ABSTRACT
A number of the basic tools of the theory of higher-order asymptotic inference, in particular Edgeworth expansions, saddlepoint expansions and Laplace's method, are discussed by Barndorff-Nielsen. Another important technique is the determination of stochastic expansions of likelihood quantities such as the maximum likelihood estimator and the log- likelihood ratio statistic. The geometrical study of the expansions could be brought much further than the author shall do, but that would require a more extensive background in differential geometry. There are two main types of expansion, observed and expected/observed. In observed expansions the coefficients consist of mixed derivatives of the log model function, while the stochastic parts consist entirely of the components of the score vector. Expected/observed expansions employ moments of log-likelihood derivatives for the coefficients and such derivatives centred at their mean values for the stochastic parts, and they are more generally applicable.
