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# Prediction intervals of loan rate for mortgage data based on bootstrapping technique: A comparative study

• *Corresponding author: Donglin Wang
• The prediction interval is an important guide for financial organizations to make decisions for pricing loan rates. In this paper, we considered four models with bootstrap technique to calculate prediction intervals. Two datasets are used for the study and $5$-fold cross validation is used to estimate performance. The classical regression and Huber regression models have similar performance, both of them have narrow intervals. Although the RANSAC model has a slightly higher coverage rate, it has the widest interval. When the coverage rates are similar, the model with a narrower interval is recommended. Therefore, the classical and Huber regression models with bootstrap method are recommended to calculate the prediction interval.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  Histogram of the noterate

Figure 2.  Box-plot for Different $R$ and Models for 1st Data

Figure 3.  Box-plot for Different $R$ and Models for 2nd Data

Table 1.  Default parameters for robust models

 Models Parameters Huber epsilon=1.35 max_iter=100 alpha=0.0001 tol=0.00001 RANSAC max_trials=100 stop_probability=0.99 loss='absolute_error' Theil Sen max_subpopulation=10000 max_iter=300 tol=0.001

Table 3.  Running time for the two datasets

 R values Model Running time (seconds) 1st Dataset 2nd Dataset R=3000 Classical 103 104 Huber 1176 1213 RANSAC 4027 4069 Theil Sen 2482 2720 R=5000 Classical 170 172 Huber 1981 2009 RANSAC 6715 6779 Theil Sen 4187 4587 R=7000 Classical 238 238 Huber 2737 2808 RANSAC 9491 9480 Theil Sen 5949 6507 R=9000 Classical 304 322 Huber 3517 3668 RANSAC 12287 12289 Theil Sen 7824 8549

Table 2.  Coverage rate for the two datasets

 R values Model Coverage rate 1st Dataset 2nd Dataset R=3000 Classical 95.18% 94.88% Huber 95.05% 94.93% RANSAC 97.89% 97.91% Theil Sen 95.05% 94.93% R=5000 Classical 95.14% 94.84% Huber 94.91% 95.01% RANSAC 97.89% 97.99% Theil Sen 95.23% 94.97% R=7000 Classical 95.14% 94.93% Huber 94.95% 94.84% RANSAC 97.84% 97.91% Theil Sen 94.86% 94.97% R=9000 Classical 95.09% 94.93% Huber 95.09% 94.84% RANSAC 97.94% 98.08% Theil Sen 95.18% 94.88%

Table 4.  Tukey test of mean of the widths for the first dataset

 R value Model Difference of Mean P-value $95\%$ Confidence Interval R=3000 Classical vs Huber -0.0103 0.3012 (-0.0255, 0.0049) RANSAC vs Huber 0.6364 0.0010 (0.6212, 0.6516) Theil Sen vs Huber 0.0862 0.0010 (0.0710, 0.1014) RANSAC vs Classical 0.6467 0.0010 (0.6315, 0.6619) Theil Sen vs Classical 0.0966 0.0010 (0.0813, 0.1118) Theil Sen vs RANSAC -0.5501 0.0010 (-0.5653, -0.5349) R=5000 Classical vs Huber -0.0099 0.3076 (-0.0246, 0.0048) RANSAC vs Huber 0.6341 0.0010 (0.6194, 0.6488) Theil Sen vs Huber 0.0871 0.0010 (0.0724, 0.1018) RANSAC vs Classical 0.6440 0.0010 (0.6293, 0.6587) Theil Sen vs Classical 0.0970 0.0010 (0.0823, 0.1117) Theil Sen vs RANSAC -0.5470 0.0010 (-0.5617, -0.5323) R=7000 Classical vs Huber -0.0134 0.0866 (-0.0280, 0.0012) RANSAC vs Huber 0.6348 0.0010 (0.6201, 0.6494) Theil Sen vs Huber 0.0833 0.0010 (0.0687, 0.0979) RANSAC vs Classical 0.6482 0.0010 (0.6335, 0.6628) Theil Sen vs Classical 0.0967 0.0010 (0.0821, 0.1113) Theil Sen vs RANSAC -0.5515 0.0010 (-0.5661, -0.5368) R=9000 Classical vs Huber -0.0151 0.0365 (-0.0295, -0.0007) RANSAC vs Huber 0.6352 0.0010 (0.6207, 0.6496) Theil Sen vs Huber 0.0843 0.0010 (0.0699, 0.0987) RANSAC vs Classical 0.6503 0.0010 (0.6358, 0.6647 Theil Sen vs Classical 0.0994 0.0010 (0.0850, 0.1138) Theil Sen vs RANSAC -0.5509 0.0010 (-0.5653, -0.5364)

Table 5.  Tukey test of mean of the widths for the second dataset

 R value Model Difference of Mean P-value $95\%$ Confidence Interval R=3000 Classical vs Huber -0.0062 0.6138 (-0.0195, 0.0071) RANSAC vs Huber 0.5572 0.0010 (0.5439, 0.5704) Theil Sen vs Huber 0.0485 0.0010 (0.0353, 0.0618) RANSAC vs Classical 0.5633 0.0010 (0.5501, 0.5766) Theil Sen vs Classical 0.0547 0.0010 (0.0414, 0.0680) Theil Sen vs RANSAC -0.5086 0.0010 (-0.5219, -0.4954) R=5000 Classical vs Huber -0.0002 0.9000 (-0.0131, 0.0127) RANSAC vs Huber 0.5600 0.0010 (0.5471, 0.5729) Theil Sen vs Huber 0.0484 0.0010 (0.0355, 0.0613) RANSAC vs Classical 0.5602 0.0010 (0.5473, 0.5731) Theil Sen vs Classical 0.0487 0.0010 (0.0357, 0.0616) Theil Sen vs RANSAC -0.5116 0.0010 (-0.5245, -0.4986) R=7000 Classical vs Huber -0.0017 0.9000 (-0.0144, 0.0111) RANSAC vs Huber 0.5609 0.0010 (0.5482, 0.5736) Theil Sen vs Huber 0.0466 0.0010 (0.0338, 0.0593) RANSAC vs Classical 0.5625 0.0010 (0.5498, 0.5753) Theil Sen vs Classical 0.0482 0.0010 (0.0355, 0.0609) Theil Sen vs RANSAC -0.5143 0.0010 (-0.5270, -0.5016) R=9000 Classical vs Huber -0.0030 0.9000 (-0.0156, 0.0096) RANSAC vs Huber 0.5624 0.0010 (0.5498, 0.5750) Theil Sen vs Huber 0.0455 0.0010 (0.0329, 0.0581) RANSAC vs Classical 0.5654 0.0010 (0.5528, 0.5780 Theil Sen vs Classical 0.0485 0.0010 (0.0359, 0.0611) Theil Sen vs RANSAC -0.5169 0.0010 (-0.5295, -0.5043)
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