Unbiasedness in point estimation. Suppose that is a continuous realvalued function defined over which is not

Unbiasedness in point estimation. Suppose that γ is a continuous realvalued function defined over which is not constant in any open subset of , and that the expectation h(θ) = Eθδ(X) is a continuous function of θ for every estimate

δ(X) of γ(θ). Then (1.11) is a necessary and sufficient condition for δ(X) to be unbiased when the loss function is the square of the error.

[Unbiasedness implies that γ2(θ

) − γ2(θ) ≥ 2h(θ)[γ(θ

) − γ(θ)] for all θ, θ

. If θ

is neither a relative minimum nor maximum of γ, it follows that there exist points

θ arbitrarily close to θ both such that γ(θ) + γ(θ

) ≥ and ≤ 2h(θ), and hence that

γ(θ) = h(θ). That this equality also holds for an extremum of γ follows by continuity, since γ is not constant in any open set.]