Table 1 Simulation designs

Uncorrelated Sparse Low-dimensional

10⁻²·I _p

10⁻²·I _q

\(10^{-2}\cdot \left [\begin {array}{ll} 0.9 & {\mathbf {0}}_{1 \times 3} \\ {\mathbf {0}}_{5 \times 1} & {\mathbf {0}}_{5 \times 3} \end {array}\right ]\)

n=100,p=6,q=4

Correlated Sparse Low-dimensional

\(10^{-2}\cdot \left [\begin {array}{ccc} 1 & 0.4 & \mathbf {0}\\ 0.4 & 1 & \mathbf {0}\\ 0 & 0 & \mathbf {I}_{4\times 4} \end {array}\right ]\)

\(10^{-2}\cdot \left [\begin {array}{ccc} 1 & 0.4 & \mathbf {0}\\ 0.4 & 1 & \mathbf {0}\\ 0 & 0 & \mathbf {I}_{2\times 2} \end {array}\right ]\)

\( 10^{-2}\cdot \left [\begin {array}{cc} 0.8 & {\mathbf {0}}_{1 \times 3} \\ {\mathbf {0}}_{5 \times 1} & {\mathbf {0}}_{5 \times 3} \end {array}\right ]\)

n=100,p=6,q=4

NonSparse Low-dimensional

10⁻²·I _p

10⁻²·I _q

10⁻²·0.1 _p×q

n=100,p=12,q=8

Sparse High-dimensional 1

10⁻¹·I _p

10⁻¹·I _q

\(10^{-1} \cdot \left [\begin {array}{cc}{\mathbf {0.45}}_{2 \times 2} & {\mathbf {0}}_{2 \times 2} \\ {\mathbf {0}}_{98 \times 2} & {\mathbf {0}}_{98 \times 2} \end {array}\right ]\)

n=100,p=100,q=4

Sparse High-dimensional 2

\(10^{-7} \cdot \left [\begin {array}{cc} \mathbf {S}_{10\times 10} & \mathbf {0} \\ \mathbf {0} & 10^{-3}\cdot \mathbf {I}_{90\times 90} \\ \end {array}\right ]\)

Σ _xx

\(10^{-7}\cdot \left [\begin {array}{cc}\mathbf {0.8}_{10\times 10} & {\mathbf {0}}_{10 \times 90} \\ {\mathbf {0}}_{90 \times 10} & {\mathbf {0}}_{90 \times 90} \end {array}\right ]\)

n=50,p=q=100

with \(\mathbf {S}_{ij}= \left \{\begin {array}{ll} 1 & if \ i=j\\ 0.8 & if \ i\neq j, \end {array}\right.\)

Sparse Ultra High-dimensional

\(10^{-7} \cdot \left [\begin {array}{cc} \mathbf {S}_{10\times 10} & \mathbf {0} \\ \mathbf {0} & 10^{-3}\cdot \mathbf {I}_{9990\times 9990} \\ \end {array}\right ]\)

Σ _xx

\(10^{-7}\cdot \left [\begin {array}{cc}\mathbf {0.8}_{10\times 10} & {\mathbf {0}}_{10 \times 9990} \\ {\mathbf {0}}_{9990 \times 10} & {\mathbf {0}}_{9990 \times 9990} \end {array}\right ]\)

n=100,p=q=10000

with \(\mathbf {S}_{ij}= \left \{\begin {array}{ll} 1 & if \ i=j\\ 0.8 & if \ i\neq j, \end {array}\right.\)