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)