Modelling Extreme Rainfall Using Adjusted Sandwich Estimator

The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100-154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.

The 5TH ISM INTERNATIONAL STATISTICAL CONFERENCE 2021 (ISM-V): Statistics in the Spotlight: Navigating the New Norm

The objective of this study is to improved inferences for a model of observed extreme data by linking the distributional of observed data with a climate change model data. We exploit the extra information from projection data of the climate models (RCM and GCM) for the period 1970-2099 under emission scenarios RCP8.5 in modelling an annual extreme rainfall that collected from 44 stations in Sabah. The joint model enables additional information about the trend from longer records of climate model to be incorporated with the limited observed extreme data at each site. The proposed models were built based on the independent likelihood over sites therefore the standard error of the parameters need to be adjusted to take into accounts the inter-site dependence. As a result, we found a common trend between the observed data and climate model and the sandwich estimator reduces the variances of the marginal characteristics and yield more accurate inferences.

Journal of Civil Engineering and Urbanism, 2020

Extreme Value Analysis (EVA) of rainfall is considered as one of the important aspects to arrive at a design value for planning, design and management of civil and hydraulic structures. This can be achieved by fitting Probability Distribution (PDs) to the series of observed annual 1-day maximum rainfall data wherein the parameters of PDs are determined by method of moments and L-Moments (LMO). In this paper, a study on comparison of Extreme Value Type-1 (EV1), Extreme Value Type-2, Generalized Extreme Value (GEV) and Generalized Pareto distributions adopted in EVA of rainfall for Anakapalli, Atchutapuram, Kasimkota and Parvada sites is carried out. The selection of best fit PD for EVA of rainfall is made through quantitative assessment by using Goodness-of-Fit (viz., Chisquare and Kolmogorov-Smirnov) and diagnostic (viz., root mean squared error) tests; and qualitative assessment by using the fitted curves of the estimated rainfall. On the basis of evaluation of EVA results through quantitative and qualitative assessments, the study indicates the extreme rainfall given by EV1 (LMO) distribution could be used for the purpose of economical design. The study also indicates the extreme rainfall obtained from GEV (LMO) distribution may be considered for the design of civil and hydraulic structure with little risk involvement.

Journal of Hydrology, 2016

The Akosombo dam is a major source of electric energy in Ghana. Considering the current increase in the demand for electricity in the country, where such an increase in demand implies more pressure on the dam, it is of key interest to study the tail behaviour of the water levels of the dam. Such a study is important because the level of water in the dam determines the amount of electricity generated. The study employed the Univariate Extreme Value Theory to model the monthly maximum and minimum water levels of the dam. The Generalized Extreme Value Distribution was fitted to the data and the Maximum likelihood estimation method was employed to estimate the model parameters. The study indicated that, the water levels cannot fall below 226.00ft which is the critical water level of the Akosombo dam. It further showed that, the lowest ever level of water the dam can attain is 226.69ft and the highest 279.07ft. The study also found that, though the water cannot fall below the critical level, there was evidence of its falling below the minimum operation head.

Science World Journal, 2020

An important statistical distribution use in modeling such extreme events is the generalized extreme value distribution while the generalized Pareto distribution is suitable in modeling threshold excesses of extreme values. In this study, monthly rainfall data from the Nigeria Meteorological Agency in Kaduna are fitted to the generalized extreme value distribution and for a suitable threshold of 251mm, threshold excesses were fitted to the generalized Pareto distribution and a return level computed for 25, 50 and 100 years return period respectively. The threshold excesses follow the Weibull distribution and are bounded above implying that there is a finite value which the maximum above the threshold cannot exceed. For the 25, 50 and 100 years return period, a return level of 350mm, 390mm and 490mm with probability of exceedances of 0.04, 0.02 and 0.01 respectively were observed. The result further show that with the increasing level of rainfall as return period increases, there is ...

Assessment of extreme rainfall is one of the important parameters for planning, design and management of hydraulic structures at the project site. This can be obtained through Extreme Value Analysis (EVA) of rainfall by fitting of probability distributions to the data series of annual 1‐day maximum rainfall. This paper illustrates the adoption of Gumbel (EV1), Frechet (EV2), 2‐parameter Log Normal (LN2) and Log Pearson Type‐3 (LP3) distributions in EVA for Tohana. Methods of moments and Maximum Likelihood Method (MLM) are used for determination of parameters of EV1, EV2, LN2 and LP3 distributions. In addition to above, order statistics approach is used for determination of parameters of EV1 and EV2. The adequacy of fitting of probability distributions is evaluated by Goodness‐of‐Fit tests viz., Anderson‐Darling and Kolmogorov‐Smirnov and diagnostic test using D‐index. By considering the design‐life of the structure, the study suggests the estimated extreme rainfall using LP3 (MLM) distribution could be used for design purposes.

Hydrological Sciences Journal, 2004

The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100-154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.

International Journal of Statistical Distributions and Applications

Extreme rainfall events have caused significant damage to agriculture, ecology and infrastructure, disruption of human activities, injury and loss of life. They have also significant social, economical and environmental consequences because they considerably damage urban as well as rural areas. Early detection of extreme maximum rainfall helps to implement strategies and measures, before they occur. Extreme value theory has been used widely in modelling extreme rainfall and in various disciplines, such as financial markets, insurance industry, failure cases. Climatic extremes have been analysed by using either generalized extreme value (GEV) or generalized Pareto (GP) distributions which provides evidence of the importance of modelling extreme rainfall from different regions of the world. In this paper, we focus on Peak Over Thresholds approach where the Poisson-generalized Pareto distribution is considered as the proper distribution for the study of the exceedances. This research considers also use of the generalized Pareto (GP) distribution with a Poisson model for arrivals to describe peaks over a threshold. The research used statistical techniques to fit models that used to predict extreme rainfall in Tanzania. The results indicate that the proposed Poisson-GP distribution provide a better fit to maximum monthly rainfall data. Further, the Poisson-GP models are able to estimate various return levels. Research found also a slowly increase in return levels for maximum monthly rainfall for higher return periods and further the intervals are increasingly wider as the return period is increasing.

Rainfall frequency analysis plays an important role in hydrologic and economic evaluation of water resources projects. It helps to estimate the return periods and their corresponding event magnitudes thereby creating reasonable design criteria. Depending on the size, life time and design criteria of the structure, different return periods are generally stipulated for adopting Extreme Value Analysis (EVA) results. This paper illustrates the use of quantitative assessment on fitting of Gumbel (EV1) and Frechet (EV2) probability distributions to the series of annual 1-day maximum rainfall (AMR) data using Goodness-of-Fit (GoF) and diagnostic tests. Order Statistics Approach (OSA) is used for determination of parameters of the distributions. Based on GoF (using Anderson-Darling and Kolmogorov-Smirnov) and diagnostic (using D-index) test results, the study identifies the EV1 distribution is better suited for EVA of rainfall for Fatehabad and Hissar.

The analysis of 27 years rainfall data of Kumulur region was conducted using two types of probability distributions, viz Gumbel distribution and generalised extreme value distribution. The method of L-moments was used for the analysis. Annual one day maximum and 2, 3,4, 5 and 7 consecutive days maximum rainfall data for 27 years was analysed and the return levels for 2, 5, 10 and 25-years were calculated using the proposed probability distribution functions. Chisquare test was conducted for comparison of the observed and expected return levels obtained using both the distributions. The statistical analysis revealed that, the annual maxima rainfall data for one day maxima and consecutive days maxima of Kumulur region fits best with the generalised extreme value distribution.