We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables. While there are very efficient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distribution-free algorithms (which must work under any arbitrary and unknown distribution), tends to be a more challenging task. When the underlying distribution is uniform, Parnas et al. (SIAM Journal on Discrete Math, 2002) give an algorithm for testing (monotone) monomials whose query complexity does not depends on n, and whose dependence on the distance parameter is (inverse) linear. In contrast, Glasner and Servedio (in Proceedings of RANDOM, 2007) prove that every distribution-free testing algorithm for monotone monomials as well as for general monomials must have query complexityΩ(n
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