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Non-standard problems stand in sharp contrast to regular statistical models where statistical estimates of certain functionals of the distribution (functions of the parameter indexing the distribution) are not-estimable at
$\sqrt{n}$ rate and exhibit non-normal asymptotic distributions. Thus, they fall outside the purview of the literature that deals with regular parametric or semi-parametric problems where $\sqrt{n}$ consistent asymptotically efficient estimates are achievable. Rates of convergence in such problems can either dip below $\sqrt{n}$ or be as fast as $n$.
I will talk about some genres of these problems that have interested me over the course of my career: problems exhibiting cube-root asymptotics ($n^{1/3}$ rate of convergence) and boundary estimation problems (where rates can be as high as $n$). Problems of this type arise in modern non-parametrics and have applications in areas like epidemiology, biomedicine, economics, natural sciences. Most recently, there have been some efforts to understand these models in the high dimensional framework.
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