On the Bootstrap of $U$ and $V$ Statistics

Arcones, Miguel A.; Giné, Evarist

doi:10.1214/aos/1176348650

Cited by 140 publications

(138 citation statements)

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“…Uniform convergence can also be derived (under mild and usual conditions) from general theory of U and V statistics (see, for example, Arcones & Gine 1992). Theorem 2 guarantees that the distribution of W 2 k (n) can be approximated by the W 2, * k (n) one and, as usual, this distribution can be approximated by using the Monte Carlo method following the algorithm:…”

Section: Bootstrap Approximationmentioning

confidence: 98%

Cramér-Von Mises Statistic for Repeated Measures

Martínez–Camblor¹,

Carleos²,

Corral³

2014

Rev. colomb. estad.

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The Cramér-von Mises criterion is employed to compare whether the marginal distribution functions of a k-dimensional random variable are equal or not. The well-known Donsker invariance principle and the KarhunenLoéve expansion is used in order to derive its asymptotic distribution. Two different resampling plans (one based on permutations and the other one based on the general bootstrap algorithm, gBA) are also considered to approximate its distribution. The practical behaviour of the proposed test is studied from a Monte Carlo simulation study. The statistical power of the test based on the Cramér-von Mises criterion is competitive when the underlying distributions are different in location and is clearly better than the Friedman one when the sole difference among the involved distributions is the spread or the shape. Both resampling plans lead to similar results although the gBA avoids the usual required interchangeability assumption. Finally, the method is applied on the study of the evolution inequality incomes distribution between some European countries along the years 2000 and 2011.Key words: Asymptotic Distribution, Bootstrap, Cramér-von Mises statistic, Hypothesis testing, Permutation test, Repeated Measures. ResumenEl criterio de Cramér-von Mises es empleado para comparar la igualdad entre las distribuciones marginales de una variable aleatoria k-dimensional. El conocido principio de invaranza de Donsker y la expansión de KarhunenLoéve se usan para derivar su distribución asintótica. Dos planes de remuestreo diferentes (uno basado en permutaciones y el otro basado en el algoritmo bootstrap general, gBA) son usados para aproximar su distribución. El comportamiento práctico del test propuesto es estudiado mediante simulaciones de Monte Carlo. La potencia estadística del test basado en el criterio

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Section: Bootstrap Approximationmentioning

confidence: 98%

Cramér-Von Mises Statistic for Repeated Measures

Martínez–Camblor¹,

Carleos²,

Corral³

2014

Rev. colomb. estad.

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“…Second, integrated variance does not follow a Markov process, ruling out as well bootstrap algorithms for Markov sequences (Rajarshi, 1990;Paparoditis and Politis, 2002;Horowitz, 2003). 3 Third, a standard nonparametric bootstrap would also fail to mimic the limiting distribution of our test statistics for they involve degenerate U-statistics (Bretagnolle, 1983;Arcones and Giné, 1992).…”

Section: Bootstrap Critical Valuesmentioning

confidence: 99%

International market links and volatility transmission

Corradi,

Distaso,

Fernandes

2012

Journal of Econometrics

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“…We may use this theorem directly to test whether two distributions are identical, given an appropriate finite sample approximation to the (1 − α)th quantile of (8). In [16], this was achieved via two strategies: by using the bootstrap [34], and by fitting Pearson curves using the first four moments [35,Section 18.8].…”

Section: Two-sample Testmentioning

confidence: 99%

A Hilbert Space Embedding for Distributions

Smola¹,

Gretton²,

Song³

et al.

Lecture Notes in Computer Science

339

491

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Abstract. We describe a technique for comparing distributions without the need for density estimation as an intermediate step. Our approach relies on mapping the distributions into a reproducing kernel Hilbert space. Applications of this technique can be found in two-sample tests, which are used for determining whether two sets of observations arise from the same distribution, covariate shift correction, local learning, measures of independence, and density estimation.Kernel methods are widely used in supervised learning [1,2,3,4], however they are much less established in the areas of testing, estimation, and analysis of probability distributions, where information theoretic approaches [5,6] have long been dominant. Recent examples include [7] in the context of construction of graphical models, [8] in the context of feature extraction, and [9] in the context of independent component analysis. These methods have by and large a common issue: to compute quantities such as the mutual information, entropy, or Kullback-Leibler divergence, we require sophisticated space partitioning and/or bias correction strategies [10,9].In this paper we give an overview of methods which are able to compute distances between distributions without the need for intermediate density estimation. Moreover, these techniques allow algorithm designers to specify which properties of a distribution are most relevant to their problems. We are optimistic that our embedding approach to distribution representation and analysis will lead to the development of algorithms which are simpler and more effective than entropy-based methods in a broad range of applications.We begin our presentation in Section 1 with an overview of reproducing kernel Hilbert spaces (RKHSs), and a description of how probability distributions can be represented as elements in an RKHS. In Section 2, we show how these representations may be used to address a variety of problems, including homogeneity testing (Section 2.1), covariate shift correction (Section 2.2), independence measurement (Section 2.3), feature extraction (Section 2.4), and density estimation (Section 2.5).

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On the Bootstrap of $U$ and $V$ Statistics

Cited by 140 publications

References 10 publications

Cramér-Von Mises Statistic for Repeated Measures

Cramér-Von Mises Statistic for Repeated Measures

International market links and volatility transmission

A Hilbert Space Embedding for Distributions

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