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 (measured according to the `1 distance averaged over the distributions) 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 1/2 (where 2 is the given distance parameter). While in the sampling model, the complexity grows roughly as m1−poly(2), where m is the number of distributions. ∗School of Computer Science, Tel Aviv University. Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported by the Israel Science Foundation grant nos. 1147/09 and 246/08 †School of Electrical Engineering, Tel Aviv University. Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported by the Israel Science Foundation grant number 246/08. ‡CSAIL, MIT, Cambridge MA 02139 and the Blavatnik School of Computer Science, Tel Aviv University. E-mail: [email protected]. Research supported by NSF grants CCF-1065125 and CCF-0728645, Marie Curie Reintegration grant PIRG03-GA-2008-231077 and the Israel Science Foundation grant nos. 1147/09 and 1675/09.