In this work, we investigate a specific continuous shrinkage prior. We assign an independent gamma-gamma prior on the local shrinkage parameter ψj:9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _j \sim {\mathrm{G}}(a,\delta _j),\quad \quad \delta _j \sim {\mathrm{G}}(b,1),$$\end{document}ψj~G(a,δj),δj~G(b,1),where G(α,β) denotes the gamma distribution with shape parameter α and scale parameter β. By using change of variables, it can be verified that placing a gamma-gamma prior on ψj is equivalent to placing a three-parameter beta (TPB) prior on the shrinkage factor τj33:10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _j \sim {\mathrm{TPB}}(a,b,\phi ),$$\end{document}τj~TPB(a,b,ϕ),where the TPB distribution has the following density function:11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x;a,b,\phi ) = \frac{{\Gamma (a + b)}}{{\Gamma (a)\Gamma (b)}}\phi ^bx^{b - 1}(1 - x)^{a - 1}\{ 1 + (\phi - 1)x\} ^{ - (a + b)},$$\end{document}f(x;a,b,ϕ)=Γ(a+b)Γ(a)Γ(b)ϕbxb-1(1-x)a-1{1+(ϕ-1)x}-(a+b),with 0 < x < 1, a > 0, b > 0 and ϕ > 0. When ϕ = 1, the TPB distribution becomes a standard Beta distribution. For a fixed value of ϕ, a controls the behavior of the TPB prior near
