A new definition of the predictive likelihood

In this paper we present a widely applicable definition of the predictive likelihood based on estimators that are either sufficient or approximately sufficient. Under regularity conditions, this predictive likelihood is shown to equal the Bayes prediction density up to terms of order O p(n-1). For the cases where this predictive likelihood is difficult to compute, an approximate predictive likelihood is provided which differs from the proposed predictive likelihood by O p(n-1). To illustrate the ideas, the approximate predictive likelihood and the Bayes prediction density are obtained for a pth order non-circular autoregression.