allows one to introduce a partial order on the set $ \Delta = \{ \delta \} $ of all non-randomized decision functions, since it is assumed that between two different decision functions $ \delta _ {1} $ and $ \delta _ {2} $ one should prefer $ \delta _ {1} $ if $ R( \theta , \delta _ {1} ) \leq R( \theta , \delta _ {2} ) $ uniformly over all $ \theta $.

If the decision function $ \delta $ is randomized, the risk of the statistical procedure is defined by the formula

$$ R( \theta , \delta ) = \int\limits _ { \mathfrak X } \int\limits _ { D } L( \theta , d) dQ _ {x} ( d) d {\mathsf P} _ \theta ( x), $$

where $ \{ Q _ {x} ( d) \} $ is the family of Markov transition probability distributions determining the randomization procedure.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2]	N.N. Chentsov, "Statistical decision rules and optimal inference" , Amer. Math. Soc. (1982) (Translated from Russian)
[3]	A. Wald, "Statistical decision functions" , Wiley (1950)