Let 0= U0,n ≤ U1,n ≤ ··· ≤ Un−1,n ≤ Un,n = 1 be an ordered sample from uniform [0,1] distribution, and Din = Ui,n −Ui−1,n, i = 1,2, . . . ,n; n = 1,2, . . . , be their spacings, and let f1n, . . . , fnn be a set of measurable functions. In this paper, the probabilities of the moderate and Cramer-type large deviation theorems for statistics Rn(D)= f1n(nD1n) + ···+ fnn(nDnn) are proved. Applicati...