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Definition: A statistic T is called complete if E[g(T)]=0 for all and some function g implies that P(g(T)=0)=1 for all.
In statistics a minimum-variance unbiased estimator (MOUE) or uniformly minimum-variance unbiased estimator (REVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.
Hence, the REVUE of is h(X(n)) = g(X(n)) + n1X(n)g(X(n)). In particular, if =, then the REVUE of is (1 + n1)X(n).
function of T(X). Therefore, X is the best unbiased estimator for µ.
An efficient estimator is the “best possible” or “optimal” estimator of a parameter of interest. The definition of “best possible” depends on one's choice of a loss function which quantifies the relative degree of undesirability of estimation errors of different magnitudes.
1 Answer. Generally, an REVUE is essentially unique. The estimator you provided is not an REVUE though, indeed it is not even unbiased!! Notice that E[1X]=1E[X]=1p provided that our random variable is a Bernoulli with parameter p.
If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficient(note that this statement does not exclude the option of a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic).
A sufficient statistic T : X T is minimal sufficient if for any sufficient statistic U : X U there is a measurable function g : UT such that T = g(U) ex|(· |)-a.s. for all .
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