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8 results on '"Wasserstein metric"'

1. Wasserstein upper bounds of [formula omitted]-norms for multivariate densities in Besov spaces.

Author

Chae, Minwoo

Subjects
*BESOV spaces, *PROBABILITY measures, *DENSITY
Abstract

While the total variation between two probability measures cannot be bounded by Wasserstein metrics in general, it is possible if they possess smooth densities. More generally, we demonstrate that L p -distances between densities in Besov spaces can be bounded by powers of the Wasserstein metrics. [ABSTRACT FROM AUTHOR]

Published
2024
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2. On quantitative bounds in the mean martingale central limit theorem.

Author

Röllin, Adrian

Subjects
*MATHEMATICAL bounds, *CENTRAL limit theorem, *DISCRETE time filters, *MARTINGALES (Mathematics), *DISTRIBUTION (Economic theory)
Abstract

We provide explicit bounds on the Wasserstein distance between discrete time martingales and the standard normal distribution. The proofs are based on a combination of Lindeberg’s and Stein’s method. [ABSTRACT FROM AUTHOR]

Published
2018
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3. Minimax rate for optimal transport regression between distributions.

Author

Ghodrati, Laya and Panaretos, Victor M.

Subjects
*DISTRIBUTION (Probability theory), *RANDOM measures, *REGRESSION analysis, *AXIOMS, *QUANTILE regression
Abstract

Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower bound that matches the convergence rate attained by the Fréchet least squares estimator. [ABSTRACT FROM AUTHOR]

Published
2023
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4. Wasserstein upper bounds of the total variation for smooth densities

Author

Stephen G. Walker and Minwoo Chae

Subjects
Statistics and Probability, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Upper and lower bounds, Sobolev space, 010104 statistics & probability, Total variation, Variation (linguistics), Bounded function, Wasserstein metric, 0101 mathematics, Statistics, Probability and Uncertainty, Probability inequality, Mathematics, Probability measure
Abstract

The total variation distance between probability measures cannot be bounded by the Wasserstein metric in general. If we consider sufficiently smooth probability densities, however, it is possible to bound the total variation by a power of the Wasserstein distance. We provide a sharp upper bound which depends on the Sobolev norms of the densities involved.

Published
2020

5. On the Wasserstein distance for a martingale central limit theorem.

Author

Fan, Xiequan and Ma, Xiaohui

Subjects
*CENTRAL limit theorem, *MARTINGALES (Mathematics), *LIMIT theorems, *RANDOM variables
Abstract

We prove an upper bound on the Wasserstein distance between normalized martingales and the standard normal random variable, which extends a result of Röllin (2018). The proof is based on a method of Bolthausen (1982). [ABSTRACT FROM AUTHOR]

Published
2020
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6. Sanov’s theorem in the Wasserstein distance: A necessary and sufficient condition

Author

Ran Wang, Xinyu Wang, and Liming Wu

Subjects
Statistics and Probability, Combinatorics, Sequence, Weak convergence, Wasserstein metric, Mathematical analysis, Polish space, Statistics, Probability and Uncertainty, Rate function, Mathematics, Sanov's theorem
Abstract

Let ( X n ) n ≥ 1 be a sequence of i.i.d.r.v.’s with values in a Polish space ( E , d ) of law μ . Consider the empirical measures L n = 1 n ∑ k = 1 n δ X k , n ≥ 1 . Our purpose is to generalize Sanov’s theorem about the large deviation principle of L n from the weak convergence topology to the stronger Wasserstein metric W p . We show that L n satisfies the large deviation principle in the Wasserstein metric W p ( p ∈ [ 1 , + ∞ ) ) if and only if ∫ E e λ d p ( x 0 , x ) d μ ( x ) + ∞ for all λ > 0 , and for some x 0 ∈ E .

Published
2010

7. Wasserstein upper bounds of the total variation for smooth densities.

Author

Chae, Minwoo and Walker, Stephen G.

Subjects
*POWER (Social sciences), *DENSITY, *SOBOLEV spaces, *PROBABILITY theory
Abstract

The total variation distance between probability measures cannot be bounded by the Wasserstein metric in general. If we consider sufficiently smooth probability densities, however, it is possible to bound the total variation by a power of the Wasserstein distance. We provide a sharp upper bound which depends on the Sobolev norms of the densities involved. [ABSTRACT FROM AUTHOR]

Published
2020
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8. On Stein-Chen factors for Poisson approximation

Author

Aihua Xia and Tim Brown

Subjects
Statistics and Probability, Logarithm, Stochastic process, Mathematical analysis, Poisson distribution, Point process, symbols.namesake, Compound Poisson distribution, Wasserstein metric, Compound Poisson process, symbols, Statistics, Probability and Uncertainty, Fractional Poisson process, Mathematics
Abstract

The Stein-Chen method for the estimation of upper bounds of the distance of the whole distribution of a point process from that of a Poisson process was investigated in Barbour and Brown (1992). A feature of the Stein-Chen approximation for random variables is the existence of “magic” factors which decrease with the mean of the Poisson distribution. Using a Wasserstein metric for process approximation, one of these factors behaves in a similar way to that for random variables but the other involves an additional logarithm. Counterexamples are presented here to show that the logarithmic factor is necessary.

Published
1995

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