Research Shonosuke Sugasawa Correction

Correction on the paper “Bayesian Estimators for Small Area Models Shrinking Both Means and Variances”

August 31, 2017

The first inequality in (12) is not correct since X^′H′₁^P′₂^P₂^H₁^X = 0. This document gives the corrected evaluation of the integral of

π(θ, γ|D) ∝^{(θ′Aθ)−(m−p−2)/2}

m

∏

i=1

γ^aiψ_i(θ_i− X_i, γ)^−(ni/2+ai),

where ψi(θi− Xi, γ) = (Xi− θi)²+ (ni− 1)S_i²+ 2biγ.

Define Ω = {θ|θ^′Aθ_{≤ 1} ⊂ R}^m. It holds that

∫

R^m×R+

π(θ, γ|D)dθdγ =

∫

Ω×R+

π(θ, γ|D)dθdγ +

∫

Ω^c×_R+

π(θ, γ|D)dθdγ.

The second term can be evaluated as

∫

Ω^c×_R+

π(θ, γ|D)dθdγ ≤ C

∫

Ω^c×_R+

m

∏

i=1

γ^aiψ_i(θ_i− X_i, γ)^−(ni/2+ai)dθdγ

≤ C

∫

R^m×R⁺ m

∏

i=1

γ^aiψ_i(θi− Xi, γ)^−(ni/2+ai)dθdγ

= C

∫ ∞ 0

m

∏

i=1

{^{∫ ∞}

−∞

γ^aiψ_i(θ_i− X_i, γ)^−(ni/2+ai)dθ_i^}dγ,

which corresponds to the last formula in (12), and it is finite. For evaluating the first term, we first note that there exists a (m − p) × m matrix H₁ such that A = H^′₁^H₁ and H₁^H′₁ = I_m−p since A is an idempotent matrix with rank(A) = m − p. By changing the variable as u₁ = H₁^θ and u_{2 = (um−p+1}, . . . , u_m)^′ with ui= θi, it follows that

∫

Ω×R⁺

π(θ, γ|D)dθdγ ≤ C^′

∫

u^′₁u_1≤1

(u^′₁^u₁)^{−(m−p−2)/2}du₁

∫ ∞ 0

m−p

∏

i=1

γ^ai{(n_i− 1)S_i²+ 2b_iγ}^−(ni/2+ai)

×

m

∏

i=m−p+1

{∫ ∞

−∞

γ^aiψ_i(u_i− X_i, γ)^−(ni/2+ai)du_i }

dγ.

Moreover, it holds that

∫

u^′₁u_1≤1

(u^′₁^u₁)^{−(m−p−2)/2}du₁ = C^′′

∫ 1 0

r^{−(m−p−2)}r^m−p−1dr <∞,

thereby the similar evaluation shows that^∫_Ω×R

+π(θ, γ|D)dθdγ is also finite.

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