26,656 research outputs found
The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models
In this contribution we give a pedagogic introduction to the newly introduced
adaptive interpolation method to prove in a simple and unified way replica
formulas for Bayesian optimal inference problems. Many aspects of this method
can already be explained at the level of the simple Curie-Weiss spin system.
This provides a new method of solution for this model which does not appear to
be known. We then generalize this analysis to a paradigmatic inference problem,
namely rank-one matrix estimation, also refered to as the Wigner spike model in
statistics. We give many pointers to the recent literature where the method has
been succesfully applied
Tail index estimation, concentration and adaptivity
This paper presents an adaptive version of the Hill estimator based on
Lespki's model selection method. This simple data-driven index selection method
is shown to satisfy an oracle inequality and is checked to achieve the lower
bound recently derived by Carpentier and Kim. In order to establish the oracle
inequality, we derive non-asymptotic variance bounds and concentration
inequalities for Hill estimators. These concentration inequalities are derived
from Talagrand's concentration inequality for smooth functions of independent
exponentially distributed random variables combined with three tools of Extreme
Value Theory: the quantile transform, Karamata's representation of slowly
varying functions, and R\'enyi's characterisation of the order statistics of
exponential samples. The performance of this computationally and conceptually
simple method is illustrated using Monte-Carlo simulations
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