# Confidence Interval Estimation for Precipitation Quantiles Based on Principle of Maximum Entropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Confidence Interval Estimation of the Quantile

#### 2.1. Estimation of Quantile

#### 2.2. Calculation of Confidence Interval

#### 2.2.1. Method of Moments (MOM)

#### 2.2.2. Maximum Likelihood (ML) Method

**I**[28]:

#### 2.2.3. Principle of Maximum Entropy (POME) Method

## 3. Asymptotic Variances of Quantile Estimators for Different Distributions

#### 3.1. Gamma Distribution

#### 3.1.1. Estimation of Asymptotic Variances by MOM and ML

#### 3.1.2. Estimation of Asymptotic Variances by POME

#### 3.2. Pearson Type 3 (P3) Distribution

#### 3.2.1. Estimation of Asymptotic Variances by MOM and ML

#### 3.2.2. Estimation of Asymptotic Variances by POME

#### 3.3. Extreme Value Type 1 (EV1) Distribution

#### 3.3.1. Estimation of asymptotic variance by MOM and ML

#### 3.3.2. Estimation of Asymptotic Variances by POME

## 4. Simulation Experiments

## 5. Application

## 6. Conclusions

- (1)
- The calculation formulas of the asymptotic variances and confidence intervals of quantiles for three distributions based on POME are given. The results of simulation experiments and the case study show that the POME method can provide an effective way for reducing the uncertainty of quantile estimators.
- (2)
- Results of the simulation experiments demonstrate that the POME method yields the smallest standard errors and the narrowest confidence intervals of quantile estimators compared with the results of MOM and ML. This may benefit from fewer sampling errors and approximation in derivation. Thus, the POME can give more accurate estimates. Furthermore, the standard errors and confidence interval widths of the quantiles increased with the return period T and decreased with the sample size.
- (3)
- Results of the case study indicate that when using different criteria for distribution selection, the results are coincidental, and the POME is the optimal method for parameter estimation. Furthermore, the POME can give more reliable precipitation quantiles since the standard errors and 95% confidence interval widths of precipitation quantiles obtained by POME are smaller than those obtained by the MOM and the ML methods.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Estimation of $\mathrm{var}\left(\overline{W}\right)$

#### Estimation of $\mathrm{cov}\left(\overline{X},\overline{W}\right)$

## References

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Case | ξ | α | β | γ | δ | Cv | Cs |
---|---|---|---|---|---|---|---|

I | 0 | 16 | 16 | 0.4 | 0.04 | 0.38 | 1.10 |

I | 15.4 | 308.8 | 10.25 | 38.5 | −0.30 | 0.36 | 0.48 |

III | 273.69 | 521.10 | 1.25 | 4.77 | −0.21 | 0.24 | 0.64 |

**Table 2.**Median of estimated quantiles (${\widehat{x}}_{T}$), standard error (St), and 95% confidence interval (CI) width from generated data; MOM = methods of moments, ML = maximum likelihood, POME = principle of maximum entropy.

Case | Sample Size n | Return Period T | MOM | ML | POME | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathit{x}}}_{\mathit{T}}$ | St | CI Width | ${\widehat{\mathit{x}}}_{\mathit{T}}$ | St | CI Width | ${\widehat{\mathit{x}}}_{\mathit{T}}$ | St | CI Width | |||

I | 20 | 10 | 1.97 | 0.22 | 0.85 | 2.08 | 0.22 | 0.85 | 2.02 | 0.20 | 0.80 |

100 | 2.83 | 0.41 | 1.60 | 3.06 | 0.38 | 1.49 | 2.97 | 0.35 | 1.37 | ||

200 | 3.09 | 0.47 | 1.83 | 3.36 | 0.43 | 1.69 | 3.26 | 0.40 | 1.55 | ||

50 | 10 | 2.00 | 0.14 | 0.56 | 2.13 | 0.15 | 0.58 | 2.06 | 0.13 | 0.52 | |

100 | 2.89 | 0.27 | 1.05 | 3.20 | 0.26 | 1.01 | 3.05 | 0.23 | 0.89 | ||

200 | 3.14 | 0.31 | 1.20 | 3.52 | 0.29 | 1.14 | 3.35 | 0.26 | 1.00 | ||

100 | 10 | 2.01 | 0.10 | 0.40 | 2.16 | 0.11 | 0.42 | 2.08 | 0.10 | 0.37 | |

100 | 2.93 | 0.19 | 0.76 | 3.24 | 0.19 | 0.73 | 3.09 | 0.16 | 0.64 | ||

200 | 3.20 | 0.22 | 0.87 | 3.56 | 0.21 | 0.82 | 3.39 | 0.18 | 0.72 | ||

1000 | 10 | 2.02 | 0.03 | 0.13 | 2.17 | 0.03 | 0.13 | 2.09 | 0.03 | 0.12 | |

100 | 2.95 | 0.06 | 0.24 | 3.27 | 0.06 | 0.23 | 3.12 | 0.05 | 0.20 | ||

200 | 3.23 | 0.07 | 0.28 | 3.60 | 0.07 | 0.27 | 3.42 | 0.06 | 0.23 | ||

II | 20 | 10 | 106.9 | 12.1 | 47.3 | 111.1 | 11.9 | 46.5 | 109.7 | 11.4 | 44.7 |

100 | 155.3 | 22.8 | 89.3 | 164.7 | 20.7 | 81.3 | 161.3 | 19.7 | 77.2 | ||

200 | 169.8 | 26.0 | 102.1 | 180.4 | 23.4 | 91.9 | 176.8 | 22.2 | 87.1 | ||

50 | 10 | 106.9 | 12.1 | 47.3 | 112.3 | 7.7 | 30.2 | 110.3 | 7.4 | 28.9 | |

100 | 155.3 | 22.8 | 89.3 | 167.8 | 13.5 | 52.8 | 163.7 | 12.8 | 50.0 | ||

200 | 169.8 | 26.0 | 102.1 | 184.4 | 15.2 | 59.7 | 179.4 | 14.4 | 56.5 | ||

100 | 10 | 107.1 | 5.5 | 21.5 | 113.0 | 5.5 | 21.6 | 110.8 | 5.3 | 20.6 | |

100 | 156.3 | 10.3 | 40.5 | 169.0 | 9.6 | 37.7 | 164.8 | 9.1 | 35.7 | ||

200 | 170.9 | 11.8 | 46.4 | 185.6 | 10.9 | 42.6 | 180.8 | 10.3 | 40.3 | ||

1000 | 10 | 107.5 | 1.7 | 6.8 | 113.5 | 1.8 | 6.9 | 111.2 | 1.7 | 6.6 | |

100 | 156.7 | 3.3 | 12.9 | 169.9 | 3.1 | 12.0 | 165.5 | 2.9 | 11.3 | ||

200 | 171.3 | 3.8 | 14.7 | 186.6 | 3.5 | 13.6 | 181.6 | 3.3 | 12.8 | ||

III | 20 | 10 | 676.4 | 58.1 | 227.6 | 685.7 | 57.2 | 224.1 | 698.5 | 57.2 | 224.4 |

100 | 905.4 | 109.5 | 429.3 | 944.9 | 99.9 | 391.8 | 958.1 | 99.0 | 388.0 | ||

200 | 974.0 | 125.2 | 490.9 | 1023.0 | 113.0 | 442.9 | 1036.1 | 111.6 | 437.6 | ||

50 | 10 | 675.0 | 36.7 | 143.9 | 700.5 | 37.8 | 148.4 | 700.2 | 36.8 | 144.3 | |

100 | 905.5 | 69.2 | 271.4 | 971.7 | 66.2 | 259.4 | 965.6 | 63.5 | 248.8 | ||

200 | 974.3 | 79.2 | 310.3 | 1053.7 | 74.8 | 293.3 | 1044.3 | 71.7 | 280.9 | ||

100 | 10 | 1065.5 | 92.4 | 362.0 | 1160.3 | 86.3 | 338.2 | 1148.5 | 82.5 | 323.5 | |

100 | 674.8 | 26.0 | 102.0 | 707.7 | 65.5 | 106.9 | 700.2 | 26.2 | 102.7 | ||

200 | 906.9 | 49.1 | 192.4 | 985.3 | 41.5 | 186.9 | 968.7 | 45.2 | 177.4 | ||

1000 | 10 | 975.7 | 56.1 | 220.0 | 1067.2 | 47.7 | 211.3 | 1048.4 | 51.1 | 200.2 | |

100 | 674.7 | 8.2 | 32.2 | 713.6 | 8.7 | 34.2 | 701.4 | 8.3 | 32.6 | ||

200 | 906.9 | 15.5 | 60.8 | 994.1 | 15.3 | 59.8 | 971.3 | 14.4 | 56.3 |

Station Name | Record Length (Year) | Mean (mm) | Standard Deviation | Coefficient of Variation | Skewness | First-Order Serial Correlation Coefficient |
---|---|---|---|---|---|---|

Changwu | 51 | 580.6 | 131.8177 | 0.2270 | 0.5070 | 3.2153 |

Lintong | 50 | 579.5 | 129.2014 | 0.2230 | 0.6299 | 3.7670 |

Meixian | 50 | 578.0 | 129.7214 | 0.2245 | 0.5828 | 3.4614 |

Tongguan | 52 | 605.5 | 143.4648 | 0.2369 | 0.5771 | 3.6438 |

**Table 4.**Parameter values of each distribution estimated by the three methods; P3 = Pearson type 3, EV1 = extreme value type 1.

Station Name | Method | Gamma | P3 | EV1 | ||||
---|---|---|---|---|---|---|---|---|

α | β | α | β | γ | α | u | ||

Changwu | MOM | 29.9282 | 19.3993 | 15.5623 | 33.4147 | 60.5782 | 102.7783 | 521.2619 |

ML | 29.0510 | 19.9851 | 16.0097 | 32.5878 | 58.8660 | 114.9989 | 518.3959 | |

POME | 29.3458 | 19.7893 | 11.1017 | 39.5620 | 141.3794 | 111.0142 | 516.5091 | |

Lintong | MOM | 28.8063 | 20.1169 | 10.0804 | 40.6938 | 169.2817 | 100.7383 | 521.3449 |

ML | 27.6873 | 20.9299 | 17.1488 | 30.6908 | 53.1797 | 113.0664 | 519.0634 | |

POME | 28.6849 | 20.2192 | 10.3139 | 40.2305 | 164.5577 | 108.5086 | 516.8608 | |

Meixian | MOM | 29.1161 | 19.8498 | 11.7762 | 37.8015 | 132.7913 | 101.1438 | 519.5689 |

ML | 28.0875 | 20.5768 | 17.7271 | 30.3383 | 40.1383 | 113.7362 | 517.1681 | |

POME | 28.7575 | 20.1089 | 10.8002 | 39.4726 | 151.6378 | 109.1477 | 514.9499 | |

Tongguan | MOM | 33.9927 | 17.8122 | 12.0090 | 41.3991 | 108.3228 | 111.8595 | 540.9202 |

ML | 32.9535 | 18.3740 | 15.0343 | 36.5741 | 55.6191 | 125.0433 | 537.9242 | |

POME | 33.7243 | 17.9653 | 10.3756 | 44.5388 | 143.3688 | 120.6552 | 535.8444 |

**Table 5.**Ordinary least square (OLS), Akaike information criterion (AIC) and quasi-optimal deterministic coefficient test (QD) values of three distributions calculated by MOM, ML, and POME.

Station Name | Method | Gamma | P3 | EV1 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

OLS | AIC | QD | OLS | AIC | QD | OLS | AIC | QD | ||

Changwu | MOM | 16.6696 | 292.9861 | 0.9837 | 16.3741 | 291.1613 | 0.9843 | 19.3927 | 308.4197 | 0.9779 |

ML | 17.5392 | 298.1729 | 0.9819 | 17.0194 | 295.1041 | 0.983 | 16.8854 | 294.2978 | 0.9833 | |

POME | 17.2128 | 296.2565 | 0.9826 | 16.0997 | 289.4375 | 0.9848 | 16.6415 | 292.8135 | 0.9837 | |

Lintong | MOM | 19.6949 | 304.0361 | 0.9763 | 18.516 | 297.8638 | 0.979 | 18.8765 | 299.792 | 0.9782 |

ML | 20.6502 | 308.7725 | 0.9739 | 20.1246 | 306.1944 | 0.9752 | 16.3975 | 285.7129 | 0.9836 | |

POME | 19.7466 | 304.2982 | 0.9762 | 18.5442 | 298.0155 | 0.979 | 16.0775 | 283.742 | 0.9842 | |

Meixian | MOM | 17.9613 | 294.8219 | 0.9804 | 16.9804 | 289.2061 | 0.9825 | 17.8405 | 294.1469 | 0.9807 |

ML | 18.9359 | 300.106 | 0.9783 | 18.5121 | 297.8426 | 0.9792 | 15.0757 | 277.3083 | 0.9862 | |

POME | 18.2446 | 296.3869 | 0.9798 | 16.8553 | 288.4664 | 0.9828 | 14.6984 | 274.7736 | 0.9869 | |

Tongguan | MOM | 20.419 | 319.7126 | 0.9793 | 20.0141 | 317.6294 | 0.9802 | 21.9677 | 327.3156 | 0.9761 |

ML | 21.2629 | 323.9242 | 0.9776 | 20.8105 | 321.6875 | 0.9785 | 19.5845 | 315.373 | 0.981 | |

POME | 20.5825 | 320.542 | 0.979 | 19.9542 | 317.3178 | 0.9803 | 19.2731 | 313.7059 | 0.9816 |

**Table 6.**Quantile estimators, standard error, and 95% confidence interval widths based on MOM, ML, and POME for the annual precipitation (mm).

Station Name | Best fitted Distribution | Return Period (Year) | MOM | ML | POME | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Quantile | Standard Error | Confidence Interval Width | Quantile | Standard Error | Confidence Interval width | Quantile | Standard Error | Confidence Interval Width | |||

Changwu | P3 | 10 | 755.0 | 31.2 | 122.1 | 753.1 | 30.5 | 119.6 | 755.7 | 31.7 | 124.3 |

20 | 814.7 | 40.7 | 159.7 | 811.9 | 38.9 | 152.3 | 817.5 | 38.4 | 150.7 | ||

50 | 885.6 | 56.1 | 220.0 | 881.9 | 51.7 | 202.5 | 891.6 | 47.3 | 185.6 | ||

100 | 935.3 | 69.1 | 270.7 | 930.8 | 62.2 | 243.8 | 943.8 | 54.0 | 211.6 | ||

200 | 982.3 | 82.8 | 324.5 | 977.1 | 73.3 | 287.2 | 993.5 | 60.6 | 237.5 | ||

500 | 1041.4 | 101.8 | 398.9 | 1035.3 | 88.5 | 347.0 | 1056.3 | 69.2 | 271.2 | ||

Lintong | EV1 | 10 | 704.4 | 31.8 | 124.7 | 724.5 | 32.2 | 126.1 | 714.0 | 30.6 | 120.0 |

20 | 748.0 | 37.5 | 146.9 | 773.5 | 37.0 | 144.9 | 761.0 | 35.0 | 137.1 | ||

50 | 820.6 | 47.4 | 185.7 | 854.9 | 45.2 | 177.3 | 839.2 | 42.5 | 166.7 | ||

100 | 984.8 | 70.7 | 277.1 | 1039.2 | 64.6 | 253.3 | 1016.0 | 60.3 | 236.2 | ||

200 | 1054.8 | 80.8 | 316.9 | 1117.8 | 73.1 | 286.4 | 1091.5 | 68.0 | 266.5 | ||

500 | 1147.3 | 94.3 | 369.7 | 1221.6 | 84.3 | 330.3 | 1191.1 | 78.2 | 306.6 | ||

Meixian | EV1 | 10 | 703.3 | 31.9 | 125.2 | 723.8 | 32.4 | 126.9 | 713.3 | 30.8 | 120.7 |

20 | 747.2 | 37.6 | 147.5 | 773.1 | 37.2 | 145.8 | 760.6 | 35.2 | 137.9 | ||

50 | 820.0 | 47.6 | 186.4 | 855.0 | 45.5 | 178.3 | 839.1 | 42.8 | 167.6 | ||

100 | 984.8 | 71.0 | 278.3 | 1040.4 | 65.0 | 254.8 | 1017.0 | 60.6 | 237.6 | ||

200 | 1055.2 | 81.2 | 318.2 | 1119.5 | 73.5 | 288.1 | 1093.0 | 68.4 | 268.0 | ||

500 | 1148.0 | 94.7 | 371.2 | 1223.9 | 84.8 | 332.3 | 1193.2 | 78.7 | 308.4 | ||

Tongguan | EV1 | 10 | 744.2 | 34.6 | 135.7 | 765.1 | 34.9 | 136.8 | 755.1 | 33.5 | 131.5 |

20 | 792.6 | 40.8 | 160.0 | 819.3 | 40.1 | 157.1 | 807.4 | 38.4 | 150.4 | ||

50 | 873.2 | 51.6 | 202.2 | 909.3 | 49.0 | 192.2 | 894.2 | 46.7 | 183.1 | ||

100 | 1055.5 | 77.0 | 301.8 | 1113.1 | 70.1 | 274.7 | 1090.9 | 66.3 | 259.9 | ||

200 | 1133.3 | 88.0 | 345.1 | 1200.1 | 79.2 | 310.6 | 1174.8 | 74.8 | 293.3 | ||

500 | 1236.0 | 102.7 | 402.5 | 1314.9 | 91.4 | 358.2 | 1285.5 | 86.1 | 337.6 |

**Table 7.**Change in the uncertainty in quantile estimators based on POME compared with the MOM and ML methods (%).

Station Name | Return Period(Year) | POME to MOM | POME to ML | ||||
---|---|---|---|---|---|---|---|

Quantile | Standard Error | Confidence Interval Width | Quantile | Standard Error | Confidence Interval Width | ||

Changwu | 10 | 0.09 | 1.74 | 1.74 | 0.34 | 3.78 | 3.78 |

20 | 0.34 | −5.63 | −5.63 | 0.68 | −1.02 | −1.02 | |

50 | 0.67 | −15.66 | −15.66 | 1.10 | −7.68 | −7.68 | |

100 | 0.91 | −21.83 | −21.83 | 1.39 | −11.88 | −11.88 | |

200 | 1.14 | −26.82 | −26.82 | 1.67 | −15.33 | −15.33 | |

500 | 1.43 | −32.02 | −32.02 | 2.02 | −18.99 | −18.99 | |

Lintong | 10 | 1.37 | −3.74 | −3.74 | −1.49 | −4.93 | −4.93 |

20 | 1.74 | −6.67 | −6.67 | −1.67 | −5.29 | −5.29 | |

50 | 2.27 | −10.23 | −10.23 | −1.92 | −5.70 | −5.70 | |

100 | 3.17 | −14.76 | −14.76 | −2.35 | −6.18 | −6.18 | |

200 | 3.48 | −15.92 | −15.92 | −2.50 | −6.29 | −6.29 | |

500 | 3.82 | −17.05 | −17.05 | −2.66 | −6.41 | −6.41 | |

Meixian | 10 | 1.41 | −3.58 | −3.58 | −1.50 | −4.95 | −4.95 |

20 | 1.79 | −6.51 | −6.51 | −1.68 | −5.32 | −5.32 | |

50 | 2.34 | −10.08 | −10.08 | −1.93 | −5.73 | −5.73 | |

100 | 3.27 | −14.62 | −14.62 | −2.37 | −6.21 | −6.21 | |

200 | 3.58 | −15.78 | −15.78 | −2.51 | −6.33 | −6.33 | |

500 | 3.93 | −16.92 | −16.92 | −2.68 | −6.44 | −6.44 | |

Tongguan | 10 | 1.47 | −3.14 | −3.14 | −1.35 | −3.92 | −3.92 |

20 | 1.86 | −5.98 | −5.98 | −1.51 | −4.20 | −4.20 | |

50 | 2.41 | −9.45 | −9.45 | −1.73 | −4.53 | −4.53 | |

100 | 3.35 | −13.88 | −13.88 | −2.11 | −4.92 | −4.92 | |

200 | 3.66 | −15.00 | −15.00 | −2.23 | −5.02 | −5.02 | |

500 | 4.01 | −16.12 | −16.12 | −2.37 | −5.11 | −5.11 |

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**MDPI and ACS Style**

Wei, T.; Song, S.
Confidence Interval Estimation for Precipitation Quantiles Based on Principle of Maximum Entropy. *Entropy* **2019**, *21*, 315.
https://doi.org/10.3390/e21030315

**AMA Style**

Wei T, Song S.
Confidence Interval Estimation for Precipitation Quantiles Based on Principle of Maximum Entropy. *Entropy*. 2019; 21(3):315.
https://doi.org/10.3390/e21030315

**Chicago/Turabian Style**

Wei, Ting, and Songbai Song.
2019. "Confidence Interval Estimation for Precipitation Quantiles Based on Principle of Maximum Entropy" *Entropy* 21, no. 3: 315.
https://doi.org/10.3390/e21030315