Minimax estimation of nonlinear functionals of a density via construction of probability measures with prescribed properties
Date(s) : 22/11/2021 iCal
14h00
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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