Implicitly defined (and easily approximated) universal constants 1.1 < an < 1.6, n = 2,3, ... , are found so that if X1, X2, ... are i.i.d. non-negative random variables and if the Tn is the set of stop rules for X1, ..., Xn, then E (max {X1, ..., Xn}) ≤ ansup {EXt : t ε Tn}, and the bound an is best possible. Similar universal constants 0 < bn < 1/4 are found so that if the (Xi) are i.i.d. random variables taking values only in [a,b], then E (max {X1, ..., Xn}) ≤ ansup {EXt : t ε Tn} + bn (b - a ), where again the bound bn is the nest possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.



Included in

Mathematics Commons



URL: https://digitalcommons.calpoly.edu/rgp_rsr/62