Higher order asymptotic theory is targeted on the development of an asymptotic
\nexpansion for the distribution function of a statistic of interest. The asymptotic
\ninference procedures are commonly based on simple characteristics of the density
\nfunction at or near a data point of interest. In particular, exponential models
\nare useful to provide accurate approximations to general statistical models.
\nTypically, to the third order the exponential approximation has three primary parameters,
\ntwo corresponding to pure model type and one for the departure from
\nan exponential model (termed a non-exponentiality term). Andrews, Fraser and
\nWong (2005) discovered that to the third order, the observed significance function
\ndoes not depend on the non-exponential term for univariate models. This finding
\nhas remarkable statistical implications for inference concerning univariate models.
\nHowever, it is not clear whether this property holds for multivariate models. In
\nthis paper we address this question, and explore the intrinsic discrepancy between
\nunivariate and multivariate models.<\/p>\n