The model identification of beta distribution based on quantiles

An enhanced two‐quantile Wilks methodology for engineering uncertainty analysis

“…To the best of our knowledge, despite the time that passed, this is the first modification proposed of the two‐constraint Wilks method 14 resolving these ramifications. This is explained by the fact that it was only relatively recently proven that two quantile constraints uniquely determine the parameters of a beta distribution, 37 with existence of that solution proven in Reference 36. The algorithm to solve for those parameters, 36 and provided in Appendix B, is integral to the two‐quantile Wilks method algorithm.…”

“…The two‐quantile Wilks algorithm starts by setting $$$ \xi =0 $$$ and solves for the parameters of the random variable $$$ Z\sim Beta\left(d,e\right) $$$ in (27) that meets the quantile equations (27) exactly. Existence of a beta distribution satisfying a lower and upper quantile constraint was first proved in Reference 36, followed by Shih 37 proving the uniqueness of that beta distribution. An algorithm was developed in Reference 36 to solve for those beta parameters $$$ d $$$ and $$$ e $$$ up to a desired level of accuracy and is provided in Appendix B.…”

“…That algorithm was developed in Reference 36 while proving the existence of a beta distribution satisfying those two quantile constraints. The uniqueness of that beta distribution solution was proven in Reference 37. By choosing the quantiles $$$ b $$$ and $$$ c $$$ reasonably close, serving as bounds for the proportion $$$ P $$$, one ensures that the sampled tolerance limits that follow enclose a reasonably well‐defined proportion of the population, as suggested by Johnson and Leone 31…”

“…That is, a beta random variable with a lesser variance 9.347 × 10 −6 . Applying the procedure set-forth by (30)-(32) yields now the four suggested integer solutions (d * , e * ) ∈ {(990, 9), (990, 10), (991, 9), (991,10)}(37) with γ = Pr(b < P < c) ∈ {89.604%, 89.984%, 89.960%, 90.014%},…”