Least-favourable distribution

An a priori distribution maximizing the risk function in a statistical problem of decision making.

Suppose that, based on a realization of a random variable with values in a sample space , , one has to choose a decision from a decision space ; it is assumed here that the unknown parameter is a random variable taking values in a sample space , . Let be a function representing the loss incurred by adopting the decision if the true value of the parameter is . An a priori distribution from the family is said to be least favourable for a decision in the statistical problem of decision making using the Bayesian approach if

where

is the risk function, representing the mean loss incurred by adopting the decision . A least-favourable distribution makes it possible to calculate the "greatest" (on the average) loss incurred by adopting . In practical work one is guided, as a rule, not by the least-favourable distribution, but, on the contrary, strives to adopt a decision that would safeguard one against maximum loss when varies; this implies a search for a minimax decision minimizing the maximum risk, i.e.

When testing a composite statistical hypothesis against a simple alternative, within the Bayesian approach, one defines a least-favourable distribution with the aid of Wald reduction, which may be described as follows. Suppose that, based on a realization of a random variable , one has to test a composite hypothesis , according to which the distribution law of belongs to a family , against a simple alternative , according to which obeys a law ; let

where is a -finite measure on and is a family of a priori distributions on . Then, for any , the composite hypothesis can be associated with a simple hypothesis , according to which obeys the probability law with density

By the Neyman–Pearson lemma for testing a simple hypothesis against a simple alternative , there exists a most-powerful test, based on the likelihood ratio. Let be the power of this test (cf. Power of a statistical test). Then the least-favourable distribution is the a priori distribution from the family such that for all . The least-favourable distribution has the property that the density of under the hypothesis is the "least distant" from the alternative density , i.e. the hypothesis is the member of the family "nearest" to the rival hypothesis . See Bayesian approach.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2]	S. Zachs, "Theory of statistical inference" , Wiley (1971)