Mostrando recursos 1 - 20 de 1.316

  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,...

  7. Fractional Brownian motion satisfies two-way crossing

    Peyre, Rémi
    We prove the following result: For $(Z_{t})_{t\in\mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, for any stopping time $\tau$, there exist arbitrarily small $\varepsilon>0$ such that $Z_{\tau+\varepsilon}

  8. Efficient Bayesian estimation and uncertainty quantification in ordinary differential equation models

    Bhaumik, Prithwish; Ghosal, Subhashis
    Often the regression function is specified by a system of ordinary differential equations (ODEs) involving some unknown parameters. Typically analytical solution of the ODEs is not available, and hence likelihood evaluation at many parameter values by numerical solution of equations may be computationally prohibitive. Bhaumik and Ghosal (Electron. J. Stat. 9 (2015) 3124–3154) considered a Bayesian two-step approach by embedding the model in a larger nonparametric regression model, where a prior is put through a random series based on B-spline basis functions. A posterior on the parameter is induced from the regression function by minimizing an integrated weighted squared distance...

  9. Asymptotic expansions and hazard rates for compound and first-passage distributions

    Butler, Ronald W.
    A general theory which provides asymptotic tail expansions for density, survival, and hazard rate functions is developed for both absolutely continuous and integer-valued distributions. The expansions make use of Tauberian theorems which apply to moment generating functions (MGFs) with boundary singularities that are of gamma-type or log-type. Standard Tauberian theorems from Feller [An Introduction to Probability Theory and Its Applications II (1971) Wiley] can provide a limited theory but these theorems do not suffice in providing a complete theory as they are not capable of explaining tail behaviour for compound distributions and other complicated distributions which arise in stochastic modelling...

  10. Non-central limit theorems for random fields subordinated to gamma-correlated random fields

    Leonenko, Nikolai; Ruiz-Medina, M. Dolores; Taqqu, Murad S.
    A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in $d$-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Itô stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.

  11. Efficiency and bootstrap in the promotion time cure model

    Portier, François; El Ghouch, Anouar; Van Keilegom, Ingrid
    In this paper, we consider a semiparametric promotion time cure model and study the asymptotic properties of its nonparametric maximum likelihood estimator (NPMLE). First, by relying on a profile likelihood approach, we show that the NPMLE may be computed by a single maximization over a set whose dimension equals the dimension of the covariates plus one. Next, using $Z$-estimation theory for semiparametric models, we derive the asymptotics of both the parametric and nonparametric components of the model and show their efficiency. We also express the asymptotic variance of the estimator of the parametric component. Since the variance is difficult to...

  12. Integrated empirical processes in $L^{p}$ with applications to estimate probability metrics

    Cárcamo, Javier
    We discuss the convergence in distribution of the $r$-fold (reverse) integrated empirical process in the space $L^{p}$, for $1\le p\le\infty$. In the case $1\le p<\infty$, we find the necessary and sufficient condition on a positive random variable $X$ so that this process converges weakly in $L^{p}$. This condition defines a Lorentz space and can be also characterized in terms of several integrability conditions related to the process $\{(X-t)^{r}_{+}:t\ge0\}$. For $p=\infty$, we obtain an integrability requirement on $X$ guaranteeing the convergence of the integrated empirical process. In particular, these results imply a limit theorem for the stop-loss distance between the empirical...

  13. Sieve maximum likelihood estimation for a general class of accelerated hazards models with bundled parameters

    Zhao, Xingqiu; Wu, Yuanshan; Yin, Guosheng
    In semiparametric hazard regression, nonparametric components may involve unknown regression parameters. Such intertwining effects make model estimation and inference much more difficult than the case in which the parametric and nonparametric components can be separated out. We study the sieve maximum likelihood estimation for a general class of hazard regression models, which include the proportional hazards model, the accelerated failure time model, and the accelerated hazards model. Coupled with the cubic B-spline, we propose semiparametric efficient estimators for the parameters that are bundled inside the nonparametric component. We overcome the challenges due to intertwining effects of the bundled parameters, and...

  14. Weak convergence of empirical copula processes indexed by functions

    Radulović, Dragan; Wegkamp, Marten; Zhao, Yue
    Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required. ¶ A novel integration by parts formula for multivariate, right-continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.

  15. On Stein’s method for products of normal random variables and zero bias couplings

    Gaunt, Robert E.
    In this paper, we extend Stein’s method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random variables. We end by proving some product normal approximation...

  16. Approximate local limit theorems with effective rate and application to random walks in random scenery

    Giuliano, Rita; Weber, Michel
    We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term. That is with explicit parameters and universal constants. We also show that our estimates allow us to recover Gnedenko and provide a version with effective bounds of Gamkrelidze’s local limit theorem. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.

  17. Pólya urn schemes with infinitely many colors

    Bandyopadhyay, Antar; Thacker, Debleena
    In this work, we introduce a class of balanced urn schemes with infinitely many colors indexed by ${\mathbb{Z} }^{d}$, where the replacement schemes are given by the transition matrices associated with bounded increment random walks. We show that the color of the $n$th selected ball follows a Gaussian distribution on ${\mathbb{R} }^{d}$ after ${\mathcal{O} }(\log n)$ centering and ${\mathcal{O} }(\sqrt{\log n})$ scaling irrespective of whether the underlying walk is null recurrent or transient. We also provide finer asymptotic similar to local limit theorems for the expected configuration of the urn. The proofs are based on a novel representation of the...

  18. Transportation and concentration inequalities for bifurcating Markov chains

    Bitseki Penda, S. Valère; Escobar-Bach, Mikael; Guillin, Arnaud
    We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful concentration inequalities. We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive function.

  19. On predictive density estimation for location families under integrated absolute error loss

    Kubokawa, Tatsuya; Marchand, Éric; Strawderman, William E.
    This paper is concerned with estimating a predictive density under integrated absolute error ($L_{1}$) loss. Based on a spherically symmetric observable $X\sim p_{X}(\Vert x-\mu\Vert^{2})$, $x,\mu \in \mathbb{R}^{d}$, we seek to estimate the (unimodal) density of $Y\sim q_{Y}(\Vert y-\mu \Vert^{2})$, $y\in \mathbb{R}^{d}$. We focus on the benchmark (and maximum likelihood for unimodal $p$) plug-in density estimator $q_{Y}(\Vert y-X\Vert^{2})$ and, for $d\geq 4$, we establish its inadmissibility, as well as provide plug-in density improvements, as measured by the frequentist risk taken with respect to $X$. Sharper results are obtained for the subclass of scale mixtures of normal distributions which include the normal...

  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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