Constrained confidence estimation of the binomial p via tail functions

A methodology is proposed for `exact' confidence estimation of the binomial parameter p when that parameter is constrained. It is shown how the technique of tail functions can be used to construct a suitable generalised Clopper-Pearson confidence interval when p is known to lie between two bounds, and how the interval can be engineered for optimality in terms of prior expected length. An example is provided which illustrates the applicability of the theory to gambling.