# Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Prediction

#### 2.1. Objective Prior

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 2.2. Posterior Analysis

**Theorem**

**3.**

**Proof.**

#### 2.3. Prediction

- Step 1a.
- Generate ${h}_{i}$ from the gamma distribution with the parameters $(s-k)$ and 1.
- Step 1b.
- Generate ${\lambda}_{i}$ and ${\theta}_{i}$ from the joint posterior distribution $\pi (\lambda ,\theta |\mathbf{x})$.
- Step 2.
- Compute$$\begin{array}{c}{y}_{i}=-\frac{1}{{\theta}_{i}}log\left(\frac{{h}_{i}}{{\lambda}_{i}}+{e}^{-{\theta}_{i}{x}_{L\left(k\right)}}\right).\hfill \end{array}$$
- Step 3.
- Repeat steps 1 and 2, N times.

## 3. Time Series Approach

- p denotes the order of the autoregressive (AR) term
- d denotes the number of differencing required to make the time series stationary
- q denotes the order of the moving average (MA) term.

## 4. Sulfur Dioxide Data

**dlm**(Petris [14]) was used to estimate the parameters and state vector in the LGM (9) with

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chandler, K.N. The distribution and frequency of record values. J. R. Stat. Soc. Ser. B
**1952**, 14, 220–228. [Google Scholar] [CrossRef] - Coles, S.G.; Tawn, J.A. A Bayesian analysis of extreme rainfall data. J. R. Stat. Soc. Ser. C
**1996**, 45, 463–478. [Google Scholar] [CrossRef] - Madi, M.T.; Raqab, M.Z. Bayesian prediction of temperature records using the Pareto model. Environmetrics
**2004**, 15, 701–710. [Google Scholar] [CrossRef] - Wergen, G.; Hense, A.; Krug, J. Record occurrence and record values in daily and monthly temperatures. Clim. Dyn.
**2013**, 42, 1275–1289. [Google Scholar] [CrossRef] [Green Version] - Seo, J.I.; Kim, Y. Statistical inference on Gumbel distribution using record values. J. Korean Stat. Soc.
**2016**, 45, 342–357. [Google Scholar] [CrossRef] - Seo, J.I.; Kang, S.B. More efficient approaches to the exponentiated half-logistic distribution based on record values. SpringerPlus
**2016**, 5, 1433–1451. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Box, G.E.P.; Jenkins, G.M.; Reinsel, G.C. Time Series Analysis: Forecasting, and Control, 3rd ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1994. [Google Scholar]
- West, M.; Harrison, J. Bayesian Forecasting and Dynamic Models, 2nd ed.; Springer: New York, NY, USA, 1997. [Google Scholar]
- Ahsanullah, M. Record Statistics; Nova Science Publishers, Inc.: New York, NY, USA, 1995. [Google Scholar]
- Jeffreys, H. Theory of Probability and Inference, 3rd ed.; Cambridge University Press: London, UK, 1961. [Google Scholar]
- Beger, J.O.; Bernardo, J.M. Estimating a product of means: Bayesian analysis with reference priors. J. Am. Stat. Assoc.
**1989**, 84, 200–207. [Google Scholar] [CrossRef] - Wang, B.X.; Ye, Z.S. Inference on the Weibull distribution based on record values. Comput. Stat. Data Anal.
**2015**, 83, 26–36. [Google Scholar] [CrossRef] - Chen, M.H.; Shao, Q.M. Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat.
**1999**, 8, 69–92. [Google Scholar] - Petris, G. dlm: Bayesian and Likelihood Analysis of Dynamic Linear Models. R Package Version 1.1-1. 2010. Available online: http://CRAN.R-project.org/package=dlm (accessed on 13 June 2020).

**Figure 2.**$95\%$ regions for ${X}_{L\left(i\right)}^{rep}$ and scatter plots between $E\left({X}_{L\left(i\right)}^{rep}\right)$ and the observed lower record values.

**Figure 3.**Estimated kernel density for ${X}_{L\left(s\right)}\mid {x}_{L\left(19\right)}$ under the priors ${\pi}_{JR}(\lambda ,\theta )$ and ${\pi}_{R1}(\lambda ,\theta )$.

Minimum | Maximum | Mean | Median | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|---|

2815 | 31,218 | 13.181 | 11.011 | 9.065 | 0.544 | −1.147 |

$\widehat{\mathit{\lambda}}$ | ${\widehat{\mathit{\lambda}}}_{\mathit{JR}}$ | ${\widehat{\mathit{\lambda}}}_{\mathit{R}1}$ | $\widehat{\mathit{\sigma}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{JR}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{R}1}$ | |
---|---|---|---|---|---|---|

Estimate | 26.240 | 26.289 | 24.409 | 10.366 | 10.911 | 11.611 |

$95\%$ ETs | (14.411, 38.070) | (15.785, 39.653) | (14.874, 36.755) | (5.930, 14.801) | (6.991, 16.989) | (7.450, 18.242) |

$95\%$ HPD | - | (15.082, 38.560) | (13.784, 35.388) | - | (6.493, 15.992) | (6.859, 17.039) |

Mean | Median | $95\%$ PI | $80\%$ PI | |||
---|---|---|---|---|---|---|

ETs | HPD | ETs | HPD | |||

${X}_{L\left(20\right)}^{JR}\mid {x}_{L\left(19\right)}$ | 2.359 | 2.560 | (0.634, 2.945) | (1.115, 2.960) | (1.566, 2.899) | (2.009, 2.960) |

${X}_{L\left(20\right)}^{R1}\mid {x}_{L\left(19\right)}$ | 2.275 | 2.507 | (0.339, 2.945) | (0.905, 2.960) | (1.384, 2.894) | (1.870, 2.960) |

${X}_{L\left(20\right)}^{ARIMA}$ | 2.139 | - | (0.082, 4.195) | - | (0.794, 3.483) | - |

${X}_{L\left(20\right)}^{DLM}$ | 2.357 | - | (0.535, 4.179) | - | (1.165, 3.548) | - |

MAD | MSE | MAPE | |
---|---|---|---|

ARIMA(0,2,1) | 0.649 | 0.927 | 0.062 |

DLM | 0.664 | 0.817 | 0.062 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, Y.; Seo, J.I.
Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction. *Symmetry* **2020**, *12*, 1443.
https://doi.org/10.3390/sym12091443

**AMA Style**

Kim Y, Seo JI.
Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction. *Symmetry*. 2020; 12(9):1443.
https://doi.org/10.3390/sym12091443

**Chicago/Turabian Style**

Kim, Yongku, and Jung In Seo.
2020. "Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction" *Symmetry* 12, no. 9: 1443.
https://doi.org/10.3390/sym12091443