Higher Order Laplace Integration and the Hyper-accuracy of Recent Likelihood Methods

SUMMARY We develop a third order approximation for the distribution function of an asymptotic density function on the real line with a scalar parameter; a computer implementation is available on the web. The derivation makes use of integration that is obtained by symbolic computer algebra. In applications, information of various kinds may be available concerning a density function and its norming constant, and values may be needed for the corresponding distribution function at one or various data points. The present results provide one bridge for this gap and use likelihood function derivatives at the data point of interest. Other methods include analytic integration, numerical integration, and various asymptotic and likelihood approximations, and a choice may depend on the availability and accessibility of the needed information. A brief survey of methods is included and examples indicate the computational diiculty and the level of accuracy available with various kinds of information.