Replication package for "Where L_p-Quantiles Cross"

Description

Code, numerical outputs, figures and detailed proofs for the article "Where L_p-Quantiles Cross" by Nurten Ulusay and Kemal Güler (submitted to Statistics & Probability Letters). The article shows the following for log-concave laws: - The quantile and the L_p-quantile cross exactly once, at the level where the moments of order p−1 of the inactivity time and the residual life are equal. The laws whose quantiles are the L_p-quantiles are ordered in the log-odds-rate order and in the increasing convex or increasing concave order. - The crossing of two L_p-quantile curves never lies between their L_p-medians. A skewness coefficient about the L_p-median, not the moment skewness, decides on which side of the median the crossing lies. - The crossing level of the quantile and the L_p-quantile preserves van Zwet's convex transform order. The exponential law therefore gives sharp bounds; for the quantile and the expectile the crossing level lies between 0.2032 and 0.7968. The package (one zip file) contains: - LpCrossing_DetailedProofs.pdf: step-by-step proofs of every result in the article and further examples. - Python scripts that compute every number reported in the article, each with its output file (*_output.txt). - Scripts that redraw both figures. Running sh run_all.sh after pip install -r requirements.txt regenerates every output and figure. The run takes about one hour. The tested versions are Python 3.13 with numpy 2.5.3, scipy 1.18.1, matplotlib 3.11.2 and mpmath 1.3.0. README.md maps each statement in the article to the script that checks it.

Steps to reproduce

1. Install Python 3.10 or later, then run pip install -r requirements.txt. 2. Run sh run_all.sh in the package folder. It overwrites every *_output.txt file and regenerates fig1.pdf and fig_geom.pdf. 3. Compare the regenerated files with the supplied ones; they should be identical. To run a single check, use python3 <script>.py. README.md lists which script supports which statement.

Institutions

• Nevşehir Hacı Bektaş Veli University

  Nevşehir Province, Nevşehir
• Rauf Denktas University

  Nicosia, Nicosia

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