An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

Frank Oertel

doi:10.1515/demo-2015-0008

An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

Dependence Modeling ◽

10.1515/demo-2015-0008 ◽

2015 ◽

Vol 3 (1) ◽

Author(s):

Frank Oertel

Keyword(s):

Distribution Functions ◽

Left Invertibility ◽

Quantile Functions

AbstractWe revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

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QUANTILE-BASED STUDY OF (DYNAMIC) INACCURACY MEASURES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964819000019 ◽

2019 ◽

Vol 34 (2) ◽

pp. 183-199

Author(s):

Suchandan Kayal ◽

Rajesh Moharana ◽

S. M. Sunoj

Keyword(s):

Distribution Function ◽

Explicit Form ◽

Statistical Models ◽

Random Variables ◽

Distribution Functions ◽

Function Approach ◽

Present Communication ◽

Alternative Approach ◽

Quantile Functions

AbstractIn the present communication, we introduce quantile-based (dynamic) inaccuracy measures and study their properties. Such measures provide an alternative approach to evaluate inaccuracy contained in the assumed statistical models. There are several models for which quantile functions are available in tractable form, though their distribution functions are not available in explicit form. In such cases, the traditional distribution function approach fails to compute inaccuracy between two random variables. Various examples are provided for illustration purpose. Some bounds are obtained. Effect of monotone transformations and characterizations are provided.

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Estimation and inference for distribution functions and quantile functions in treatment effect models

Journal of Econometrics ◽

10.1016/j.jeconom.2013.03.010 ◽

2014 ◽

Vol 178 ◽

pp. 383-397 ◽

Cited By ~ 28

Author(s):

Stephen G. Donald ◽

Yu-Chin Hsu

Keyword(s):

Treatment Effect ◽

Distribution Functions ◽

Quantile Functions ◽

Treatment Effect Models

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Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions

ISRN Probability and Statistics ◽

10.1155/2013/412958 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 7

Author(s):

Michael Short

Keyword(s):

Binomial Distribution ◽

Quantile Function ◽

Distribution Functions ◽

Cumulative Distribution ◽

Exponential Inequalities ◽

Normal Approximations ◽

Universal Inequalities ◽

Exact Evaluation ◽

Exponential Bounds ◽

Quantile Functions

The exact evaluation of the Poisson and Binomial cumulative distribution and inverse (quantile) functions may be too challenging or unnecessary for some applications, and simpler solutions (typically obtained by applying Normal approximations or exponential inequalities) may be desired in some situations. Although Normal distribution approximations are easy to apply and potentially very accurate, error signs are typically unknown; error signs are typically known for exponential inequalities at the expense of some pessimism. In this paper, recent work describing universal inequalities relating the Normal and Binomial distribution functions is extended to cover the Poisson distribution function; new quantile function inequalities are then obtained for both distributions. Exponential bounds—which improve upon the Chernoff-Hoeffding inequalities by a factor of at least two—are also obtained for both distributions.

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Computer processing of high-resolution images of periodic specimens

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100141834 ◽

1986 ◽

Vol 44 ◽

pp. 2-5

Author(s):

W. Chiu ◽

M.F. Schmid ◽

T.-W. Jeng

Keyword(s):

High Resolution ◽

Fourier Transforms ◽

Complex Structure ◽

Distribution Functions ◽

Multiple Image ◽

Cryo Electron Microscopy ◽

Structure Factors ◽

Unit Cells ◽

High Resolution Images ◽

Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

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