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Web-based Supplementary Materials for “Bias-Corrected
Diagonal Discriminant Rules for High-dimensional Data
Classification”
Song Huang
Program of Computational Biology and Bioinformatics,
Yale University, New Haven, Connecticut, U.S.A.
Tiejun Tong
Department of Applied Mathematics,
University of Colorado, Boulder, Colorado, U.S.A.
Hongyu Zhao
Department of Epidemiology and Public Health,
Department of Genetics,
Yale University, New Haven, Connecticut, U.S.A.
hongyu.zhao@yale.edu
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Web Appendix A: E(Lˆ
k1), E(Lˆ
k2) and E(Lˆ
k)
Note that ˆμki ∼ N(μki, σ2
ki/nk), ˆσ
2
ki ∼ σ
2
kiχ
2
nk−1
/(nk − 1), and ˆμki and ˆσ
2
ki are independent
of each other. We have
E(Lˆ
k1) = X
p
i=1
E
3(yi − μˆki)
2
σˆ
2
ki
́
=
X
p
i=1
E(yi − μki + μki − μˆki)
2E
1
σˆ
2
ki
=
X
p
i=1
3
(yi − μki)
2 +
σ
2
ki
nk
́ nk − 1
(nk − 3)σ
2
ki
=
nk − 1
nk − 3
Lk1 +
(nk − 1)p
nk(nk − 3).
For Lˆ
k2, note that E(ln χ
2
ν
) = Ψ(ν/2) + ln(2) (Tong and Wang, 2007), we have
E(Lˆ
k2) = X
p
i=1
E(ln ˆσ
2
ki)
=
X
p
i=1
E
3
ln((nk − 1)ˆσ
2
ki
σ
2
ki
) + ln( σ
2
ki
nk − 1
)
́
=
X
p
i=1
3
Ψ(nk − 1
2
) + ln(2) + ln( σ
2
ki
nk − 1
)
́
= Lk2 + p
3
Ψ(nk − 1
2
) − ln(nk − 1
2
)
́
.
For Lˆ
k, note that ˆσ
2
i ∼ σ
2
i χ
2
n−K/(n − K), we have
E(Lˆ
k) = X
p
i=1
E
3(yi − μˆki)
2
σˆ
2
i
́
=
X
p
i=1
E(yi − μki + μki − μˆki)
2E
1
σˆ
2
i
=
X
p
i=1
3
(yi − μki)
2 +
σ
2
ki
nk
́ n − K
(n − K − 2)σ
2
i
=
n − K
n − K − 2
Lk1 +
(n − K)p
(n − K − 2)nk
.
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Web Figure 1: Comparison of methods with Σ1 6= Σ2 and ρ =
0.3 (sample version)
n1
log(Bias
2
)
4 8 12 16 20 24 28 32 36 40
2 4 6 8 10 12
BQDA
DQDA
BLDA
DLDA
n1
log(MSE)
4 8 12 16 20 24 28 32 36 40
6 7 8 9 10 11 12
n1
CWA (%)
4 8 12 16 20 24 28 32 36 40
50 60 70 80 90
p
log(Bias
2
)
10 200 400 600 800 1000
0 2 4 6 8
p
log(MSE)
10 200 400 600 800 1000
4 5 6 7 8 9
p
CWA (%)
10 200 400 600 800 1000
50 60 70 80 90
Figure 1: Comparison of bias-corrected discriminant scores with the original ones (sample
version) when Σ1 6= Σ2 and ρ = 0.3. Left column: p = 100. Right column: n1 = 20 and
n2 = 100.
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