The QFlex Distribution

Quantile-parameterized distributions (QPDs) are widely used in decision analysis because expert judgments are naturally expressed in terms of quantiles. Existing QPD families such as the Metalog offer considerable flexibility but lack analytic monotonicity guarantees, requiring ex post numerical repair. This paper introduces the QFlex distribution, a new QPD system constructed entirely from monotone transformations of valid quantile functions. QFlex interleaves powers of exponential, reflected-exponential, and centered-uniform quantile bases, yielding a flexible expansion that remains monotone under simple coefficient conditions. We show that QFlex has a generically full-rank design matrix with universal full rank through order K = 6, can interpolate any finite set of quantile assessments, and converges to the target quantile function as the number of terms increases. When all coefficients are nonnegative, QFlex is strictly increasing and unimodal; additional modes arise only in a structured and controllable manner. A comprehensive comparison across approximately 3,500 Pearson-system distributions demonstrates that QFlex generally matches or exceeds the accuracy of the Metalog distribution at moderate orders while offering explicit analytic monotonicity conditions. A synthetic example illustrates QFlex’s practical advantage: valid fits can be enforced directly through simple coefficient constraints, avoiding the complex post-fit monotonicity repair required by the Metalog distribution.