ESTIMATION UP TO A CHANGE-POINT

Consider the problem of estimating mu, based on the observation of Y0, Y1,...,Y(n), where it is assumed only that Y0, Y1,...,Y(kappa) iid N( mu, sigma2) for some unknown kappa. Unlike the traditional change-poin t problem, the focus here is not on estimating kappa, which is now a n uisance parameter. When it is known that kappa = k, the sample mean Y( k)BAR = SIGMA0(k)Y(i)/(k + 1), provides, in addition to wonderful effi ciency properties, safety in the sense that it is minimax under square d error loss. Unfortunately, this safety breaks down when kappa is unk nown; indeed if k > kappa, the risk of Y(k)BAR is unbounded. To addres s this problem, a generalized minimax criterion is considered whereby each estimator is evaluated by its maximum risk under Y0, Y1,...,Y(kap pa) iid N(mu, sigma2) for each possible value of kappa. An essentially complete class under this criterion is obtained. Generalizations to o ther situations such as variance estimation are illustrated.