New Results for Pearson Type III Family of Distributions and Application in Wireless Power Transfer

Distribution‐Based Model Evaluation and Diagnostics: Elicitability, Propriety, and Scoring Rules for Hydrograph Functionals

“…Suppose we use the Pearson type III distribution $$P={\mathcal{P}}_{\text{III}}(\mu ,{\sigma }^{2},\rho )$$ for log‐transformed annual maxima discharges. If we reparameterize the location μ , shape ρ and scale σ 2 of $${\mathcal{P}}_{\text{III}}(\mu ,{\sigma }^{2},\rho )$$ to ξ = μ − 2 σ / ρ , a = 4/ ρ 2 and $$b=\frac{1}{2}\sigma \vert \rho \vert $$, respectively, then the PDF of P simplifies to (Hosking & Wallis, 1997; Tegos et al., 2022) $$${f}_{P}(x,\xi ,a,b)=\frac{\vert x-\xi {\vert }^{a-1}}{{b}^{a}{\Gamma }(a)}\mathrm{exp}(-{b}^{-1}\vert x-\xi \vert ),$$$ where $$x,\xi ,a,b\in \mathbb{R}$$, a > 0 and b > 0. If ρ > 0 then x ∈ ( ξ , ∞ ), otherwise for ρ < 0 we yield x ∈ (− ∞ , ξ ).…”

“…Suppose that the distribution forecast P follows a univariate Pearson type III distribution P III (μ,σ 2 ,ρ) with mean μ, variance σ 2 and skewness ρ. If we reparametrize the PIII distribution and define ξ = μ − 2σ/ρ, a = 4/ρ 2 and b = 1 2 σ|ρ| as location, shape and scale parameters, respectively, then the CDF of P = P III (ξ, a, b) simplifies to (Tegos et al, 2022) Water Resources Research…”

“…Suppose we use the Pearson type III distribution P = P III (μ,σ 2 ,ρ) for logtransformed annual maxima discharges. If we reparameterize the location μ, shape ρ and scale σ 2 of P III (μ,σ 2 ,ρ) to ξ = μ − 2σ/ρ, a = 4/ρ 2 and b = 1 2 σ|ρ|, respectively, then the PDF of P simplifies to (Hosking & Wallis, 1997;Tegos et al, 2022)…”