57 research outputs found

### On ergodic two-armed bandits

A device has two arms with unknown deterministic payoffs and the aim is to
asymptotically identify the best one without spending too much time on the
other. The Narendra algorithm offers a stochastic procedure to this end. We
show under weak ergodic assumptions on these deterministic payoffs that the
procedure eventually chooses the best arm (i.e., with greatest Cesaro limit)
with probability one for appropriate step sequences of the algorithm. In the
case of i.i.d. payoffs, this implies a "quenched" version of the "annealed"
result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004)
1424--1454] by the law of iterated logarithm, thus generalizing it. More
precisely, if $(\eta_{\ell,i})_{i\in \mathbb {N}}\in\{0,1\}^{\mathbb {N}}$,
$\ell\in\{A,B\}$, are the deterministic reward sequences we would get if we
played at time $i$, we obtain infallibility with the same assumption on
nonincreasing step sequences on the payoffs as in Lamberton, Pag\`{e}s and
Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d.
assumption by the hypothesis that the empirical averages
$\sum_{i=1}^n\eta_{A,i}/n$ and $\sum_{i=1}^n\eta_{B,i}/n$ converge, as $n$
tends to infinity, respectively, to $\theta_A$ and $\theta_B$, with rate at
least $1/(\log n)^{1+\varepsilon}$, for some $\varepsilon >0$. We also show a
fallibility result, that is, convergence with positive probability to the
choice of the wrong arm, which implies the corresponding result of Lamberton,
Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] in the
i.i.d. case.Comment: Published in at http://dx.doi.org/10.1214/10-AAP751 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Semiparametric estimation of a two-component mixture model

Suppose that univariate data are drawn from a mixture of two distributions
that are equal up to a shift parameter. Such a model is known to be
nonidentifiable from a nonparametric viewpoint. However, if we assume that the
unknown mixed distribution is symmetric, we obtain the identifiability of this
model, which is then defined by four unknown parameters: the mixing proportion,
two location parameters and the cumulative distribution function of the
symmetric mixed distribution. We propose estimators for these four parameters
when no training data is available. Our estimators are shown to be strongly
consistent under mild regularity assumptions and their convergence rates are
studied. Their finite-sample properties are illustrated by a Monte Carlo study
and our method is applied to real data.Comment: Published at http://dx.doi.org/10.1214/009053606000000353 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Semiparametric topographical mixture models with symmetric errors

Motivated by the analysis of a Positron Emission Tomography (PET) imaging
data considered in Bowen et al. (2012), we introduce a semiparametric
topographical mixture model able to capture the characteristics of dichotomous
shifted response-type experiments. We propose a local estimation procedure,
based on the symmetry of the local noise, for the proportion and locations
functions involved in the proposed model. We establish under mild conditions
the minimax properties and asymptotic normality of our estimators when Monte
Carlo simulations are conducted to examine their finite sample performance.
Finally a statistical analysis of the PET imaging data in Bowen et al. (2012)
is illustrated for the proposed method.Comment: 19 figure

### Semiparametric mixtures of symmetric distributions

We consider in this paper the semiparametric mixture of two distributions
equal up to a shift parameter. The model is said to be semiparametric in the
sense that the mixed distribution is not supposed to belong to a parametric
family. In order to insure the identifiability of the model it is assumed that
the mixed distribution is symmetric, the model being then defined by the mixing
proportion, two location parameters, and the probability density function of
the mixed distribution. We propose a new class of M-estimators of these
parameters based on a Fourier approach, and prove that they are square root
consistent under mild regularity conditions. Their finite-sample properties are
illustrated by a Monte Carlo study and a benchmark real dataset is also studied
with our method

### A simple variance inequality for U-statistics of a Markov chain with applications

We establish a simple variance inequality for U-statistics whose underlying
sequence of random variables is an ergodic Markov Chain. The constants in this
inequality are explicit and depend on computable bounds on the mixing rate of
the Markov Chain. We apply this result to derive the strong law of large number
for U-statistics of a Markov Chain under conditions which are close from being
optimal

### Semiparametric topographical mixture models with symmetric errors

Motivated by the analysis of a Positron Emission Tomography (PET) imaging data considered in Bowen et al. (2012), we introduce a semiparametric topographical mixture model able to capture the characteristics of dichotomous shifted response-type experiments. We propose a local estimation procedure, based on the symmetry of the local noise, for the proportion and locations functions involved in the proposed model. We establish under mild conditions the minimax properties and asymptotic normality of our estimators when Monte Carlo simulations are conducted to examine their finite sample performance. Finally a statistical analysis of the PET imaging data in Bowen et al. (2012) is illustrated for the proposed method

### The Logarithmic Sobolev Constant of The Lamplighter

We give estimates on the logarithmic Sobolev constant of some finite
lamplighter graphs in terms of the spectral gap of the underlying base. Also,
we give examples of application

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