On an Optimal Stopping Problem of Gusein-Zade
STANFORD UNIV CA DEPT OF STATISTICS
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We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r 25, we give the limiting as n approaches infinity optimal risk probability of not selecting one of the r best and the limiting optimal proportion of individuals to let go by before being willing to stop. The complete limiting form of the optimal stopping rule is presented for each r up to r 10, and for r 15, 20 and 25. We show that, for large n and r, the optimal risk is approximately l-t to the rth power where t is approx. .2834 is obtained as the root of a function which is the solution to a certain differential equation and the optimal stopping rule tau sub r,n lets approximately tn arrivals go by then stops almost immediately in the sense that tau sub r,nn approaches t in probability as n approaches infinity, r approaches infinity.
- Statistics and Probability