Bayesian nonparametric statistical inference for Poisson point processes

Summary A random measure is said to be selected by a weighted gamma prior probability if the values it assigns to disjoint sets are independent gamma random variables with positive multipliers. If the intensity measure of a nonhomogeneous Poisson point process is selected by a weighted gamma prior probability and if a sample is drawn from the Poisson point process having this intensity measure, then the posterior random intensity measure given the observations is also selected by a weighted gamma prior probability. If the measure space is Euclidean and if the true intensity measure is continuous and finite, the centered posterior process, rescaled by the square root of the sample size, will converge weakly in Skorohod topology to a Wiener process subject to a change of time scale.This research was supported in part by the National Science Foundation Grants MCS 77-10376 and MCS 75-14194     

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.     

Asymptotic normality of the nearest neighbor hazard estimates

1.IntroductionSupposethatXI)')Xu,'beani.i.d.sequenceofrandomvariableswithdistributionfunctionF(x)anddensityfunctionf(x).TOestimatethedensityfunctionatxbasedonthefirstnobservations,thekerneltypeestimategiveswhereK')isakernelfunction,R.(x)isabandwidthsequenceandFi')isanestimateofF,usuallytakentobetheempiricaldistributionfunction.Formoredetails,see[21.Inmedicalapplications,itisoftenmoreimportanttoestimatethehazardfunctiondefinedbyA(x)~f(x)/(1--F(x)).IfXrepresentsalife-time,thenA(x)repre…     

Projection estimates for harmonic measure

Stochastic proofs of the Beurling projection theorem and the Hall projection theorem for harmonic measure are given. Somed-dimensional versions (for alld>1) which follow from this     

Bayesian robustness under a skew-normal class of prior distribution

We develop a global sensitivity analysis to measure the robustness of the Bayesian estimators with respect to a class of prior distributions. This class arises when we consider multiplicative contamination of a base prior distribution. A similar structure was presented by van der Linde [12]. Some particular specifications for this multiplicative contamination class coincide with well known families of skewed distributions. In this paper, we explore the skew-normal multiplicative contamination class for the prior distribution of the location parameter of a normal model. Results of a Bayesian conjugation and expressions for some measures of distance between posterior means and posterior variance are obtained. We also elaborate on the behavior of the posterior means and of the posterior variances through a simulation study.     

Interval estimates for posterior probabilities in a multivariate normal classification model

This paper is devoted to the asymptotic distribution of estimators for the posterior probability that a p-dimensional observation vector originates from one of k normal distributions with identical covariance matrices. The estimators are based on training samples for the k distributions involved. Observation vector and prior probabilities are regarded as given constants. The validity of various estimators and approximate confidence intervals is investigated by simulation experiments.     

Localisation and weighted inequalities for spherical Fourier means

In this work, we establish certain equivalences between the localisation properties with respect to spherical Fourier means of the support of a given Borel measure and the L ²-rate of decay of the Fourier extension operator associated to it. This, in turn, is intimately connected with the property that the X-ray transform of the measure be uniformly bounded. Geometric properties of sets supporting such a measure are studied. The research of this paper has been partially supported by the EU Comission via the network HARP, and by MEC Grant MTM2004-00678. The first author was partially supported by a Leverhulme Study Abroad Fellowship.     

Error estimates for Miller's algorithm

Summary A rigorous error analysis is given of both truncation and rounding errors in Miller's algorithm for three-term scalar recursions and 2×2 matrix-vector recursions. The error bounds are shown to be very realistic and this will be supported by examples. The results are generalized to recursions of higher order.     

Local approximations on manifolds and weighted estimates

An approach proposed by the author for representing spaces of local functions (s.l.f.) by means of an abstract function with values in the dual space is developed; thanks to this generalization of s.l.f. to the case of a nonuniform net and manifolds are obtained. Moreover, some weighted estimates of approximation by regular s.l.f. are established, and simple proofs are given of theorems of approximation by regular s.l.f. in the spaces of S. L. Sobolev and O. V. Besov.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 125–138, 1984.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

Adaptive methods for boundary integral equations: Complexity and convergence estimates

This paper is concerned with developing numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates. This is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for PDEs. We shall show how to realize asymptotically optimal complexity in the present context of global operators. ``Asymptotically optimal' means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given. The theoretical findings are supported and quantified by the first numerical experiments.

     

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular, uniform, Wigner, Weyl, Karamata and hypergeometric types are explicitly identified. Some illustrative examples are given.     

The hazard of being an English football league manager: empirical estimates for three recent league seasons

This paper uses data drawn from the English Football League to model separate hazard rates for club managers for the 2001/2, 2002/3 and 2003/4 seasons. On average over the three seasons, approximately one-third of managers involuntarily exited employment status with their club. We model the hazard using a standard logistic model exploiting information on the spell at risk rather than the individual. The role of neglected heterogeneity is also examined using random and fixed effects logistic models within a discrete-time setting. League position at the start of the spell at risk is found to be the most important determinant of a manager's involuntary exit. A variety of individual specific human capital and other covariates are found to be unimportant in determining the hazard and no role for unobservable heterogeneity as captured by random effects is detected.     

Estimating the Finite Population Versions of Mean Residual Life-Time Function and Hazard Function Using Bayes Method

In this paper we introduce a finite population version of the mean residual life-time (MRL) function and the hazard function, and study Bayesian estimation of these functions. The unknown parameter is the complete set (y1,...;,yN) of lifetimes of the N units which constitute the complete population. A hierarchical type prior is used, where the yi's are assumed conditionally independent given a random parameter . The data consists of a random sample of n values of yi. The Bayes estimators of MRL and hazard functions, respectively, are then obtained as the posterior expectations of the unknown functions.     

Uniform Mixtures Via Posterior Means

The problem of identification of uniform mixtures via posterior means is studied. For linear posterior means a complete solution is given. It determines a family of prior distributions involving beta of both kinds and gamma. Identifiability via any consistent posterior mean is also investigated.     

Geodesic metrics on fractals and applications to heat kernel estimates

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of pr...     

Some properties of the bivariate lognormal distribution for reliability applications

In this paper, we study the bivariate lognormal distribution from a reliability point of view. The conditional distribution of X given Y > y is found to be log‐skew normal. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient, and its value in the case of our model is derived. The probability distributions, in the case of series and parallel systems, are derived, and the monotonicity of their failure rates is discussed. Three real applications of the bivariate lognormal distribution are provided, two from financial economics and one from reliability. Copyright © 2012 John Wiley & Sons, Ltd.     

Inverse inequality estimates with symbolic computation

In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound, given in “p- and hp-finite element methods” (Schwab, 1998), by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.     

Lp estimates for-(e)-equation on generalized complex ellipsoids

The estimate of a holomorphic supporting function for the generalized complex ellipsoid in is Cn given, This domain is not decoupled. By using this estimate, the best possible Lp estimates for the -(e)-equation and some results of function theory on generalized complex ellipsoids are proved.