Applications of Unbiased Estimation Theory

Abstract

The choice of a decision rule δ(x) in constructing an estimator of a parameter θ is significantly defined by a further usage of the estimator δ(X) (see [102], [167], and [222]). If the unique estimator of a parameter θ is under consideration, then it is much more important, for example, to get the estimator with a minimal square risk R(δ,θ) = (δ(X) −θ)². In this case it should not be necessarily unbiased. The contrary is the case when, say, there is a set of N estimators T _j = 1,2,…N, considering which one ought to construct a joint estimator of θ. Such as situation, for example, is typical in modern experimental physics where in order to enlarge statistics based on N estimators T_j (for example, a weighted-mean estimator). If the estimators T_j are biased, i.e. E T_j ≠ θ, then a mean squared deviation of the error d = T − θ will decrease √N times and its bias will be approximately equal to the bias of an individual estimator, and for sufficiently large N the value of T, with a large probability, may appear to differ from the value θ by more than several standard deviations, i.e. it will no longer be a “good” estimator of θ. In such a situation, unbiased estimators should be used.