A.P.O. Rules are Asymptotically non Deficient for Estimation with Squared Error Loss

The problem considered is sequential estimation of the mean 0 of a one-parameter exponential family of distributions with squared error loss for estimation error and a cost c > 0 for each of an i.i.d, sequence of potential observations X 1, X 2 . . . . . A Bayesian approach is adopted, and natural conjugate prior distributions are assumed. For this problem, the asymptotically pointwise optimal (A.P.O.) procedure continues sampling until the posterior variance of 0 is less than c(ro+n ), where n is the sample size and r o is the fictitous sample size implicit in the conjugate prior distribution. It is known that the A.P.O. procedure is Bayes risk efficient, under mild integrability conditions. In fact, the Bayes risk of both the optimal and A.P.O. procedures are asymptotic to 2Vo]/C, as c--,0, where V o is the prior expectation of the standard deviation of X 1, given 0. Here the A.P.O. rule is shown to be asymptotically non-deficient, under stronger regularity conditions: that is, the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of a single observation, as c ~ 0 . The result is illustrated in the exponential and Bernoulli cases, and extended to the case of a normal distribution with both the mean and variance unknown.