Bernoulli experiment

of size

The special case of a statistical experiment (cf. also Probability space; Statistical experiments, method of) consisting of a set of probability measures on a -algebra of subsets of a set , where (, the set of natural numbers), is the -algebra of all subsets of and . Here, the probability measure describes the probability

for a given probability of success that will be observed. Clearly, decision-theoretical procedures associated with Bernoulli experiments are based on the sum of observations because of the corresponding sufficient and complete data reduction (cf. [a2] and [a3]). Therefore, uniformly most powerful, as well as uniformly most powerful unbiased, level tests for one-sided and two-sided hypotheses about the probability of success are based on (cf. [a2]; see also Statistical hypotheses, verification of). Moreover, based on the quadratic loss function, the sample mean

is admissible on account of the Rao–Cramér inequality (cf. [a3]) and the estimator (cf. also Statistical estimator)

is minimax by means of equalizer decision rules (cf. [a2]). Furthermore, the Lehmann–Scheffé theorem implies that is a uniform minimum-variance unbiased estimator (an UMVU estimator; cf. also Unbiased estimator) for the probability of success (cf. [a2] and [a3]).

All UMVU estimators, as well as all unbiased estimators of zero, might be characterized in connection with Bernoulli experiments by introducing the following notion for general statistical experiments : A , being square-integrable for all , is called an UMVU estimator if

for all . The covariance method tells that is a UMVU estimator if and only if , , for all unbiased estimators of zero, i.e. if , (cf. [a3]). In particular, the covariance method implies the following properties of UMVU estimators:

i) (uniqueness) , , UMVU estimators with , , implies -a.e. for all .

ii) (linearity) , UMVU estimators, ( the set of real numbers), , implies that is also an UMVU estimator.

iii) (multiplicativity) , , UMVU estimators with or bounded, implies that is also an UMVU estimator.

iv) (closedness) , , UMVU estimators satisfying for some and all implies that is an UMVU estimator.

In the special case of a Bernoulli experiment of size one arrives by the property of uniqueness i) and the property of linearity ii), together with an argument based on interpolation polynomials, at the following characterization of UMVU estimators: is a UMVU estimator if and only if one of the following conditions is valid:

v) is a polynomial in , , , of degree not exceeding ;

vi) is symmetric (permutation invariant).

Moreover, the set of all real-valued parameter functions admitting some with , , coincides with the set consisting of all polynomials in of degree not exceeding . In particular, is an unbiased estimator of zero if and only if its symmetrization , defined by

vanishes. Therefore, the set consisting of all estimators is equal to the direct sum , where , stands for and is equal to . In particular, , and .

If one is interested, in connection with general statistical experiments , only in locally minimum-variance unbiased estimators at some , one might start from satisfying

Then the covariance method yields again the properties of uniqueness, linearity and closedness (with respect to ), whereas the property of multiplicativity does not hold, in general, for locally minimum-variance unbiased estimators; this can be illustrated by infinite Bernoulli experiments, where the probability of success is equal to , as follows.

Let () be the special statistical experiment with , coinciding with the set of all subsets of , and being the set of all binomial distributions with integer-valued parameter and probability of success (cf. also Binomial distribution). Then the covariance method, together with an argument based on interpolation polynomials, yields the following characterization of locally optimal unbiased estimators: is locally optimal at for all ( fixed) among all estimators with , , if and only if is a polynomial in of degree not exceeding . In particular, is a UMVU estimator if and only if is already deterministic. Moreover, the property of multiplicativity of locally optimal unbiased estimators is not valid.

There is also the following version of the preceding characterization of locally optimal unbiased estimators for realizations of independent, identically distributed random variables with some binomial distribution , , as follows. Let , let be the set of all subsets of , let , where denotes the -fold direct product of having the binomial distribution . Then is locally optimal at for all ( fixed) among all estimators with , , if is a symmetric polynomial in and a polynomial in keeping the remaining variables , fixed, , of degree not exceeding . In particular, for the sample mean

is not locally optimal at for any and some fixed .

Finally, there are also interesting results about Bernoulli experiments of size with varying probabilities of success, which, in connection with the randomized response model (cf. [a1]), have the form , , with , , fixed and . Then there exists an UMVU estimator for based on if and only if or for all . In this case

is a UMVU estimator for .

If the probabilities of success are functions , , with as parameter space, there exists a symmetric and sufficient data reduction of if and only if there are functions , such that

In particular, the sample mean is sufficient in this case.

References

[a1]	A. Chaudhuri, R. Mukerjee, "Randomized response" , M. Dekker (1988)
[a2]	T.S. Ferguson, "Mathematical statistics: a decision theoretic approach" , Acad. Press (1967)
[a3]	E.L. Lehmann, "Theory of point estimation" , Wiley (1983)