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We derive the maximum likelihood estimator of the binomial success probability when the number of trials is known, but the number of successes is only known to fall in some interval. This estimator has a surprisingly simple form. Specifically, the maximum likelihood estimate of the log odds of success is the mean of the maximum likelihood estimates of the log odds of success that are obtained by letting the number of successes take on each integer value in the interval. An application is suggested.
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Frey et al. (2008) studied this question.
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