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Economic data are often generated by stochastic processes that take place incontinuous time, though observations may occur only at discrete times. Forexample, electricity and gas consumption take place in continuous time. Datagenerated by a continuous time stochastic process are called functional data.This paper is concerned with comparing two or more stochastic processes thatgenerate functional data. The data may be produced by a randomized experimentin which there are multiple treatments. The paper presents a method for testingthe hypothesis that the same stochastic process generates all the functionaldata. The test described here applies to both functional data and multipletreatments. It is implemented as a combination of two permutation tests. Thisensures that in finite samples, the true and nominal probabilities that eachtest rejects a correct null hypothesis are equal. The paper presents upper andlower bounds on the asymptotic power of the test under alternative hypotheses.The results of Monte Carlo experiments and an application to an experiment onbilling and pricing of natural gas illustrate the usefulness of the test.

Permutation Tests for Equality of Distributions of Functional Data