Robust bandwidth selection in semiparametric partly linear regression models: Monte Carlo study and influential analysis

initial bandwidth0.25 0.30 0.35 0.38 0.40 0.43 0.45

LS 0.0385 0.0368 0.0371 0.0344 0.0359 0.037 0.0373 C0

LTS 0.0591 0.0598 0.0714 0.0623 0.058 0.062 0.0623GM 0.036 0.0364 0.0413 0.0372 0.0372 0.038 0.0379LS 0.0875 0.0891 0.1271 0.1225 0.1054 0.0973 0.0941 C1

LTS 0.0718 0.0699 0.0881 0.0735 0.073 0.0734 0.0672GM 0.0449 0.0439 0.0471 0.0483 0.0491 0.045 0.0415LS 1.2196 1.164 1.1079 1.0822 1.0454 1.0293 0.9688 C2

LTS 0.0699 0.0697 0.0659 0.0701 0.0691 0.0677 0.0714GM 0.0466 0.044 0.0452 0.0463 0.0455 0.0458 0.0463

Table 6: Estimation of the regression regression function g. Median of MedSE(g) whenusing the differentiating approach.

MSE(g) MedSE(g)LS GM LTS LS GM LTS

C0 0.099 0.1848 0.1005 0.0392 0.0599 0.0405C1 9.698 0.1963 0.1209 0.0863 0.0692 0.0468C2 18.5513 0.3477 0.1203 1.3116 0.0857 0.0493

Table 7: Estimation of the regression regression function g. Mean of MSE(g) and medianof MedSE(g) when using the cross-validation.

Classical selector Robust selector

-0.5

0.5

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-0.5

0.5

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-0.5

0.5

1.0

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-0.5

0.5

1.0

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-0.5

0.5

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-0.5

0.5

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

C2

C1

C0

Figure 1: Boxplots of log(h/hopt

)using the differentiating approach.

28

Classical selector Robust selector-0

.50.

51.

01.

5

0.25 0.30 0.35 0.38 0.40 0.43 0.45 -0.5

0.5

1.0

1.5

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-1.0

0.0

1.0

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-1.0

0.0

1.0

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-10

12

0.25 0.30 0.35 0.38 0.40 0.43 0.45

-10

12

0.25 0.30 0.35 0.38 0.40 0.43 0.45

C0

C1

C2

Figure 2: Boxplots of log(h/hopt

)using the robust local polynomials.

(a) (b) (c)

-0.5

0.0

0.5

1.0

C0 C1 C2

-0.5

0.0

0.5

1.0

C0 C1 C2

-1.5

-1.0

-0.5

0.0

0.5

1.0

C0 C1 C2

Figure 3: Boxplots of log(h/hopt

)for the robust data–driven bandwidths (a) and (b) plug–

in bandwidths with initial bandwidth 0.4 (a) using the differentiating approach (b) usingthe local polynomial method and (c) robust cross–validation bandwidths.

29

(a) (b)

t0.0 0.2 0.4 0.6 0.8 1.0

-8-6

-4-2

02

46

g(t)+β φ(t)

g(t)

t0.0 0.2 0.4 0.6 0.8 1.0

02

4

φ(t)

Figure 4: Generated Data Set. The dashed line corresponds to the nonparametric compo-nent g while the solid one to the regression function γ(t) = g(t) + βφ(t) in (a). In (b), thesolid line corresponds to φ(t).

30

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

050

100

150

200

250

t=0.10

Classical Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

010

020

030

040

050

060

0

t=0.10

Classical Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

050

100

150

200

250 t=0.10

Classical Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

010

020

030

040

050

060

070

0

t=0.10

Classical Plug-in Bandwidth

Figure 5: EIF (0.10, x, y) and EIF1(0.10, x, y) for the classical bandwidth selector, using the differentiating approach, ((a1)and (a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

31

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

67

89

1011

t=0.10

Robust Plug-in Bandwidth

t=0.10

Robust Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

1011

1213

1415

1617

t=0.10

Robust Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

56

78

910

1112

t=0.10

Robust Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

68

1012

1416

18

t=0.10

Robust Plug-in Bandwidth

Figure 6: EIF (0.10, x, y) and EIF1(0.10, x, y) for the robust bandwidth selector, using the differentiating approach, ((a1) and(a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

32

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

1020

3040

5060

70

t=0.50

Classical Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

020

4060

8010

012

014

0

t=0.50

Classical Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

010

2030

4050

60

t=0.50

Classical Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

020

4060

8010

012

0

t=0.50

Classical Plug-in Bandwidth: Local Polinomial

Figure 7: EIF (0.50, x, y) and EIF1(0.50, x, y) for the classical bandwidth selector, using the differentiating approach, ((a1)and (a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

33

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

56

78

910

1112

t=0.50

Robust Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

810

1214

1618

t=0.50

Robust Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

67

89

1011

t=0.50

Robust Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

910

1112

1314

1516

t=0.50

Robust Plug-in Bandwidth: Local Polinomial

Figure 8: EIF (0.50, x, y) and EIF1(0.50, x, y) for the robust bandwidth selector, using the differentiating approach, ((a1) and(a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

34

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

1020

3040

5060

7080

t=0.90

Classical Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

020

4060

8010

012

014

0

t=0.90

Classical Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

010

2030

4050

6070

t=0.90

Classical Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

020

4060

8010

012

014

0

t=0.90

Classical Plug-in Bandwidth: Local Polinomial

Figure 9: EIF (0.90, x, y) and EIF1(0.90, x, y) for the classical bandwidth selector, using the differentiating approach, ((a1)and (a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

35

(a1) (a2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

510

15 t=0.90

Robust Plug-in Bandwidth

-40

-20

0

20

40

x

-40

-20

0

20

40

y

05

1015

2025

30

t=0.90

Robust Plug-in Bandwidth

(b1) (b2)

-40

-20

0

20

40

x

-40

-20

0

20

40

y

56

78

910

11

t=0.90

Robust Plug-in Bandwidth: Local Polinomial

-40

-20

0

20

40

x

-40

-20

0

20

40

y

810

1214

16

t=0.90

Robust Plug-in Bandwidth: Local Polinomial

Figure 10: EIF (0.90, x, y) and EIF1(0.90, x, y) for the robust bandwidth selector, using the differentiating approach, ((a1)and (a2), respectively) and using the local polynomial approach, ((b1) and (b2), respectively).

36

k = 1 k = 3

t0.2 0.4 0.6 0.8

1012

1416

1820

t0.2 0.4 0.6 0.8

68

1014

18

k = 5 k = 7

t0.2 0.4 0.6 0.8

510

1520

25

t0.2 0.4 0.6 0.8

510

1520

25

k = 9 k = 11

t0.2 0.4 0.6 0.8

010

2030

40

t0.2 0.4 0.6 0.8

010

2030

4050

Figure 11: The solid lines correspond to EIF (t, 10, 10) while the dashed lines (− · −) withempty circles to EIF1(t, 10, 10) for the robust plug–in selector based on the differentiatingapproach.

37

k = 1 k = 3

t0.2 0.4 0.6 0.8

810

1214

t0.2 0.4 0.6 0.8

02

46

810

12

k = 5 k = 7

t0.2 0.4 0.6 0.8

010

2030

4050

60

t0.2 0.4 0.6 0.8

010

020

030

040

0

k = 9 k = 11

t0.2 0.4 0.6 0.8

010

2030

4050

60

t0.2 0.4 0.6 0.8

010

2030

4050

Figure 12: The solid lines correspond to EIF (t, 10, 10) while the dashed lines (− · −) withempty circles to EIF1(t, 10, 10) for the robust plug–in selector based on the the robust localpolynomial approach.

38

k = 1 k = 3

t0.2 0.4 0.6 0.8

6080

100

140

t0.2 0.4 0.6 0.8

050

100

150

200

k = 5 k = 7

t0.2 0.4 0.6 0.8

5010

015

020

025

0

t0.2 0.4 0.6 0.8

4060

8010

012

014

0

k = 9 k = 11

t0.2 0.4 0.6 0.8

4060

8010

012

014

0

t0.2 0.4 0.6 0.8

020

6010

014

0

Figure 13: The solid lines correspond to EIF (t, 10, 10) while the dashed lines (− · −) withempty circles to EIF1(t, 10, 10) for the robust cross–validation selector.

39