Third order asymptotics: connections among test quantities

Saddlepoint methods, extended to distribution functions, can provide highly accurate tail probabilities for testing real parameters in exponential models. For extensions, asymptotic connections among various test quantities are needed. For five quantities, the maximum likelihood departure standardized by observed and expected information, the score function standardized by observed and expected information, and the signed square root of the likelihood ratio statistic, the needed connections to third order are recorded. Their use is illustrated by a simple integration proof of the Lugannani and Rice formula.