Fixed precision estimation of the maximal value of a bounded random variable

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The object of this paper is to survey the methods of fixed precision estimation of the maximal value of a bounded random variable. In particular the paper gives solutions to this problem for a class of distributions with unknown scale parameter (section 2) and for a class of distributions with certain features of symmetry (section 3). The sequential procedures solving both subproblems are not only asymptotically consistent and asympto-tically efficient in the sense of Chow and Robbins (like that presented in section 4), but they assure the exact consistency. Moreover, in section 5, the case of the uniform distribution and the problem of finding the optimal stopping rule in this case are discussed in detail.