Median unbiasedness.

(i) A real number m is a median for the random variable Y if P{Y ≥ m} ≥ 1 2 , P{Y ≤ m} ≥ 1 2 . Then all real a1, a2 such that m ≤ a1 ≤ a2 or m ≥ a1 ≥ a2 satisfy E|Y − a1| ≤ E|Y − a2|.

(ii) For any estimate δ(X) of γ(θ), let m−(θ) and m+(θ) denote the infimum and supremum of the medians of δ(X), and suppose that they are continuous functions of θ. Let γ(θ) be continuous and not constant in any open subset of

Ω. Then the estimate δ(X) of γ(θ) is unbiased with respect to the loss function L(θ,

d) = |γ(θ)−d| if and only if γ(θ) is a median of δ(X) for each θ. An estimate with this property is said to be median-unbiased.