1,720,999 research outputs found
Approximating weighted chi-square distributions
No full textDegenerate U− and V−statistics have asymptotic distributions that are given by an infinite weighted sum of χ2 random variables, whose weights are the eigenvalues of an integral operator whose solution is in general very difficult. We provide a new computational approximation of the eigenvalues and of the asymptotic distribution. As an illustration, we show that our algorithm is able to recover the tabulated asymptotic distribution of the Cramér-von Mises statistic
Consistency of quasi-maximum likelihood estimators in ARCH models
No full textThe aim of this paper is to propose a new approach to the proof of consistency of quasi-maximum likelihood estimators in ARCH models. The new proof uses epigraphical convergence, a kind of convergence that has proved to be useful in Operations Research, Optimization and Statistics, and asymptotic mean stationarity, that is the most general concept allowing for an Ergodic Theorem to hold
Approximation of robust and chance-constrained programs
No full textWe consider scenario approximation of problems given by the optimization of a function over a constraint that is too difficult to be handled but can be efficiently approximated by a finite collection of constraints corresponding to alternative scenarios. The covered programs include min-max games, and semi-infinite, robust and chance-constrained programming problems. We prove convergence of the solutions of the approximated programs to the given ones, using mainly epigraphical convergence, a kind of variational convergence that has demonstrated to be a valuable tool in optimization problems
Computational Aspects of Cui-Freeden Statistics for Equidistribution on the Sphere
No full textIn this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (SIAM J. Sci. Comput., 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations
Numerical Properties of Generalized Discrepancies on Spheres of Arbitrary Dimension
No full textQuantifying uniformity of a configuration of points on the sphere is an interesting topic that is receiving growing attention in numerical analysis. An elegant solution has been provided by Cui and Freeden [J. Cui, W. Freeden, Equidistribution on the sphere, SIAM J. Sci. Comput. 18 (2) (1997) 595–609], where a class of discrepancies, called generalized discrepancies and originally associated with pseudodifferential operators on the unit sphere in R3, has been introduced. The objective of this paper is to extend to the sphere of arbitrary dimension this class of discrepancies and to study their numerical properties. First we show that generalized discrepancies are diaphonies on the hypersphere. This allows us to completely characterize the sequences of points for which convergence to zero of these discrepancies takes place. Then we discuss the worst-case error of quadrature rules and we derive a result on tractability of multivariate integration on the hypersphere. At last we provide several versions of Koksma–Hlawka type inequalities for integration of functions defined on the sphere
Analytic Hierarchy Process, A Psychometric Approach
No full textThe Analytic Hierarchy Process, or AHP for short, is a decision method that is often used in Management and Economics. In this paper, we study real data that we obtained from a simple experiment. Then, from the analysis of these data, we point out that most pathological behaviours of the AHP correspond to phenomena already encountered in Psychometric Choice Theory
Bootstrap confidence sets for the Aumann mean of a random closed set
No full textThe objective is to develop a reliable method to build confidence sets for the Aumann mean of a random closed set as estimated through the Minkowski empirical mean. First, a general definition of the confidence set for the mean of a random set is provided. Then, a method using a characterization of the confidence set through the support function is proposed and a bootstrap algorithm is described, whose performance is investigated in Monte Carlo simulations
Comparison of Approximations for Compound Poisson Processes
No full textThe aim of this paper is to provide a comparison of the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this theory it is usually supposed that a portfolio of clients is at risk for a time period of length t. The claims take place according to a Poisson process, so that the number of claims is a Poisson random variable N. Each single claim is an independent replication of the random variable X, representing the claim severity. The object of study is the cumulative distribution function of the random sum of N independent replications of X, i.e. a compound Poisson process representing the aggregate claim or total claim amount process in a period of length t. Due to the complexity of its computation, several approximation methods for this cdf have been proposed in the literature.
In this paper, we only consider approximations using the lower order moments of the involved distributions. This requirement rules out the Esscher approximation as well as methods of exact computation based on a preliminary discretization of the claim distribution (Panjer recursion, FFT). Therefore, we consider the Normal, Edgeworth, NP2, NP2a, Adjusted NP2, NP3, Wilson-Hilferty, Haldane A and B, Lognormal, Gamma, Translated Gamma, Bowers Gamma, Inverse Gaussian and Gamma-IG approximations. For these fifteen approximations put forward in the literature, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as the observation period diverges to infinity. Using these expansions, several statements concerning the quality of these approximations can find theoretical support, while other statements can be disproved on the same grounds. At last we investigate numerically the accuracy of the proposed formulas
Statistical Properties of Generalized Discrepancies and Related Quantities
No full textWhen testing that a sample of n points in the unit hypercube [0,1]d comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell (1996, 1998) introduced the so-called generalized Lp−discrepancies. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness of fit tests. The aim of this paper is to derive the strong and weak asymptotic properties of these statistics
Estimation in Discrete Parameter Models
No full textIn some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points. In this case, we speak of discrete parameter models. Even though the problem is quite old and has interesting connections with testing and model selection, asymptotic theory for these models has hardly ever been studied. Therefore, we discuss consistency, asymptotic distribution theory, information inequalities and their relations with efficiency and superefficiency for a general class of m-estimators
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