Unbiasedness in point estimation. Suppose that is a continuous real-valued function defined over which is

Unbiasedness in point estimation. Suppose that γ is a continuous real-valued function defined over Ω which is not constant in any open subset of

Ω, 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.]