For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments.
Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose.
Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics.
[1] Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues.
Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam.