Minimax estimation of nonlinear functionals of a density via construction of probability measures with prescribed properties

Oleg Lepski

Date(s) : 22/11/2021   iCal
14 h 00 min

In this talk I discuss the minimax estimation of integral-type functionals of a probability density. In the first part  I present the construction of  the lower bound of minimax risk on an arbitrary set of functions. This construction is mostly based on the following principles.
First, the original original estimation is reduced  to a problem of testing two composite hypotheses for mixture distributions which are obtained by imposing prior probability measures with intersecting supports on parameters of a
functional family. These couple of measures should possess several properties and their construction,
basing on the notion of the best approximation of continuous functions, has an independent interest. In particular, the moment matching technique can be mentioned in this context.
The second idea is related to construction of a specific parameterized family of densities on
which the lower bound of the minimax risks is established.
The third one is related to the analysis of the so-called Bayesian likelihood ratio. The multivariate density model on $R^d$ requires development of the original technique.

In the second part of the talk I discuss  the application of  the proposed approach to the estimation of $L_p$-norm of a density, $1<p<infty$. The considered functional class  is the intersection of usual $L_q$-ball, $1<qleqinfty$ with a ball in an arbitrary anisotropic Nikolskii semi-metric. Some unusual phenomena related to the decay of minimax risk will be presented.


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