Higher order asymptotics: an intrinsic difference between univariate and multivariate models

Higher order asymptotic theory is targeted on the development of an asymptotic
expansion for the distribution function of a statistic of interest. The asymptotic
inference procedures are commonly based on simple characteristics of the density
function at or near a data point of interest. In particular, exponential models
are useful to provide accurate approximations to general statistical models.
Typically, to the third order the exponential approximation has three primary parameters,
two corresponding to pure model type and one for the departure from
an exponential model (termed a non-exponentiality term). Andrews, Fraser and
Wong (2005) discovered that to the third order, the observed significance function
does not depend on the non-exponential term for univariate models. This finding
has remarkable statistical implications for inference concerning univariate models.
However, it is not clear whether this property holds for multivariate models. In
this paper we address this question, and explore the intrinsic discrepancy between
univariate and multivariate models.