Unbiasedness in interval estimation. Confidence intervals I = (L, L) are unbiased for estimating with loss

Unbiasedness in interval estimation. Confidence intervals I = (L, L¯)

are unbiased for estimating θ with loss function L(θ, I) = (θ − L)2 + (L¯ − θ)2 provided E[ 1 2 (L + L¯)] = θ for all θ, that is, provided the midpoint of I is an unbiased estimate of θ in the sense of (1.11).