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Conditional Expectation as Quantile Derivative

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Dirk Tasche

Probability · Submitted 19 Apr 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile.

how it is stated · asserted 0.98 read the record →

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arXiv IDmath/0104190 (version 1)
Pith IDFR7BFEL5, Pith's permanent record of this paper and its review
CitationDirk Tasche. "Conditional Expectation as Quantile Derivative." arXiv:math/0104190 (2001). https://pith.science/paper/FR7BFEL5
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,922 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.44 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of the variables for the so-called expected shortfall (first or higher moments) of the combination. Finally, we study in some more detail the coherence properties of the expected shortfall in case it is defined as a first conditional moment. Key words: quantile; value-at-risk; quantile derivative; conditional expectation; expected shortfall; conditional value-at-risk; coherent risk measure.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. background 0.92 0.21 0.02 unclear 0.71
s1 We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. contribution 0.68 0.37 0.04 asserted 0.70
s2 It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. result 1.00 0.81 0.83 asserted 0.98
s3 Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of the variables for the so-called expected shortfall (first or higher moments) of the combination. result 0.67 0.94 0.15 asserted 1.00
s4 Finally, we study in some more detail the coherence properties of the expected shortfall in case it is defined as a first conditional moment. contribution 0.98 0.90 0.07 asserted 0.77
s5 Key words: quantile; value-at-risk; quantile derivative; conditional expectation; expected shortfall; conditional value-at-risk; coherent risk measure. background 0.83 0.05 0.02 unclear 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
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  • P abstract
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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of the variables for the so-called expected shortfall (first or higher moments) of the combination. Finally, we study in some more detail the coherence properties of the expected shortfall in case it is defined as a first conditional moment. Key words: quantile; value-at-risk; quantile derivative; conditional expectation; expected shortfall; conditional value-at-risk; coherent risk measure.

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