Choosing the best of the current crop
Gregory Campbell, Stephen M. Samuels · Advances in Applied Probability · 1981
A best choice problem is presented which is intermediate between the constraints of the ‘no-information’ problem (observe only the sequence of relative ranks) and the demands of the ‘full-information’ problem (observations from a known continuous distribution). In the intermediate problem prior information is available in the form of a ‘training sample’ of sizemand observations are the successive ranks of thencurrent items relative to their predecessors in both the current and training samples. Optimal stopping rules for this problem depend on m andnessentially only throughm + n; and, asm/(m + n) →t, their success probabilities,P*(m, n), converge rapidly to explicitly derived limitsp*(t) which are the optimal success probabilities in an infinite version of the problem. For fixedn, P*(m, n) increases withmfrom the ‘no-information’ optimal success probability to the ‘full-information’ value for sample sizen.And astincreases from 0 to 1,p*(t) increases from the ‘no-information’ limite–1≍ 0·37 to the ‘full-information’ limit ≍0·58. In particularp*(0·5) ≍ 0·50.