Testing similar means

We consider the problem of testing a basic property of collections of distributions: having similar means. Namely, the algorithm should accept collections of distributions in which all distributions have means that do not differ by more than some given parameter and should reject collections that are relatively far from having this property. By "far" we mean that it is necessary to modify the distributions in a relatively significant manner (according to some predetermined distance measure) so as to obtain the property. We study this problem in two models. In the first model (the query model) the algorithm may ask for samples from any distribution of its choice, and in the second model (the sampling model) the distributions from which it gets samples are selected randomly. We provide upper and lower bounds in both models. In particular, in the query model, the complexity of the problem is polynomial in 1/ε (where ε is the given distance parameter), while in the sampling model, the complexity grows roughly as m¹-poly(ε), where m is the number of distributions.