# Bayesian Predictive Analysis of Natural Disaster Losses

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Natural Losses in the US

#### 2.1. The Data

#### 2.2. The i.i.d. Assumption for Loss Severity

#### 2.3. Non-Parametric Distribution of Loss Severity

## 3. Composite Models

#### 3.1. Three Composite Distributions

#### 3.2. Model Selection for Loss Severity

- 1.
- Sort the sample of the natural disaster damage losses in an increasing order, i.e., ${x}_{1}<{x}_{2}<...,<{x}_{n}$, where n is the sample size. Let ${n}^{*}$ be the size of the partial sample of the first ${n}^{*}$ losses ${x}_{1},{x}_{2},\dots ,{x}_{{n}^{*}}$. Start from ${n}^{*}=1$.
- 2.
- Compute the maximum likelihood estimates $\widehat{\theta}$ and $\widehat{\beta}$ as in Table 3 for the given ${n}^{*}$. If $\widehat{\theta}$ is in between ${x}_{{n}^{*}}\le \widehat{\theta}\le {x}_{{n}^{*}+1}$, we found ${n}^{*}$; otherwise, increase ${n}^{*}$ by 1.
- 3.
- Repeat Step 2 for ${n}^{*}=2,3,\dots ,$ till ${x}_{{n}^{*}}\le \widehat{\theta}\le {x}_{{n}^{*}+1}$. The ML estimates of the parameters are found based on the correct ${n}^{*}$.

## 4. The Bayesian Estimate

#### 4.1. Bayesian Estimator of LN-Pareto

- Sort the sample of size n in increasing order, i.e., ${x}_{1}<{x}_{2}<...,<{x}_{n}$ and let ${n}^{*}$ be the size of the partial sample of the first ${n}^{*}$ losses ${x}_{1},{x}_{2},\dots ,{x}_{{n}^{*}}$. Start from ${n}^{*}=1$.
- Compute the Bayes estimate $\beta $ via (8) for the given ${n}^{*}$.
- Compute the conditional Bayes estimate of $\theta $ via (7), given ${\widehat{\beta}}_{Bayes}$ from Step 2. If ${x}_{{n}^{*}+1}\le {\widehat{\theta}}_{Bayes|\beta}\le {x}_{{n}^{*}+1}$, then we found ${n}^{*}$. otherwise, increase ${n}^{*}$ by 1.
- Repeat Step 2 and 3 for ${n}^{*}=2,3,\dots $ till ${x}_{{n}^{*}+1}\le {\widehat{\theta}}_{Bayes|\beta}\le {x}_{{n}^{*}+1}$, and we found the correct ${n}^{*}$.

#### 4.2. Validation by Simulation

#### 4.3. Bayesian Estimates of Three Composite Models

## 5. Risk Measures

#### Value at Risk and Tailed Value at Risk

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Number and Loss Amounts of Natural Events from 1900 to 2016

**Table A1.**The number of occurrence of the natural events breakdown by time, type of natural events from 1900 to 2016.

1900 to 1980 | 1980 to 2016 | 1900 to 2016 | % of All Events | |
---|---|---|---|---|

Drought | $\underset{(9.09\%)}{1}$ | $\underset{(90.91\%)}{10}$ | 11 | 4.26% |

Earthquake | $\underset{(43.75\%)}{14}$ | $\underset{(56.25\%)}{18}$ | 32 | 12.40% |

Epidemic | $\underset{(0.00\%)}{0}$ | $\underset{(100.00\%)}{4}$ | 4 | 1.55% |

Extreme temperature | $\underset{(29.03\%)}{9}$ | $\underset{(70.97\%)}{22}$ | 31 | 12.02% |

Flood | $\underset{(32.69\%)}{17}$ | $\underset{(67.31\%)}{35}$ | 52 | 20.16% |

Landslide | $\underset{(40.00\%)}{2}$ | $\underset{(60.00\%)}{3}$ | 5 | 1.94% |

Storm | $\underset{(58.70\%)}{54}$ | $\underset{(41.30\%)}{38}$ | 92 | 35.66% |

Volcanic activity | $\underset{(0.00\%)}{0}$ | $\underset{(100.00\%)}{1}$ | 1 | 0.39% |

Wildfire | $\underset{(16.67\%)}{5}$ | $\underset{(83.33\%)}{25}$ | 30 | 11.63% |

Total | $\underset{(39.53\%)}{102}$ | $\underset{(60.47\%)}{156}$ | 258 | 100.00% |

**Table A2.**The adjusted total damage (’0000000 US$) breakdown by time, type of natural events from 1900 to 2016.

1900 to 1979 | 1980 to 2016 | 1900 to 2016 | % of All Damages | |
---|---|---|---|---|

Drought | $\underset{(0.00\%)}{0}$ | $\underset{(100.00\%)}{4371.623471}$ | 4371.623471 | 3.62% |

Earthquake | $\underset{(19.08\%)}{1521.333617}$ | $\underset{(80.92\%)}{6454.101764}$ | 7975.435381 | 6.60% |

Epidemic | $\underset{(0.00\%)}{0}$ | $\underset{(0.00\%)}{0}$ | 0 | 0.00% |

Extreme temperature | $\underset{(43.24\%)}{1633.100124}$ | $\underset{(56.76\%)}{2143.756251}$ | 3776.856375 | 3.13% |

Flood | $\underset{(30.94\%)}{4260.932883}$ | $\underset{(69.06\%)}{9511.711943}$ | 13,772.64483 | 11.40% |

Landslide | $\underset{(0.00\%)}{0}$ | $\underset{(100.00\%)}{2.027634158}$ | 2.027634158 | 0.00% |

Storm | $\underset{(11.57\%)}{10,143.84049}$ | $\underset{(88.43\%)}{77,499.01392}$ | 87,642.85441 | 72.54% |

Volcanic activity | $\underset{(0.00\%)}{0}$ | $\underset{(100.00\%)}{250.4927427}$ | 250.4927427 | 0.21% |

Wildfire | $\underset{(8.34\%)}{253.0904446}$ | $\underset{(91.66\%)}{2782.634167}$ | 3035.724612 | 2.51% |

Total | $\underset{(14.74\%)}{17,812.29756}$ | $\underset{(85.26\%)}{103,015.3619}$ | 120,827.6595 | 100.00% |

## Appendix B. Damage Losses from Natural Events from 1980 to 2016

Year | # of Losses | $1\mathbf{st}$ Loss | $2\mathbf{nd}$ Loss | $3\mathbf{th}$ Loss | $4\mathbf{th}$ Loss | $5\mathbf{th}$ Loss | … | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

1980 | 6 | 1019.45 | 87.38 | 58.25 | 2504.93 | 5825.41 | 2504.93 | ||||

1981 | 2 | 1056.14 | 1217.20 | ||||||||

1982 | 8 | 572.04 | 2487.12 | 84.31 | 99.48 | 248.71 | 104.46 | 497.42 | 1243.56 | ||

1983 | 9 | 7229.13 | 2409.71 | 74.70 | 15.06 | 1265.10 | 722.91 | 240.97 | 313.26 | 36.15 | |

1984 | 10 | 1683.05 | 80.85 | 2309.98 | 69.30 | 46.20 | 39.27 | 80.85 | 265.65 | 69.30 | 1385.99 |

1985 | 10 | 2453.60 | 2007.49 | 3345.82 | 26.99 | 758.39 | 1784.44 | 446.11 | 22.31 | 516.59 | 0.89 |

1986 | 5 | 1.58 | 3832.23 | 87.59 | 65.70 | 54.75 | |||||

1987 | 7 | 450.01 | 8.45 | 242.96 | 33.80 | 122.54 | 21.13 | 10.56 | |||

1988 | 0 | ||||||||||

1989 | 4 | 13,548.78 | 10,839.03 | 735.51 | 967.77 | ||||||

1990 | 6 | 23.32 | 73.45 | 183.63 | 918.16 | 64.27 | 82.63 | ||||

1991 | 9 | 59.03 | 2643.25 | 52.86 | 4405.41 | 52.86 | 1762.17 | 1497.84 | 1762.17 | 590.33 | |

1992 | 8 | 128.30 | 171.07 | 145.41 | 8553.35 | 5132.01 | 153.96 | 171.07 | 45,332.75 | ||

1993 | 7 | 315.58 | 207.62 | 166.09 | 8304.74 | 19,931.38 | 1660.95 | 12.46 | |||

1994 | 7 | 161.95 | 3.24 | 404.87 | 48,584.41 | 1133.64 | 809.74 | 3.40 | |||

1995 | 12 | 196.86 | 3307.18 | 4724.55 | 1330.75 | 1102.39 | 3149.70 | 4724.55 | 157.48 | 15.75 | 15.75 |

3149.70 | 4724.55 | ||||||||||

1996 | 6 | 1070.78 | 5200.92 | 13.00 | 2294.52 | 764.84 | 30.59 | ||||

1997 | 17 | 269.17 | 3.74 | 2.99 | 366.37 | 74.77 | 373.84 | 224.31 | 299.07 | 747.69 | 149.54 |

224.31 | 299.07 | 89.72 | 747.69 | 747.69 | 2243.06 | 7476.85 | |||||

1998 | 25 | 2061.41 | 2945.61 | 1472.44 | 406.39 | 690.57 | 6294.66 | 220.87 | 147.24 | 88.35 | 6.63 |

73.62 | 0.88 | 92.03 | 1472.44 | 2.94 | 1774.21 | 544.80 | 663.33 | 92.03 | 295.22 | ||

2208.65 | 736.22 | 397.56 | 92.03 | 295.22 | |||||||

1999 | 18 | 648.28 | 144.06 | 3976.11 | 288.84 | 1440.62 | 216.09 | 132.54 | 100.84 | 10,084.33 | 288.84 |

144.06 | 1440.62 | 10.08 | 90.04 | 288.12 | 0.43 | 1584.68 | 432.19 | ||||

2000 | 16 | 627.20 | 292.69 | 2090.65 | 139.38 | 39.72 | 11.29 | 1393.77 | 231.37 | 69.69 | 125.44 |

305.24 | 13.94 | 27.88 | 487.82 | 1533.15 | 696.88 | ||||||

2001 | 12 | 338.80 | 31.17 | 2.44 | 8131.24 | 17.62 | 27.10 | 13.55 | 40.66 | 5.42 | 9.49 |

4.07 | 2710.41 | ||||||||||

2002 | 16 | 533.65 | 267.49 | 6.67 | 17.34 | 5.34 | 2935.05 | 26.68 | 267.49 | 1334.11 | 26.68 |

400.23 | 933.88 | 2668.23 | 601.02 | 8.81 | 4402.57 | ||||||

2003 | 13 | 6521.93 | 32.61 | 5217.54 | 138.26 | 4395.78 | 260.88 | 521.75 | 22.17 | 65.22 | 4565.35 |

4.43 | 2739.21 | 260.88 | |||||||||

2004 | 14 | 381.17 | 5.72 | 1397.61 | 889.39 | 76.23 | 0.22 | 20,328.81 | 79.41 | 13,976.06 | 22,869.91 |

10,164.40 | 2.67 | 1.27 | 635.28 | ||||||||

2005 | 11 | 307.23 | 245.78 | 36.87 | 430.12 | 6.55 | 430.12 | 2740.48 | 19,662.63 | 17,573.48 | 301.08 |

122.89 | |||||||||||

2006 | 20 | 1428.61 | 714.31 | 1904.82 | 308.34 | 14.29 | 535.73 | 101.19 | 1190.51 | 8.33 | 19.05 |

119.05 | 18.45 | 39.12 | 29.76 | 113.10 | 357.15 | 29.76 | 178.58 | 107.15 | 428.58 | ||

2007 | 15 | 32.41 | 578.77 | 810.28 | 150.48 | 2893.85 | 364.63 | 1157.54 | 578.77 | 162.06 | 405.14 |

2315.08 | 347.26 | 347.26 | 694.52 | 347.26 | |||||||

2008 | 16 | 501.63 | 2.23 | 780.32 | 1783.58 | 113.70 | 1337.69 | 200.65 | 33,442.22 | 1449.16 | 2229.48 |

668.84 | 1114.74 | 1226.21 | 11,147.41 | 122.62 | 401.31 | ||||||

2009 | 12 | 185.71 | 1901.83 | 111.87 | 268.49 | 2796.80 | 559.36 | 1230.59 | 671.23 | 2237.44 | 1118.72 |

950.91 | 1678.08 | ||||||||||

2010 | 7 | 2586.57 | 2971.80 | 13.76 | 2201.33 | 110.07 | 1651.00 | 550.33 | |||

2011 | 14 | 2133.97 | 195.26 | 11,736.86 | 14,937.82 | 2027.28 | 7789.01 | 1066.99 | 3200.96 | 800.24 | 3734.45 |

4908.14 | 213.40 | 2133.97 | 8535.90 | ||||||||

2012 | 22 | 182.94 | 1620.30 | 1881.64 | 219.52 | 181.89 | 627.21 | 2090.71 | 52,267.70 | 2.09 | 52.27 |

4181.42 | 209.07 | 522.68 | 5226.77 | 4704.09 | 104.54 | 3554.20 | 1463.50 | 1986.17 | 731.75 | ||

219.52 | 20,907.08 | ||||||||||

2013 | 26 | 1648.42 | 1133.29 | 309.08 | 3193.82 | 309.08 | 2163.55 | 22.05 | 515.13 | 927.24 | 2.06 |

25.76 | 25.76 | 2.06 | 334.84 | 180.30 | 309.08 | 1957.50 | 10.30 | 1339.34 | 2.06 | ||

206.05 | 103.03 | 2266.58 | 103.03 | 103.03 | 1133.29 | ||||||

2014 | 19 | 2.03 | 2027.63 | 101.38 | 3953.89 | 273.73 | 66.91 | 1622.11 | 709.67 | 172.35 | 101.38 |

253.45 | 91.24 | 212.90 | 253.45 | 1622.11 | 760.36 | 2534.54 | 2230.40 | 20.28 | |||

2015 | 28 | 172.14 | 506.31 | 1417.66 | 961.98 | 1012.62 | 162.02 | 1417.66 | 2734.06 | 658.20 | 101.26 |

81.01 | 2.03 | 708.83 | 101.26 | 961.98 | 151.89 | 1417.66 | 2.03 | 1721.45 | 101.26 | ||

273.41 | 141.77 | 911.35 | 607.57 | 405.05 | 151.89 | 3037.85 | 1822.71 | ||||

2016 | 25 | 550.00 | 125.00 | 3900.00 | 2000.00 | 2400.00 | 1000.00 | 1100.00 | 300.00 | 1000.00 | 150.00 |

50.00 | 10,000.00 | 100.00 | 600.00 | 550.00 | 10,000.00 | 1200.00 | 275.00 | 20.00 | 1200.00 | ||

100.00 | 2300.00 | 1600.00 | 1200.00 | 2300.00 |

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1 | |

2 | |

3 | The CPI data was download from Bureau of Labor Statistics https://data.bls.gov/pdq/SurveyOutputServlet. |

**Figure 1.**The number and damage costs of natural events from 1900 to 2016 in the US. (

**a**) The number of natural events; (

**b**) Total damage in 2016 dollars (’0000000 US$).

**Figure 2.**The percentage of the number of occurrence and damage costs of natural disasters to the total occurrence and damage losses by different types of natural events before and after 1980 in the US. (

**a**) Percent of the number of occurrence; (

**b**) Percent of the damage costs.

**Figure 4.**The histogram and the empirical distribution with 95% confidence bands of the CPI adjusted natural losses from 1980 to 2016 in the US. (

**a**) The histogram of the CPI adjusted natural losses; (

**b**) The empirical distribution with 95% confidence bands of the CPI adjusted natural losses.

**Figure 5.**The cumulative distribution functions of three composite models with various parameter values.

Pairs | Kendall Tau Statistics (p-Value) | Spearman Statistics (p-Value) |
---|---|---|

${X}_{1},{X}_{5}$ | $\underset{(0.002)}{-0.373}$ | $\underset{(0.001)}{-0.546}$ |

${X}_{7},{X}_{21}$ | $\underset{(0.0051)}{0.393}$ | $\underset{(0.003)}{0.566}$ |

${X}_{9},{X}_{16}$ | $\underset{(0.009)}{-0.575}$ | $\underset{(0.0034)}{-0.769}$ |

${\mathit{f}}_{1}(\mathit{x})$ | ${\mathit{f}}_{2}(\mathit{x})$ | The Composite pdf $\mathit{f}(\mathit{x})$ | |
---|---|---|---|

Exp-Pareto | ${f}_{1}(x)=\lambda {e}^{-\lambda x}$, $x>0,\lambda >0$ | ${f}_{2}(x)=\frac{{\displaystyle \alpha {\theta}^{\alpha}}}{{\displaystyle {x}^{\alpha +1}}},$ $x\ge \theta >0,\alpha >0$ | $f(x)=\left\{\begin{array}{cc}{\displaystyle \frac{0.775}{\theta}{e}^{\frac{-1.35x}{\theta}},}\hfill & 0<x\le \theta \hfill \\ {\displaystyle \frac{0.2{\theta}^{0.35}}{{x}^{1.35}},}\hfill & \theta \le x<\infty \hfill \end{array}\right.$ |

IG-Pareto | $f}_{1}(x)=\frac{{\beta}^{\alpha}{x}^{-\alpha -1}{e}^{-\beta /x}}{\Gamma (\alpha )$, $x>0,\alpha >0,\beta >0$ | ${f}_{2}(x)=\frac{a{\theta}^{a}}{{x}^{a+1}},$ $x\ge \theta ,a>0,\theta >0$ | $f(x)=\left\{\begin{array}{cc}{\displaystyle \frac{c{(k\theta )}^{\alpha}{x}^{-\alpha -1}{e}^{\frac{-k\theta}{x}}}{\Gamma (\alpha )},}\hfill & 0<x\le \theta \hfill \\ {\displaystyle \frac{c(\alpha -k){\theta}^{\alpha -k}}{{x}^{\alpha -k+1}},}\hfill & \theta \le x<\infty \hfill \end{array}\right.$ where $\alpha =0.308289$, $k=0.144351$, $c=0.711384$ |

LN-Pareto | ${f}_{1}(x)=\frac{{e}^{-\frac{1}{2}{(\frac{lnx-\mu}{\sigma})}^{2}}}{x\sigma \sqrt{2\pi}},$ $x>0,\sigma >0$ | ${f}_{2}(x)=\frac{\alpha {\theta}^{\alpha}}{{x}^{\alpha +1}},$ $x\ge \theta ,\alpha >0,\theta >0$ | $f(x)=\left\{\begin{array}{cc}{\displaystyle \frac{\beta {\theta}^{\beta}{e}^{-0.5{(\frac{\beta}{k})}^{2}{ln}^{2}(x/\theta )}}{(1+\Phi (k)){x}^{\beta +1}},}\hfill & 0<x\le \theta \hfill \\ {\displaystyle \frac{\beta {\theta}^{\beta}}{(1+\Phi (k)){x}^{\beta +1}},}\hfill & \theta \le x<\infty \hfill \end{array}\right.$ where $k=0.372238898$ |

Composite Model | ML Estimates of Parameters |
---|---|

Exp-Pareto | $\widehat{\theta}=\frac{1.35{n}^{*}{\overline{x}}_{{n}^{*}}}{1.35{n}^{*}-0.35n}$, where $\overline{x}}_{{n}^{*}}=\frac{1}{{n}^{*}}\sum _{i=1}^{{n}^{*}}{x}_{i$. |

IG-Pareto | $\widehat{\theta}=\frac{{n}^{*}\alpha +(\alpha -k){n}^{*}}{kS}$, where $S=\sum _{i=1}^{{n}^{*}}{x}_{i}^{-1}$, $\alpha =0.308289$, $k=0.144351$. |

LN-Pareto | If ${n}^{*}=1$, $\widehat{\theta}={x}_{1}{\left(\prod _{i=1}^{{n}^{*}}{x}_{i}/{x}_{1}\right)}^{{k}^{2}}$, $\widehat{\beta}={n}^{*}{\left(\sum _{i=1}^{{n}^{*}}ln({x}_{i}/{x}_{1}\right)}^{-1}$, where $k=0.372238898$; otherwise, $\widehat{\theta}=exp\left(\frac{n{k}^{2}}{{n}^{*}\widehat{\beta}}\right){\left(\prod _{i=1}^{{n}^{*}}{x}_{i}\right)}^{\frac{1}{{n}^{*}}}$, $\widehat{\beta}=\frac{{\displaystyle {k}^{2}B+\sqrt{{k}^{4}{B}^{2}+4{n}^{*}n{k}^{2}A}}}{2A}$, where $A={n}^{*}\sum _{i=1}^{{n}^{*}}{(ln{x}_{i})}^{2}-{(\sum _{i=1}^{{n}^{*}}ln{x}_{i})}^{2}$, $B=n\sum _{i=1}^{{n}^{*}}ln{x}_{i}-{n}^{*}\sum _{i=1}^{n}ln{x}_{i}$. |

**Table 4.**The ML estimates of three composite models and three non-composite models for the severity of natural events in the US.

Model | ML Estimates of Parameters | $\mathit{NLL}$ | $\mathit{AIC}$ | $\mathit{BIC}$ |
---|---|---|---|---|

Exp-Pareto | $\widehat{\theta}=25.561$ | 2698.02 | 5398.05 | 5402.19 |

${n}^{*}=189$ | ||||

IG-Pareto | $\widehat{\theta}=2.86262$ | 2719.35 | 5440.7 | 5444.83 |

${n}^{*}=64$ | ||||

LN-Pareto | $\widehat{\theta}=20.94751406$ | 2327.74 | 4659.49 | 4667.76 |

$\widehat{\beta}=0.22044516$ | ||||

${n}^{*}=174$ | ||||

Exponential $X\sim $ Exp($\lambda $) | $\widehat{\lambda}=5.32\times {10}^{-3}$ | $2881.23$ | $5764.47$ | $5768.61$ |

Inverse Gamma | $\widehat{\alpha}=0.254669$ | 2813.29 | 5630.58 | 5638.85 |

$X\sim $ IG( $\alpha ,\beta $) | $\widehat{\beta}=0.527914$ | |||

Lognormal | $\widehat{\mu}=3.50972$ | 3058.66 | 6121.31 | 6129.58 |

$X\sim $ LN( $\mu ,\sigma $) | $\widehat{\sigma}=2.1659785$ |

$\theta =5$ $\beta =0.5$ (${a}_{1}=100,{b}_{1}=0.005,{c}_{1}=1.33231$) | |||||||||

n | ${\overline{\widehat{\theta}}}_{ML}$ | ${\u03f5}_{{\theta}_{ML}}$ | ${\overline{\widehat{\beta}}}_{ML}$ | ${\u03f5}_{{\beta}_{ML}}$ | ${\overline{\widehat{\theta}}}_{Bayes}$ | ${\u03f5}_{{\theta}_{Bayes}}$ | ${\overline{\widehat{\beta}}}_{Bayes}$ | ${\u03f5}_{{\beta}_{Bayes}}$ | |

20 | 7.8604 | 4.0640 | 0.4747 | 0.1611 | 5.6933 | 2.0959 | 0.4641 | 0.0402 | |

50 | 7.8942 | 3.8006 | 0.4435 | 0.1121 | 5.7941 | 1.2265 | 0.4338 | 0.0688 | |

100 | 7.6421 | 3.6415 | 0.4333 | 0.1056 | 5.8745 | 1.1874 | 0.4058 | 0.0960 | |

$\theta =20\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta =0.5$ (${a}_{1}=560,{b}_{1}=0.001,{c}_{1}=2.71861$) | |||||||||

n | ${\overline{\widehat{\theta}}}_{ML}$ | ${\u03f5}_{{\theta}_{ML}}$ | ${\overline{\widehat{\beta}}}_{ML}$ | ${\u03f5}_{{\beta}_{ML}}$ | ${\overline{\widehat{\theta}}}_{Bayes}$ | ${\u03f5}_{{\theta}_{Bayes}}$ | ${\overline{\widehat{\beta}}}_{Bayes}$ | ${\u03f5}_{{\beta}_{Bayes}}$ | |

20 | 21.8966 | 12.2540 | 0.5878 | 0.2626 | 20.5062 | 5.5390 | 0.5341 | 0.0345 | |

50 | 19.9977 | 4.9775 | 0.5738 | 0.2431 | 20.5262 | 3.3892 | 0.5025 | 0.0065 | |

100 | 19.5730 | 3.6691 | 0.5467 | 0.1830 | 21.2182 | 3.0585 | 0.4629 | 0.0379 | |

$\theta =5\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta =1.5$ (${a}_{1}=3000,{b}_{1}=0.0005,{c}_{1}=1.57865$) | |||||||||

n | ${\overline{\widehat{\theta}}}_{ML}$ | ${\u03f5}_{{\theta}_{ML}}$ | ${\overline{\widehat{\beta}}}_{ML}$ | ${\u03f5}_{{\beta}_{ML}}$ | ${\overline{\widehat{\theta}}}_{Bayes}$ | ${\u03f5}_{{\theta}_{Bayes}}$ | ${\overline{\widehat{\beta}}}_{Bayes}$ | ${\u03f5}_{{\beta}_{Bayes}}$ | |

20 | 5.342 | 1.2443 | 1.510 | 0.3811 | 5.076 | 0.3877 | 1.480 | 0.02047 | |

50 | 5.041 | 0.5606 | 1.528 | 0.2181 | 5.038 | 0.2526 | 1.451 | 0.0492 | |

100 | 5.035 | 0.2344 | 1.516 | 0.1389 | 5.111 | 0.2196 | 1.406 | 0.0941 | |

$\theta =20\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta =1.5$ ( ${a}_{1}=3000,{b}_{1}=0.0005,{c}_{1}=2.96494$) | |||||||||

n | ${\overline{\widehat{\theta}}}_{ML}$ | ${\u03f5}_{{\theta}_{ML}}$ | ${\overline{\widehat{\beta}}}_{ML}$ | ${\u03f5}_{{\beta}_{ML}}$ | ${\overline{\widehat{\theta}}}_{Bayes}$ | ${\u03f5}_{{\theta}_{Bayes}}$ | ${\overline{\widehat{\beta}}}_{Bayes}$ | ${\u03f5}_{{\beta}_{Bayes}}$ | |

20 | 20.698 | 3.3162 | 1.560 | 0.4157 | 20.556 | 1.8798 | 1.460 | 0.04030 | |

50 | 20.763 | 3.0164 | 1.468 | 0.2479 | 20.460 | 1.0797 | 1.404 | 0.09601 | |

100 | 20.218 | 2.6237 | 1.466 | 0.2839 | 20.637 | 0.9563 | 1.324 | 0.1762 |

Model | Prior Distributions | Bayesian Estimates | $\mathit{NLL}$ | $\mathit{AIC}$ | $\mathit{BIC}$ |
---|---|---|---|---|---|

Exp-Pareto | $\theta \sim $ Inverse-Gamma(10, 5) | $\widehat{\theta}=23.1451$ ${n}^{*}=183$ | 2697.57 | 5397.13 | 5401.27 |

IG-Pareto | $\theta \sim $ Gamma(50, 1) | $\widehat{\theta}=4.3818$ ${n}^{*}=77$ | 2699.17 | 5400.34 | 5404.48 |

LN-Pareto | $\theta \sim $ LN(1.61352, 2.857) $\beta \sim $ Gamma(20,500, 1.1 × 10${}^{-6}$) | $\widehat{\theta}=19.2316$ $\widehat{\beta}=0.220173$ ${n}^{*}=168$ | 2327.63 | 4659.27 | 4667.54 |

Models | ${\mathit{VaR}}_{0.85}(\mathit{x})$ | ${\mathit{LTVaR}}_{0.85}(\mathit{x})$ | |
---|---|---|---|

Bayesian Estimation | Exp-Pareto | 1073 | 30,723 |

IG-Pareto | 58,212 | 98,041 | |

LN-Pareto | 11,070 | 75,860 | |

ML Estimation | Exp-Pareto | 1185 | 31,770 |

IG-Pareto | 38,029 | 94,619 | |

LN-Pareto | 11,963 | 76,946 |

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**MDPI and ACS Style**

Deng, M.; Aminzadeh, M.; Ji, M.
Bayesian Predictive Analysis of Natural Disaster Losses. *Risks* **2021**, *9*, 12.
https://doi.org/10.3390/risks9010012

**AMA Style**

Deng M, Aminzadeh M, Ji M.
Bayesian Predictive Analysis of Natural Disaster Losses. *Risks*. 2021; 9(1):12.
https://doi.org/10.3390/risks9010012

**Chicago/Turabian Style**

Deng, Min, Mostafa Aminzadeh, and Min Ji.
2021. "Bayesian Predictive Analysis of Natural Disaster Losses" *Risks* 9, no. 1: 12.
https://doi.org/10.3390/risks9010012