Analysis of the quality of approximations in the statistical analysis of multivariate observations

The formal construction of second-order Chebyshev-Edgeworth expansions for statistics from samples with random sample sizes is further developed. The asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. Non-random, random or mixed normalisation factors for the statistics lead to different limit distributions with related approximation functions. Moreover, new results for the sample median are proved.