Asymptotic behavior of large deviations of empirical distribution functions (df's) is considered. Borovkov (1967) and Hoadley (1967) obtained results for functionals continuous in the sup norm topology on the set of df's. Groeneboom, Oosterhoff, and Ruymgaart (1979) extended this to functionals continuous in a stronger $\tau$-topology. This result is now extended to functionals that are $\tau$-continuous only on a particular useful subset of df's. Applications to the Anderson-Darling statistic and linear combinations of order statistics are considered. We begin by correcting the work of Abrahamson (1967); from this the role of the key weight function $\psi(t) = -\log t(1 - t)$ is discovered. It is then exploited to the end indicated above, and it is considered as a weight function in tests of fit.
"Large Deviations of Goodness of Fit Statistics and Linear Combinations of Order Statistics." Ann. Probab. 9 (6) 971 - 987, December, 1981. https://doi.org/10.1214/aop/1176994268