Asymptotic equivalence for nonparametric regression experiments with random design

where ζi ∼ N (0, 1), and the Xi are marginally i.i.d. with density g on [0, 1]. This nonparametric regression problem with random design is a more realistic model than assuming the data will be observed on a regular grid. How much of a difference does it make whether the Xi are randomly placed given that we have a sufficiently large number of observations? Our approach will be to find a statistic that is approximately sufficient in the sense of Le Cam and is observed at regular fixed intervals over the design space. One motivation for this analysis is the wavelet smoothing techniques (as in Kovac & Silverman (2000) for example) that map regression data onto a regular grid before applying the wavelet transform.