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# Estimation of normal distribution parameters and its application to carbonation depth of concrete girder bridges

• * Corresponding author: Yuan Li
• Taking carbonation depth uncertainty into account is key to approach durability analysis of concrete girder bridges in a probabilistic way. The Normal distribution has been widely used to represent the probability distribution of carbonation depth. In this study, two new methods such as Least Squares method and Bayesian Quantile method, are used to estimate the parameters of the Normal distribution. These two considered methods are also compared with the commonly used Maximum Likelihood method via an extensive numerical simulation and three real carbonation depth data examples based on performance measures such as, K-S test, RMSE and ${\text{R}}^{2}$. The numerical study reveals that the Least Squares method is the best one for estimating the parameters of the Normal distribution. Statistical analysis of real carbonation depth data sets are presented to demonstrate the applicability and the conclusion of the simulation results.

Mathematics Subject Classification: 37A50.

 Citation: • • Table 1.  Comparison of the estimation methods

 Maximum likelihood method Bayesian Quantile method Least Squares method $n$ Parameter $\mu$ $\sigma$ $\mu$ $\sigma$ $\mu$ $\sigma$ 10 mean 0.12214 1.15672 0.11672 1.16318 0.09491 1.08113 RMSE 0.26513 0.35772 0.25617 0.36147 0.19817 0.27136 KS 0.35337 0.32109 0.24578 R$^{2}$ 0.83298 0.84576 0.88978 20 mean 0.07571 1.10291 0.06984 1.08983 0.05116 1.05886 RMSE 0.18364 0.24536 0.19225 0.22139 0.14281 0.18775 KS 0.26355 0.28776 0.19771 R$^{2}$ 0.90137 0.89516 0.92335 30 mean 0.05319 1.06572 0.05187 1.07102 0.04785 1.04213 RMSE 0.15361 0.21369 0.14793 0.20398 0.11251 0.15720 KS 0.15367 0.13476 0.09877 R$^{2}$ 0.95226 0.96237 0.97562 50 mean 0.04367 1.05318 0.04412 1.05277 0.03918 1.03889 RMSE 0.11623 0.15617 0.10987 0.15726 0.08273 0.12918 KS 0.12981 0.13287 0.08726 R$^{2}$ 0.96314 0.96512 0.98715 100 mean 0.03647 1.04891 0.03265 1.04912 0.02797 1.03276 RMSE 0.07629 0.13912 0.07292 0.14021 0.05172 0.09885 KS 0.08398 0.08203 0.06512 R$^{2}$ 0.97651 0.97261 0.99143 200 mean 0.02674 1.03628 0.02556 1.03719 0.02102 1.01493 RMSE 0.05728 0.07635 0.05276 0.07682 0.04729 0.05112 KS 0.06729 0.07102 0.05112 R$^{2}$ 0.98112 0.98372 0.99557 300 mean 0.01839 1.02987 0.01821 1.02898 0.01315 1.01011 RMSE 0.03672 0.05729 0.03629 0.05827 0.02791 0.03174 KS 0.05237 0.05311 0.04986 R$^{2}$ 0.99108 0.99203 0.99778 500 mean 0.00587 1.00532 0.00526 1.00516 0.00338 1.00201 RMSE 0.02392 0.03738 0.02371 0.03276 0.01818 0.01679 KS 0.03129 0.03063 0.02701 R$^{2}$ 0.99536 0.99277 0.99913 1000 mean 0.00161 1.00114 0.00108 1.00112 0.00036 1.00008 RMSE 0.01307 0.02119 0.01298 0.02101 0.00737 0.00082 KS 0.01112 0.01134 0.00601 R$^{2}$ 0.99821 0.99903 0.99996

Table 2.  Parameter estimates, RMSE, KS and R$^{2}$ for the first data set

 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 14.7500 1.2923 0.2677 0.1912 0.8826 Bayesian Quantile method 14.6534 1.4505 0.2301 0.2171 0.8755 Least Squares method 14.5703 1.2197 0.1329 0.1162 0.9283

Table 3.  Parameter estimates, RMSE, KS and R$^{2}$ for the second data set

 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 24.5556 9.5808 1.0122 0.1175 0.9218 Bayesian Quantile method 24.6528 10.3198 0.9526 0.1013 0.9427 Least Squares method 23.5642 10.6848 0.7128 0.0816 0.9577

Table 4.  Parameter estimates, RMSE, KS and R$^{2}$ for the third data set

 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 2.9852 0.5702 0.0441 0.0966 0.9761 Bayesian Quantile method 3.0127 0.5985 0.0412 0.0843 0.9788 Least Squares method 2.9697 0.6770 0.0391 0.0498 0.9916
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