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The notion of approximately integrable linear statistical models is introduced to analyze the higher order optimality properties of some common nonparametric estimators. The approximately integrable models suggest a useful approach to a unified treatment of both regular and irregular non-parameter problems. It is shown that with such models any rate of improvement ranging from (log n) to the alpha power/squared to 1/(log...log n) to the alpha power), alpha 0, of the classical non-parametric procedures can be anticipated. Both an example of a first order asymptotically optimal estimator with the unusual rate 1/n log n and an estimator with an extremely slow unimprovable rate of convergence 1(log...log n) the alpha power are presented.