Estimation of the Loss of an Estimate

Suppose y1, . . . , yn are independent random variables. The density pθi(·) of yi is supposed to be known up to a parameter θi. Let θ̂ = ( θ̂1(y), . . . , θ̂n(y) ) be an estimate of θ constructed from the sample y = (y1, . . . , yn). Note that the estimate of θi can depend on yj , j 6= i. Let `i(·, ·) be a loss function so that `i(θi, θ̂i) represents the loss of using θ̂i as the estimate of θi. The purpose of the present paper is to introduce some methods for the estimation of the average loss L(θ, θ̂) = 1 n n ∑