This paper deals with asymptotic expansions of the distribution of the kth-largest order statistic Zn–k+1:n for the sample size n. These expansions establish higher-order approximations which hold uniformly over all Borel sets. In the particular case of the distribution of Zn–k+1:n under the uniform distribution and the exponential distribution, the approximating measures are linear combinations of ‘negative’ gamma distributions and linear combinations of extreme-value distributions. These results can be extended to the case of the joint distribution of the k largest order statistics. A numerical comparison to a different asymptotic expansion is given where the normal distribution is the leading term.
