2018/05/29 by Marcos C. S. Carreira, Carreira, Marcos Costa Santos
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #65C50 #FOS: Mathematics #Probability (math.PR) #Probability and Statistical Research #Sports Analytics and Performance #Statistical Mechanics and Entropy #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1805.11556
openalex publication_date 2018/05/29 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In "Recognizing the Maximum of a Sequence", Gilbert and Mosteller analyze a\nfull information game where n measurements from an uniform distribution are\ndrawn and a player (knowing n) must decide at each draw whether or not to\nchoose that draw. The goal is to maximize the probability of choosing the draw\nthat corresponds to the maximum of the sample. In their calculations of the\noptimal strategy, the optimal probability and the asymptotic probability, they\nassume that after a draw x the probability that the next i numbers are all\nsmaller than x is xi; but this fails to recognize that continuing the game\n(not choosing a draw because it is lower than a cutoff and waiting for the next\ndraw) conditions the distribution of the following i numbers such that their\nexpected maximum is higher then i/(i+1). The problem is now redefined with each\ndraw leading to a win, a false positive loss, a false negative loss and a\ncontinuation. An exact formula for these probabilities is deduced, both for the\ngeneral case of n-1 different indifference numbers (assuming 0 as the last\ncutoff) and the particular case of the same indifference number for all cutoffs\nbut the last. An approximation is found that preserves the main characteristics\nof the optimal solution (slow decay of win probability, quick decay of false\npositives and linear decay of false negatives). This new solution and the\noriginal Gilbert and Mosteller formula are compared against simulations, and\ntheir asymptotic behavior is studied.\n