The Poisson distribution is a standard model for analyzing numerical data, and estimating the mean parameter of this distribution is widely used in practice. So far, several confidence intervals for the mean of the Poisson distribution have been proposed, all of which are asymptotically valid, and their exact comparison is important. The probability of covering the confidence interval (L(X),U(X)) for the mean of a Poisson random variable X with an unknown parameter theta is a function of theta. Since the Poisson distribution is discrete, the probability of covering the probability function does not have a closed form and changes with theta in the parameter space. Therefore, it is very difficult to calculate the maximum, minimum, and mean of the covering probabilities exactly for the confidence intervals of theta parameter.
A method for calculating the minimum and mean of the covering probabilities of confidence intervals with ascending bounds for theta parameter was presented by Wang [11]. This paper presents a method for accurately calculating the maximum probability of covering confidence intervals with ascending bounds for the unknown parameter theta in the Poisson distribution. Decision-making will be more reliable if the confidence intervals are compared simultaneously based on the maximum, minimum, and average probability of their covering.