In statistical decision theory
, where we are faced with the problem of estimating a deterministic parameter (vector)
. An estimator
is called minimax
if its maximal risk
is minimal among all estimators of
. In a sense this means that
is an estimator which performs best in the worst possible case allowed in the problem.
Consider the problem of estimating a deterministic (not Bayesian
from noisy or corrupt data
related through the conditional probability distribution
. Our goal is to find a "good" estimator
for estimating the parameter
, which minimizes some given risk function
. Here the risk function is the expectation
of some loss function
with respect to
. A popular example for a loss function is the squared error loss
, and the risk function for this loss is the mean squared error
Unfortunately in general the risk cannot be minimized, since it depends on the unknown parameter itself (If we knew what was the actual value of , we wouldn't need to estimate it). Therefore additional criteria for finding an optimal estimator in some sense are required. One such criterion is the minimax criteria.
: An estimator
is called minimax
with respect to a risk function
if it achieves the smallest maximum risk among all estimators, meaning it satisfies
Least favorable distribution
Logically, an estimator is minimax when it is the best in the worst case. Continuing this logic, a minimax estimator should be a Bayes estimator
with respect to a prior least favorable distribution of
. To demonstrate this notion denote the average risk of the Bayes estimator
with respect to a prior distribution
: A prior distribution
is called least favorable if for any other distribution
the average risk satisfies,
Theorem : If , then:
- is minimax.
- If is a unique Bayes estimator, it is also the unique minimax estimator.
- is least favorable.
Conclusion: If an estimator has constant risk, it is minimax. Note that it is not a necessary condition.
Example: Consider the problem of estimating the mean of dimensional Gaussian random vector, . The Maximum likelihood (ML) estimator for in this case is simply , and it risk is
So the risk is constant, and therefore the ML estimator is minimax. Nonetheless, minimaxity does not always imply admissibility. In fact in this example, the ML estimator is known to be inadmissible (not admissible) whenever . The famous James-Stein estimator dominates the ML whenever . Though both estimators have the same risk when , and they are both minimax, the James-Stein Estimator has smaller risk for any finite . This fact is illustrated in the following figure.
The reason for that is that the ML estimator is not an actual Bayes estimator, but rather the limit of such estimators.
Definition : A sequence of prior distributions , is called least favorable if for any other distribution ,
Theorem 2 : If and , then .
- is minimax.
- The sequence is least favorable.
Notice that no uniqueness is guaranteed here. For example, the ML estimator from the previous example may be attained as the limit of Bayes estimators with respect to a uniform prior, with increasing support and also with respect to a zero mean normal prior with increasing variance. So neither the resulting ML estimator is unique minimax not the least favorable prior is unique.
In general it is difficult, often even impossible to determine the minimax estimator. Nonetheless, in many cases a minimax estimator has been determined.
Example 1, Bounded Normal Mean: When estimating the Mean of a Normal Vector , where it is known that . The Bayes estimator with respect to a prior which is uniformly distributed on the edge of the bounding sphere is known to be minimax whenever . The analytical expression for this estimator is
, is the modified Bessel function
of the first kind of order
Example 2, Unfair Coin
: Consider the problem of estimating the "success" rate of a Binomial
. This may be viewed as estimating the rate at which an unfair coin
falls on "heads" or "tails". In this case the minimax estimator is the Bayes estimator with respect to a Beta
, and the analytical expression for it is
- E. L. Lehmann and G. Casella (1998), Theory of Point Estimation, 2nd ed. New York: Springer-Verlag.
- F. Perron and E. Marchand (2002), "On the minimax estimator of a bounded normal mean," Statistics and Probability Letters 58: 327-333.
- J. O. Berger (1985), Statistical Decision Theory and Bayesian Analysis, 2nd ed. New York: Springer-Verlag. ISBN 0-387-96098-8.
- C. Stein (1981), "Estimation of the mean of a multivariate normal distribution," Ann. Stat. 9 (6): 1135-1151.