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Statistical Theory and Related Fields Volume 5, 2021 - Issue 1
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Articles

Efficient estimation of smoothing spline with exact shape constraints

Vincent Chana Department of Statistics, University of Wisconsin-Madison, Madison, WI, USAView further author information
,
Kam-Wah Tsuia Department of Statistics, University of Wisconsin-Madison, Madison, WI, USAView further author information
,
Yanran Weib Department of Statistics, Virginia Tech, Blacksburg, VA, USAORCID Iconhttps://orcid.org/0000-0002-6745-7326View further author information
ORCID Icon,
Zhiyang Zhangb Department of Statistics, Virginia Tech, Blacksburg, VA, USAView further author information
&
Xinwei Dengb Department of Statistics, Virginia Tech, Blacksburg, VA, USACorrespondence[email protected]
ORCID Iconhttps://orcid.org/0000-0002-1560-2405View further author information
ORCID Icon
Pages 55-69 | Received 14 Feb 2019, Accepted 24 Jan 2020, Published online: 07 Feb 2020

Figures & data

Figure 1. Comparison of the five true functions used in simulation studies.

Figure 1. Comparison of the five true functions used in simulation studies.

Table 1. Simulation studies comparing SCSS and other estimators under different functions and errors settings.Averaged MSPE (×100) and its standard error (×100) over 500 repetitions are reported. NA entries correspond to methods with no applicable settings.

Figure 2. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and convex constraint.

Figure 2. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and convex constraint.

Table 2. Simulation studies measuring the convergence of SCSS based on the integrated mean squared errors.

Figure 3. Comparison between unconstrained (SS) and monotone constrained (SCSS) smoothing spline for the Auto MPG Data. The response is mpg, modelled as a function of weight.

Figure 3. Comparison between unconstrained (SS) and monotone constrained (SCSS) smoothing spline for the Auto MPG Data. The response is mpg, modelled as a function of weight.

Figure 4. Comparison between unconstrained (SS) and monotone constrained (SCSS) smoothing spline for the Auto MPG Data. The response is mpg, modelled as a function of displacement.

Figure 4. Comparison between unconstrained (SS) and monotone constrained (SCSS) smoothing spline for the Auto MPG Data. The response is mpg, modelled as a function of displacement.

Figure A1. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under normal error.

Figure A1. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under normal error.

Figure A2. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under t error.

Figure A2. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under t error.

Figure A3. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under beta error.

Figure A3. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(20x2+x3)/3000 in Example 4 under beta error.

Figure A4. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error.

Figure A4. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error.

Figure A5. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under t error.

Figure A5. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under t error.

Figure A6. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under beta error.

Figure A6. The estimator percentiles (2.5% and 97.5%) of SCSS for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under beta error.

Figure A7. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and monotone constraint.

Figure A7. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and monotone constraint.

Figure A8. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and mixed constraint.

Figure A8. SCSS convergence for function f(x)=(ex/20−e−10/20)/(e10/20−e−10/20) in Example 5 under normal error and mixed constraint.

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