. Author manuscript; available in PMC: 2020 Apr 1.

Published in final edited form as: Stat Biosci. 2017 Dec 6;11(1):47–90. doi: 10.1007/s12561-017-9208-x

Algorithm 2.

VR procedure

Input: matrices Z, X, positive semi-definite matrix Q with rank r < p;

1: Let U be orthogonal matrix from (3.1) and define matrix of nonzero singular values, Σ := diag(s₁,…, s_r);

2: Denote by A last p−r columns of ZU and by ℤ first r columns of ZU;

3: Put 𝕏 := [X A] and consider minimization problem with the objective

{‖ y - X \tilde{β} - ℤ d ‖}_{2}^{2} + λ d^{⊤} \sum d

, for β̃ ∈ ℝ^m+p⁻^r and d ∈ ℝ^r ;

4: Estimate σ² and

σ_{d}^{2}

in equivalent LMM, i.e. the model y = 𝕏β̃ +ℤd +ε, where

d ~ N (0, σ_{d}^{2} \sum^{- 1})

and ε ~𝒩(0,σ²I_n), and define

λ : = {\hat{σ}}^{2} / {\hat{σ}}_{d}^{2}

;

5: Find estimates β̂^VR and b̂^VR by applying formula (2.4).