For fixed $\theta$, let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables having density $f_\theta(x)$. Using a sequential Bayes decision theoretic approach we consider the problem of estimating any strictly monotone function $g(\theta)$ when the error incurred by a wrong estimate is measured by squared error loss and the sampling cost is $c$ units per observation. A heuristic stopping rule is suggested. It is shown that the excess risk which results when using it is bounded above by terms of order $c$.
"Bayesian Sequential Estimation." Ann. Statist. 5 (5) 955 - 968, September, 1977. https://doi.org/10.1214/aos/1176343951