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

Abstract: For some ranges of its parameters and arguments, the series for Tweedie probability density functions are sometimes exceedingly difficult to sum numerically. Existing numerical implementations utilizing inversion techniques and properties of stable distributions can cope with these problems, but no single one is successful in all cases. In this work we investigate heuristically the nature of the problem, and show that it is not related to the order of summation of the terms. Using a variable involved in the analytical proof of convergence of the series, the critical parameter for numerical non-convergence (“alpha”) is identified, and an heuristic criterion is developed to avoid numerical non-convergence for a reasonably large sub-interval of the latter. With these practical rules, simple summation algorithms provide sufficiently robust results for the calculation of the density function and its definite integrals. These implementations need to utilize high-precision arithmetic, and are programmed in the Python programming language. A thorough comparison with existing R functions allows the identification of cases when the latter fail, and provide further guidance to their use.