Complete asymptotic expansions for normal extremes

Hall (1979) was the first to derive the rate of convergence for normal extremes. Many authors have followed up the work of Hall, but complete asymptotic expansions have not been known for normal extremes. Here, we derive such expansions for the first time. The expansions are single infinite sums of terms involving Bell polynomials and Stirling numbers. The usefulness of the expansions over the results in Hall is illustrated computationally.