ASYMPTOTIC NONEQUIVALENCE OF NONPARAMETRIC EXPERIMENTS WHEN THE SMOOTHNESS INDEX IS 1/2 By

An example is provided to show that the natural asymptotic equivalence does not hold between any pairs of three nonparametric experiments: density problem, white noise with drift and nonparametric regression, when the smoothness index of the unknown nonparametric function class is 1/2. 1. Introduction. There have recently been several papers demonstrating the global asymptotic equivalence of certain nonparametric problems. See especially Brown and Low (1996), who established global asymptotic equivalence of the usual white-noise-with-drift problem to the nonparametric regression problem, and Nussbaum (1996), who established global asymptotic equivalence to the nonparametric density problem. In both these instances the results were established under a smoothness assumption on the unknown nonparametric drift, regression or density function. In both cases such functions were assumed to have smoothness coefficient α > 1/2, for example, to satisfy