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An SPRT-like method is developed for the problem of selecting the best of k multinomial parameter estimation procedures when only one observation per the k estimation procedures is possible but the k estimation procedures can be repeated many times. The multinomial probability mass function considered is fx1,x2, &ldots;xn= px1<rm>1p x22&cdots;p xnn where S xi=1 and xi=0 or 1 and parameters p1,<hsp sp="0.167"> p2,&ldots;<rm>pn satisfy Sp i=1and all p i>=0. It is assumed that there are two or more procedures for estimating parameters p1,t, p2,t, &cdots; , pn(t),t for each observation t. The number of parameters n(t) can depend upon the observation number t. An example of this would be competing procedures for estimating the probability of a horse winning a race. The parameter pj,t would represent an estimate of the probability of horse j winning race t. Race t can be run only once and hence only one observation can be obtained for the k estimation procedures for race t. However, the k estimation procedures can be repeated for different races. Other examples of competing multinomial parameter estimation procedures would include different methods of estimating the probability of a financial market being up in a given time period or different forecasts of the probability of precipitation. A new procedure for estimating probabilities at a racetrack is developed and the SPRT-like method is used to compare this new procedure to the existing theory that an entry's probability of winning is equal to the fraction of the win pool bet on that entry. |
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