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

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).