Predictive fit for natural exponential families

The paper examines predictive distributions, concentrating on measuring their fit to the true distribution by average KulLback–Leibler divergence. The notion of an ‘averaged bootstrap’ predictive distribution is introduced. This predictive distribution is shown to be asymptotically superior to the estimative distribution, in terms of average Kullback–Leibler divergence, when the true distribution is in a natural exponential family. Small-sample results are presented for the Poisson and binomial distributions which suggest that the bootstrap distribution performs well in these cases.