Practical rules for summing the series of the Tweedie probability density function with high-precision arithmetic

Area of Science:

• Computational statistics
• Numerical analysis
• Probability theory

Background:

• Tweedie probability density functions (PDFs) present numerical summation challenges for certain parameter ranges.
• Existing numerical methods, including inversion techniques and stable distribution properties, have limitations and are not universally successful.
• The summation order of terms is not the cause of numerical non-convergence.

Purpose of the Study:

• To investigate the nature of numerical non-convergence in Tweedie PDF series summation.
• To identify the critical parameter responsible for numerical non-convergence.
• To develop practical criteria and algorithms for robust numerical computation of Tweedie PDFs and their integrals.

Main Methods:

• Heuristic investigation of the numerical summation problem for Tweedie PDFs.
• Identification of a critical parameter ('alpha') related to analytical convergence proofs.
• Development of an heuristic criterion to avoid numerical non-convergence.
• Implementation of simple summation algorithms using high-precision arithmetic in Python.

Main Results:

• A critical parameter ('alpha') for numerical non-convergence was identified.
• A practical heuristic criterion was developed to ensure numerical convergence for a significant range of parameters.
• Simple summation algorithms using high-precision arithmetic in Python provide robust results for PDF calculation and integration.
• Comparison with existing R functions highlights cases where they fail, offering guidance for their use.

Conclusions:

• The identified heuristic criterion and Python-based algorithms offer a robust solution for numerically calculating Tweedie probability density functions and their definite integrals.
• These methods overcome limitations of existing numerical implementations, particularly in challenging parameter ranges.
• The findings provide valuable guidance for researchers and practitioners working with Tweedie distributions.