96 research outputs found
Nonparametric regression with martingale increment errors
We consider the problem of adaptive estimation of the regression function in
a framework where we replace ergodicity assumptions (such as independence or
mixing) by another structural assumption on the model. Namely, we propose
adaptive upper bounds for kernel estimators with data-driven bandwidth
(Lepski's selection rule) in a regression model where the noise is an increment
of martingale. It includes, as very particular cases, the usual i.i.d.
regression and auto-regressive models. The cornerstone tool for this study is a
new result for self-normalized martingales, called ``stability'', which is of
independent interest. In a first part, we only use the martingale increment
structure of the noise. We give an adaptive upper bound using a random rate,
that involves the occupation time near the estimation point. Thanks to this
approach, the theoretical study of the statistical procedure is disconnected
from usual ergodicity properties like mixing. Then, in a second part, we make a
link with the usual minimax theory of deterministic rates. Under a beta-mixing
assumption on the covariates process, we prove that the random rate considered
in the first part is equivalent, with large probability, to a deterministic
rate which is the usual minimax adaptive one
Statistical inference versus mean field limit for Hawkes processes
We consider a population of individuals, of which we observe the number
of actions as time evolves. For each couple of individuals , may or
not influence , which we model by i.i.d. Bernoulli-random variables,
for some unknown parameter . Each individual acts autonomously at
some unknown rate and acts by mimetism at some rate depending on the
number of recent actions of the individuals which influence him, the age of
these actions being taken into account through an unknown function
(roughly, decreasing and with fast decay). The goal of this paper is to
estimate , which is the main charateristic of the graph of interactions, in
the asymptotic , . The main issue is that the mean
field limit (as ) of this model is unidentifiable, in that it
only depends on the parameters and . Fortunately, this mean
field limit is not valid for large times. We distinguish the subcritical case,
where, roughly, the mean number of actions per individual increases
linearly and the supercritical case, where increases exponentially.
Although the nuisance parameter is non-parametric, we are able, in
both cases, to estimate without estimating in a nonparametric
way, with a precision of order , up to some
arbitrarily small loss. We explain, using a Gaussian toy model, the reason why
this rate of convergence might be (almost) optimal
On the false discovery proportion convergence under Gaussian equi-correlation
We study the convergence of the false discovery proportion (FDP) of the
Benjamini-Hochberg procedure in the Gaussian equi-correlated model, when the
correlation converges to zero as the hypothesis number grows to
infinity. By contrast with the standard convergence rate holding
under independence, this study shows that the FDP converges to the false
discovery rate (FDR) at rate in this
equi-correlated model
Testing the finiteness of the support of a distribution: a statistical look at Tsirelson's equation
We consider the following statistical problem: based on an i.i.d.sample of
size n of integer valued random variables with common law m, is it possible to
test whether or not the support of m is finite as n goes to infinity? This
question is in particular connected to a simple case of Tsirelson's equation,
for which it is natural to distinguish between two main configurations, the
first one leading only to laws with finite support, and the second one
including laws with infinite support. We show that it is in fact not possible
to discriminate between the two situations, even using a very weak notion of
statistical test
On Monte-Carlo tree search for deterministic games with alternate moves and complete information
We consider a deterministic game with alternate moves and complete
information, of which the issue is always the victory of one of the two
opponents. We assume that this game is the realization of a random model
enjoying some independence properties. We consider algorithms in the spirit of
Monte-Carlo Tree Search, to estimate at best the minimax value of a given
position: it consists in simulating, successively, well-chosen matches,
starting from this position. We build an algorithm, which is optimal, step by
step, in some sense: once the first matches are simulated, the algorithm
decides from the statistics furnished by the first matches (and the a
priori we have on the game) how to simulate the -th match in such a way
that the increase of information concerning the minimax value of the position
under study is maximal. This algorithm is remarkably quick. We prove that our
step by step optimal algorithm is not globally optimal and that it always
converges in a finite number of steps, even if the a priori we have on the game
is completely irrelevant. We finally test our algorithm, against MCTS, on
Pearl's game and, with a very simple and universal a priori, on the games
Connect Four and some variants. The numerical results are rather disappointing.
We however exhibit some situations in which our algorithm seems efficient
Asymptotic lower bounds in estimating jumps
We study the problem of the efficient estimation of the jumps for stochastic
processes. We assume that the stochastic jump process is
observed discretely, with a sampling step of size . In the spirit of
Hajek's convolution theorem, we show some lower bounds for the estimation error
of the sequence of the jumps . As an intermediate result,
we prove a LAMN property, with rate , when the marks of the
underlying jump component are deterministic. We deduce then a convolution
theorem, with an explicit asymptotic minimal variance, in the case where the
marks of the jump component are random. To prove that this lower bound is
optimal, we show that a threshold estimator of the sequence of jumps based on the discrete observations, reaches the minimal variance of
the previous convolution theorem.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ515 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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