Optimal estimation of linear functionals in irregular nonparametric models

It is well known that irregular (i.e., not Hellinger-differentiable) parametric models show unusual asymptotic behaviour (e.g., faster rates of convergence, non-Gaussian limit laws). In the case of nonparametric regression $Y_i=f(x_i)+\epsilon_i$ with irregular errors, e.g. $\epsilon_i\sim \Gamma(\alpha,\lambda)$, faster rates of convergence can be obtained for tail indices $\alpha<2$ using a classical local parametric (e.g. polynomial) method. Surprisingly, for estimating linear functionals like $\int_0^1 f(x)dx$ the theory and methods are completely different from both, the parametric irregular and the nonparametric regular theory. In particular, plug-in methods are not rate-optimal and optimal estimators are unbiased even though involving a bandwidth tuning parameter. In addition, a nonparametric MLE (over Hölder classes) is computationally feasible and reveals fascinating properties in connection with stochastic geometry. Nonparametric sufficiency and completeness arguments can be applied to show its non-asymptotic optimality in a fundamental Poisson point process model. The theory is motivated by volatility estimation based on best bid and ask prices in stock markets.

(work in progress with Leonie Selk, Hamburg)

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