A Cramer type large deviation theorem is proved under alternatives as well as under hypothesis for the generalized linear rank statistic which includes as special cases (unsigned) linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, and a rank combinatorial statistic.
Ralescu, S. S.,
& Puri, M. L.
(1985). Cramer Type Large Deviations for Generalized Rank Statistics. Annals of Probability, 13 (1), 115-125.