In this paper, we have established a unified framework of multistage parameter estimation. We demonstrate that a wide variety of statistical problems such as fixed-sample-size interval estimation, point estimation with error control, bounded-width confidence intervals, interval estimation following hypothesis testing, construction of confidence sequences, can be cast into the general framework of constructing sequential random intervals with prescribed coverage probabilities. We have developed exact methods for the construction of such sequential random intervals in the context of multistage sampling. In particular, we have established inclusion principle and coverage tuning techniques to control and adjust the coverage probabilities of sequential random intervals. We have obtained concrete sampling schemes which are unprecedentedly efficient in terms of sampling effort as compared to existing procedures.
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