a research paper with supplementary materials - software, notebooks
by Martina Hančová, Andrej Gajdoš, Jozef Hanč
martina.hancova@upjs.sk
At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with the different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction, and linear mixed models.
The results of our numerical study, with emphasis on using open data science tools, demonstrate that our open tool implemented in high-performance Python(with Numba)
is exponentially fast, highly accurate, and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our tool outperformed the speed of the analytical
computation based on Tricomi's
The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology as well as in financial mathematics and risk analysis.
This is a pre-print of an article published in Journal of Statistical Computation and Simulation. The final revised published version is available at: https://www.tandfonline.com/doi/full/10.1080/00949655.2021.2023873.
A pre-print, the first version is available at https://arxiv.org/abs/2105.04427.
The notebooks folders contain our open codes, Jupyter and Mathematica notebooks from the entire numerical study which are detailed records of our computing with explaining narratives ilustrating explored concepts and methods.
Notebooks can be studied and viewed statically in Jupyter nbviewer 
with full visualisation. If there is a need, they can be also viewed directly on Github also as a raw code.
For interactive executing Jupyter notebooks as live documents without any need to install or compile the software use CoCalc providing interactive computing with our Jupyter notebooks.
All source code is distributed under the MIT license.
This work was supported by the Slovak Research and Development Agency under the contract no. APVV-17-0568 and the Internal Research Grant System of Faculty of Science, P. J. Šafárik University in Košice - project vvgs-pf-2020-1423.
The developed algorithms and open science digital tools were embedded and became a starting point for computational, simulation and theoretical reseach in our current projects supported by the Slovak Research and Development Agency under the Contract no. APVV-21-0216 and APVV-21-0369.