arXiv 10 Oct 2021 · Econometrics · 1 citations (OpenAlex)
arXiv:2110.04787 · PDF · DOI · OpenAlex · Extracted main text
Beyond the new results mentioned hereafter, this article aims at familiarizing researchers working in applied fields -- such as physics or economics -- with notions or formulas that they use daily without always identifying all their theoretical features or potentialities. Various situations where the L1-norm distance E|X-Y| between real-valued random variables intervene are closely examined. The axiomatic surrounding this distance is also explored. We constantly try to build bridges between the concrete uses of E|X-Y| and the underlying probabilistic model. An alternative interpretation of this distance is also examined, as well as its relation to the Gini index (economics) and the Lukaszyk-Karmovsky distance (physics). The main contributions are the following: (a) We show that under independence, triangle inequality holds for the normalized form E|X-Y|/(E|X| + E|Y|). (b) In order to present a concrete advance, we determine the analytic form of E|X-Y| and of its normalized expression when X and Y are independent with Gaussian or uniform distribution. The resulting formulas generalize relevant tools already in use in areas such as physics and economics. (c) We propose with all the required rigor a brief one-dimensional introduction to the optimal transport problem, essentially for a L1 cost function. The chosen illustrations and examples should be of great help for newcomers to the field. New proofs and new results are proposed.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | M. Thorpe (2018) Introduction to optimal transport | 0.874 | 5 | 2 | 100% |
| 2 | W. Gangbo (2004) An introduction to the mass transport theory and its applications | 0.843 | 3 | 3 | 100% |
| 3 | S. Rachev, S. Stoyanov, and F. Fabozzi (2007) Advanced Stochastic Modes, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures | 0.843 | 3 | 3 | 100% |
| 4 | S. Lukaszyk (2004) A new concept of probability metric and its applications in approximation of scattered data sets | 0.811 | 4 | 2 | 100% |
| 5 | A. F. Karr (1993) Probability | 0.737 | 3 | 3 | 67% |
| 6 | S. Yitzhaki (1998) More than a dozen alternative ways of spelling Gini | 0.737 | 3 | 2 | 100% |
| 7 | S. Yitzhaki and E. Schechtman (2013) The Gini Methodology, A Primer on a Statistical Methodology | 0.737 | 3 | 2 | 100% |
| 8 | M. Deza and E. Deza (2014) Encyclopedia of Distances | 0.644 | 2 | 2 | 100% |
| 9 | L. Kantorovich (1942) On the transport of masses | 0.644 | 2 | 2 | 100% |
| 10 | S. Rachev and L. Rueschendorf (1998) Mass Transportation Problems: VolumeI. Theory Vol.1 | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 38 scored citations.