Quantile-based power-series expansions of the Johnson distribution system

The Johnson system of distributions provides a flexible framework for modeling continuous random variables through monotonic transformations of the standard normal distribution. The system is typically presented in transformation form, which obscures the analytic structure of its quantile functions. This paper introduces a novel formulation: explicit power-series expansions for the Johnson SU, SL, and SB quantile functions, expressed in terms of a centered standard normal quantile function. These Johnson Power-Series Expansions (JPSE) provide a new analytic foundation for the Johnson system and facilitate structured quantile-based modeling. The SU and SB expansions consist of odd powers, but only SU has strictly positive coefficients, ensuring monotonicity for all truncation orders. In contrast, the SB expansion involves alternating-sign coefficients and may violate monotonicity under truncation. The SL expansion includes both even and odd powers with positive coefficients, requiring careful selection of truncation order to preserve validity. We derive convergence rates, remainder bounds, and sufficient conditions for monotonicity, and demonstrate through Q–Q and density comparisons that low-order truncations yield accurate approximations across the Johnson families. These results support the development of quantile-parameterized distribution systems for applications in expert elicitation and empirical quantile fitting.