ASYMPTOTIC OTI MALITY OF CERTAIN

can be observed; they are known to be identically distributed with unknown mean and variance u, a2(u) respectively. We assume further that the common distribution is of exponential type, so that the sample mean X is a sufficient statistic. The problem is to estimate u taking into account the cost of sampling, which is assumed to be linear in the sample size, n. For the case of a fixed-size sample, with squared-error loss function, the minimum risk (expected loss) is obtained, after rescaling if necessary, as minnE(n + A(X - u)) = minn(n + Aa2(u)/n) - 2A 2a(u), and is attained at no A 2a(u). Since u is unknown, this optimum value is not available; and for any fixed n, if a(u) is unbounded, the risk n + Aa2(u)/n can be much larger