Testing multivariate uniformity and its applications

Liang, Jiajuan; Fang, Kai‐Tai; Hickernell, Fred J.; Li, Runze

doi:10.1090/s0025-5718-00-01203-5

Cited by 48 publications

(67 citation statements)

References 24 publications

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“…This phenomenon is clearly observed in the simulations of Section 3 below as well as in those given by Liang et al (2001). At first sight, this could seem a bit surprising as it contradicts the well-known "curse of dimensionality" phenomenon which affects many statistical procedures.…”

Section: Theorem 2 Every Convex Polyhedron (With Deepest Point At 0)mentioning

confidence: 69%

“…A different approach is followed in the paper by Liang, Fang, Hickernell and Li (2001) where several tests are proposed for testing uniformity on [0, 1] d . These tests are based on non-trivial number-theoretic considerations and can be used even in high-dimensional cases.…”

Section: Introductionmentioning

confidence: 99%

“…The basic idea is to use some "discrepancy measures" in order to evaluate the distance between expectations of type E(f (X)), f being a real function on [0, 1] d and their Monte Carlo approximations. The asymptotic behaviour of these discrepancies under the uniformity assumption is derived in Liang et al (2001) and used to define several asymptotic uniformity tests. They can be implemented in any dimension d, with a reasonable performance in terms of both type I (significance) and type II (power) errors.…”

Section: Introductionmentioning

confidence: 99%

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Testing multivariate uniformity: The distance‐to‐boundary method

Berrendero¹,

Cuevas²,

Vazquez-Grande³

2006

Can J Statistics

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Given a random sample taken on a compact domain S ⊂ R d , the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties, namely: Unlike most available methods, it is feasible and particularly suitable for high dimensional data, it is distribution-free for a wide range of choices of S, it can be adapted to the case that the support S is unknown and also allows for one-sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well-known "curse of dimensionality". The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive simulation study is given in order to compare the performance of our methods with some recent alternative procedures. The conclusions suggest that the proposed test provides quite a satisfactory balance between statistical power, computational simplicity, and flexibility of application for different dimensions and supports. Title in French: we can supply thisRésumé : Given a random sample taken on a compact domain S ⊂ R d , the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties, namely: Unlike most available methods, it is feasible and particularly suitable for high dimensional data, it is distribution-free for a wide range of choices of S, it can be adapted to the case that the support S is unknown and also allows for one-sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well-known "curse of dimensionality". The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive simulation study is given in order to compare the performance of our methods with some recent alternative procedures. The conclusions suggest that the proposed test provides quite a satisfactory balance between statistical power, computational simplicity, and flexibility of application for different dimensions and supports.

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Section: Theorem 2 Every Convex Polyhedron (With Deepest Point At 0)mentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Testing multivariate uniformity: The distance‐to‐boundary method

Berrendero¹,

Cuevas²,

Vazquez-Grande³

2006

Can J Statistics

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“…Using an argument based on the probability integral transform (see e.g. [1], for the result and [27], for applications to Stochastic Programming) and on its multivariate generalization ( [29]; see [20], p. 354, for an application to Numerical Analysis), it is possible to create random vectors with arbitrary distributions: however, this method often requires the evaluation of complex functions (quantile functions of conditional distributions) and can be quite time-demanding in practice. Moreover, quasi-Monte Carlo methods require more stringent hypotheses on the behavior of the function g: indeed, while any Lebesgue integrable function can be integrated using Monte Carlo algorithms, low discrepancy point sets require the function to be Riemann integrable (however, in order to derive a convergence rate it also has to be of bounded variation in the sense of HardyKrause).…”

Section: Example 3 (Quasi-monte Carlo)mentioning

confidence: 99%

“…Set N 1 x∈X k≥1 N 1 (x, k), where we recall that X is a dense countable subset of X. Inequality (12) is valid for ω ∈ Ω\N 1 , k ≥ 1 and x ∈ X ; moreover, it remains valid for any x ∈ X because each side of (12) defines a Lipschitz function of x, with Lipschitz constant k. Then, taking the supremum, with respect to k, in both sides of (12) and using formula (20) together with the monotone convergence theorem, we obtain (9).…”

Section: Proofsmentioning

confidence: 99%

Approximation of Stochastic Programming Problems

Choirat

Hess

Seri

Monte Carlo and Quasi-Monte Carlo Methods 2004

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Summary. In Stochastic Programming, the aim is often the optimization of a criterion function that can be written as an integral or mean functional with respect to a probability measure P. When this functional cannot be computed in closed form, it is customary to approximate it through an empirical mean functional based on a random Monte Carlo sample. Several improved methods have been proposed, using quasi-Monte Carlo samples, quadrature rules, etc. In this paper, we propose a result on the epigraphical approximation of an integral functional through an approximate one. This result allows us to deal with Monte Carlo, quasi-Monte Carlo and quadrature methods. We propose an application to the epi-convergence of stochastic programs approximated through the empirical measure based on an asymptotically mean stationary (ams) sequence. Because of the large scope of applications of ams measures in Applied Probability, this result turns out to be relevant for approximation of stochastic programs through real data.

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Koksma–Hlawka Inequality

Hickernell¹

2004

Encyclopedia of Statistical Sciences

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The Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points, \documentclass{minimal}\usepackage{amsmath}\begin{document}$D(\{\vec{x}_{i}\})$\end{document} , and the variation of the integrand, V ( g ). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.

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Testing multivariate uniformity and its applications

Cited by 48 publications

References 24 publications

Testing multivariate uniformity: The distance‐to‐boundary method

Testing multivariate uniformity: The distance‐to‐boundary method

Approximation of Stochastic Programming Problems

Koksma–Hlawka Inequality

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