Sequential Estimation of a Truncation Parameter

Consider the problem of sequentially estimating the parameter θ of a uniform distribution on (0, θ). It is shown that if the loss is measured by squared error plus a linear cost function, and the prior on θ is given by a Pareto distribution, a simple heuristic procedure incurs an excess risk above the Bayes risk not greater than the cost of three observations. The result is extended to a wider class of distributions which includes the uniform and to a wide class of priors.