Let U1, U2, . . . , Un–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as
$${G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}$$
with notation U0 = 0, Un = 1, where
$${N^\prime=\left\lfloor n/s\right\rfloor}$$
is the integer part of n/s. Let
$${ N=\left\lceil n/s\right\rceil}$$
be the smallest integer greater than or equal to n/s, fm (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic
$${f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}$$
is proved.