Recursos de colección

Bernoulli

1. Simulation of hitting times for Bessel processes with non-integer dimension

Deaconu, Madalina; Herrmann, Samuel
In this paper, we complete and improve the study of the simulation of the hitting times of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work (Ann. Appl. Probab. 23 (2013) 2259–2289), the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method, called walk on moving spheres algorithm (WoMS), was based mainly on the explicit formula for the distribution of the hitting time and on the connection between the Bessel process and the Euclidean norm...

2. Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials

Fox, Colin; Parker, Albert
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss–Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and generalizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique...

3. Some monotonicity properties of parametric and nonparametric Bayesian bandits

Yu, Yaming
One of two independent stochastic processes (arms) is to be selected at each of $n$ stages. The selection is sequential and depends on past observations as well as the prior information. The objective is to maximize the expected future-discounted sum of the $n$ observations. We study structural properties of this classical bandit problem, in particular how the maximum expected payoff and the optimal strategy vary with the priors, in two settings: (a) observations from each arm have an exponential family distribution and different arms are assigned independent conjugate priors; (b) observations from each arm have a nonparametric distribution and different...

4. The failure of the profile likelihood method for a large class of semi-parametric models

Beutner, Eric; Bordes, Laurent; Doyen, Laurent
We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector $\mathbf{{\theta}}$, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde{\boldsymbol {\theta}}$, say, with probability $p$. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we...

5. A proof of the Shepp–Olkin entropy concavity conjecture

Hillion, Erwan; Johnson, Oliver
We prove the Shepp–Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin’s original conjecture, to consider Rényi and Tsallis entropies.

6. Adaptive estimation for bifurcating Markov chains

Bitseki Penda, S. Valère; Hoffmann, Marc; Olivier, Adélaïde
In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^{p}$-loss error, for $1\leq p<\infty$. We prove that our estimators are (nearly) optimal in a minimax sense. As an application,...

Peyre, Rémi

20. Influence functions for penalized M-estimators

Avella-Medina, Marco
We study the local robustness properties of general nondifferentiable penalized M-estimators via the influence function. More precisely, we propose a framework that allows us to define rigorously the influence function as the limiting influence function of a sequence of approximating estimators. We show that it can be used to characterize the robustness properties of a wide range of sparse estimators and we derive its form for general penalized M-estimators including lasso and adaptive lasso type estimators. We prove that our influence function is equivalent to a derivative in the sense of distribution theory.

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