Comparison of tests can be made in a local way using the concept of Pitman efficiency or in a non-local way using for instance Bahadur efficiency. In the local case the level of significance is kept fixed, whereas the non-local comparison typically deals with very small levels. These points of view may be seen as the extreme points of view. Here a possibly more realistic intermediate approach is introduced: as the number of observations tends to infinity the level is sent to zero, but not so very fast, in that way filling the gap between the Pitman and Bahadur approaches. The theory is exemplified by results on likelihood ratio tests, locally most powerful tests, and simple linear rank tests.
"Intermediate Efficiency, Theory and Examples." Ann. Statist. 11 (1) 170 - 182, March, 1983. https://doi.org/10.1214/aos/1176346067