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. Author manuscript; available in PMC: 2020 Sep 1.
Published in final edited form as: J Multivar Anal. 2019 Feb 19;173:145–164. doi: 10.1016/j.jmva.2019.02.007

Table 1:

Simulation results for GSP-based and SP-based methods for estimating the population eigenvalues, cosine of the angles between sample and population eigenvectors, correlations between sample and population PC scores, and the asymptotic shrinkage factors. Each cell has empirical bias (%) with coefficients of variations (%) in parentheses.

Settings Method Eigenvalue Angle Correlation Shrinkage
No. 1 2 1 2 1 2 1 2
1 n = 500
p = 5000
σ2 = 4
ρ = 0.8		 SP 5.27
(2.37)		 18.27
(3.11)		 6.52
(0.32)		 34.07
(0.60)		 3.83
(0.03)		 23.33
(0.08)		 5.32
(0.60)		 17.88
(1.06)
λ-GSP 0.43
(2.67)		 0.95
(5.27)		 0.53
(0.77)		 3.28
(6.26)		 0.33
(0.31)		 2.79
(4.69)		 0.47
(0.92)		 0.58
(3.31)
d-GSP 0.47
(2.67)		 0.69
(5.45)		 0.47
(0.77)		 2.48
(6.70)		 0.24
(0.31)		 2.11
(5.07)		 0.51
(0.92)		 0.31
(3.51)
2 n = 500
p = 5000
σ2 = 1
ρ = 0.7		 SP 0.10
(0.90)		 0.46
(1.27)		 0.16
(0.04)		 0.44
(0.08)		 0.08
(0.001)		 0.24
(0.003)		 0.18
(0.08)		 0.39
(0.16)
λ-GSP −0.04
(0.90)		 0.04
(1.28)		 0.01
(0.04)		 0.004
(0.10)		 0.01
(0.03)		 0.01
(0.01)		 0.03
(0.08)		 −0.03
(0.18)
d-GSP −0.004
(0.90)		 0.10
(1.28)		 0.03
(0.04)		 0.03
(0.10)		 0.004
(0.03)		 0.01
(0.01)		 0.07
(0.08)		 0.03
(0.18)
3 n = 500
p = 5000
σ2 = 7.5
ρ = 0.8		 SP 25.68
(2.54)		 — 64.06
(0.52)		 — 46.50
(0.07)		 — 26.41
(0.90)		 —
λ-GSP 2.92
(5.7)		 — 12.62
(11.90)		 — 10.95
(10.13)		 — 3.47
(4.20)		 —
d-GSP 2.45
(5.74)		 — 12.25
(10.52)		 — 10.87
(8.58)		 — 3.00
(4.24)		 —
4 n = 500
p = 5000
σ2 = 4
ρ = 0		 SP 0.05
(1.58)		 −0.26
(2.35)		 0.06
(0.23)		 −0.06
(0.53)		 0.03
(0.02)		 0.05
(0.08)		 0.07
(0.43)		 −0.22
(0.90)
λ-GSP 0.03
(1.58)		 −0.35
(2.35)		 0.02
(0.24)		 −0.18
(0.54)		 0.01
(0.02)		 −0.02
(0.09)		 0.04
(0.43)		 −0.31
(0.91)
d-GSP 0.16
(1.58)		 −0.12
(2.35)		 0.10
(0.23)		 −0.03
(0.53)		 0.01
(0.02)		 0.02
(0.09)		 0.18
(0.42)		 −0.08
(0.90)