The Median Value of a Continuous Function

Bivens, Irl C.; Klein, Benjamin G.

doi:10.4169/math.mag.88.1.39

Cited by 3 publications

(1 citation statement)

References 3 publications

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“…its levels with respect to Lebesgue measure. More precisely, it is shown by Bivens and Klein [9] that…”

Section: Circular Optimal Transport: Alternative Representation and N...mentioning

confidence: 96%

The Statistics of Circular Optimal Transport

Hundrieser¹,

Klatt²,

Munk³

2021

Preprint

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Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh's test, the most powerful invariant test for von Mises alternatives. For regimes with many modes the circular OT test is less powerful which is explained by the shape of the corresponding transport plan.

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“…its levels with respect to Lebesgue measure. More precisely, it is shown by Bivens and Klein [9] that…”

Section: Circular Optimal Transport: Alternative Representation and N...mentioning

confidence: 96%

The Statistics of Circular Optimal Transport

Hundrieser¹,

Klatt²,

Munk³

2021

Preprint

View full text Add to dashboard Cite

show abstract

The Statistics of Circular Optimal Transport

Hundrieser

Klatt

Munk

2022

Forum for Interdisciplinary Mathematics

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Application of Optimized Fractional Grey Model-Based Variable Background Value to Predict Electricity Consumption

Liu

Lao

et al. 2021

Fractals

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In recent years, the grey-based models with fractional accumulation have received extensive attention by scholars and have been widely used in various fields. However, the existing rough construction of the background value in the fractional grey model impairs its predictive performance to some extent. To address this problem, this paper reconstructs a dynamic background value for the fractional grey model by the composite integral median theorem, as a result, a novel fractional grey model-based variable background value (denoted as OFAGM(1,1) for short) is proposed. In particular, the Particle Swarm Optimization algorithm (PSO) is then employed to determine the optimum for the fractional order and the background value coefficients. Based on electricity consumption of two regions of China (i.e. Beijing, Inner Mongolia), the superiority of the proposed model has been verified in comparison with other benchmark models, the electricity consumption of Beijing is predicted to reach [Formula: see text][Formula: see text]Kwh in 2022 and that of Inner Mongolia will reach [Formula: see text][Formula: see text]Kwh in 2023. This study also takes the relative growth rate and the doubling time into account for these two regions that facilities comparison of the regions’ performance.

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The Median Value of a Continuous Function

Cited by 3 publications

References 3 publications

The Statistics of Circular Optimal Transport

The Statistics of Circular Optimal Transport

The Statistics of Circular Optimal Transport

Application of Optimized Fractional Grey Model-Based Variable Background Value to Predict Electricity Consumption

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