instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Near-exact distributions closer to exact distributions than common asymptotic distributions
Seminario IMUS

We are all quite familiar with the concept of asymptotic distribution. For some sets of statistics, as it is for example the case with the likelihood ratio test statistics, mainly those used in Multivariate Analysis, methods of building such asymptotic distributions have been developed , which are nowadays seen as “standard”, as it is the case of the methods in the seminal paper by Box (Box, GEP (1949) A General Distribution Theory for a Class of Likelihood Criteria. Biometrika, 36, 317-346 ).
However, such asymptotic distributions quite commonly yield approximations which fall  short of the precision we need and may also exhibit some problems when some parameters in the exact distributions grow large, as it is indeed the case with many asymptotic distributions commonly used in Multivariate Analysis when the number of variables involved grows even just moderately large, being yet the case that they never show an asymptotic behavior for increasing numbers of variables.
The pertinent question is thus the following one: are we willing to pay a bit more in terms of a more elaborate structure for the approximating distribution, anyway keeping it much manageable in terms of allowing for a quite easy computation of p-values and quantiles, in order to obtain much better approximations, which will even be asymptotic for increasing numbers of variables?
If our answer to the above question is affirmative, then we are ready to enter the world of the near-exact distributions.
Near-exact distributions are asymptotic distributions developed under a new concept of approximating distributions. Based on a decomposition (i.e., a factorization or a split in two or more terms) of the characteristic function of the statistic being studied, or of the characteristic function of its logarithm, they are asymptotic distributions which lie much closer to the exact distribution than common asymptotic distributions.
If we are able to keep untouched a good part of the original structure of the exact distribution of the random variable or statistic being studied, we may in this way obtain a much better approximation, which not only does not exhibit anymore the problems referred above which occur with some asymptotic distributions, but which on top of this exhibits extremely good performances even for very small sample sizes and large numbers of variables involved, being asymptotic not only for increasing sample sizes but also (opposite to what happens with the common asymptotic distributions) for increasing values of the number of variables involved.