The weak convergence of the sample df is studied under a given sequence of alternative hypotheses when parameters are estimated from the data. For a general class of estimators it is shown that the sample df, when normalised, converges weakly to a specified normal process. The results are specialised to the case of efficient estimation.
"Weak Convergence of the Sample Distribution Function when Parameters are Estimated." Ann. Statist. 1 (2) 279 - 290, March, 1973. https://doi.org/10.1214/aos/1176342365