Method Of Probability Weighted Moments Research Articles

Statistical Inference and Application of Asymmetrical Generalized Pareto Distribution Based on Peaks-Over-Threshold Model

Generalized Pareto distribution (GPD), an asymmetrical distribution, primarily models exceedances over a high threshold in many applications. Within the peaks-over-threshold (POT) framework, we consider a new GPD parameter estimation method to estimate a common tail risk measure, the value at risk (VaR). The proposed method is more suitable for the POT framework and makes full use of data information. Specifically, our estimation method builds upon the generalized probability weighted moments method and integrates it with the nonlinear weighted least squares method. We use exceedances for the GPD, minimizing the sum of squared differences between the sample and population moments of a function of GPD random variables. At the same time, the proposed estimator uses three iterations and assigns weight to further improving the estimated performance. Under Monte Carlo simulations and with a real heavy-tailed dataset, the simulation results show the advantage of the newly proposed estimator, particularly when VaRs are at high confidence levels. In addition, by simulating other heavy-tailed distributions, our method still exhibits good performance in estimating misjudgment distributions.

Modeling of Extreme Temperature Using min-Generalized Extreme Value Distribution

Extreme value theory (EVT) is one of major importance in many fields of applications where extreme values may appear and have detrimental effects and finally Generalized Extreme Value (GEV) distribution model is found as the best fitted distribution model. Extreme minimum temperature using 40 years of data is studied. Minimum of four different time periods (monthly, quarterly, half yearly and yearly) are fitted to the minimum Generalized Extreme Value (MGEV) distribution. The first objective of this study is to describe and model the behaviour of extreme minimum temperature in Sebha by using the MGEV distribution. The second object, we determine the effect of using different size blocks, and illustrate the best four different time (Monthly, Quarterly, Half yearly and yearly) selection periods that are suitable for modelling with the MGEV distribution. The method of probability-weighted moments (PWM) has been used to estimate the unknown parameters; and its corresponding Deviance Test (DT) approach to test the goodness of fit. The results show that minimum Weibull distribution as special case of MGEV distribution is the most an appropriate choice of all selection periods (quarterly half yearly and yearly) are significant except only monthly of all three periods are significant to be fitted by minimum Frechet model. We will use the R programming with packages of fExtreme, evir and ismev to calculate, parameter estimation, testing and diagnostic plots.

The principle of maximum entropy and the probability-weighted moments for estimating the parameters of the Kumaraswamy distribution.

Since Shannon’s formulation of the entropy theory in 1940 and Jaynes’ discovery of the principle of maximum entropy (POME) in 1950, entropy applications have proliferated across a wide range of different research areas including hydrological and environmental sciences. In addition to POME, the method of probability-weighted moments (PWM), was introduced and recommended as an alternative to classical moments. The PWM is thought to be less impacted by sampling variability and be more efficient at obtaining robust parameter estimates. To enhance the PWM, self-determined probability-weighted moments was introduced by (Haktanir 1997). In this article, we estimate the parameters of Kumaraswamy distribution using the previously mentioned methods. These methods are compared to two older methods, the maximum likelihood and the conventional method of moments techniques using Monte Carlo simulations. A numerical example based on real data is presented to illustrate the implementation of the proposed procedures.

New Zealand design wind speeds, directional and lee-zone multipliers proposed for AS/NZS 1170.2:2021

The study aims to estimate design wind speeds and associated directional multipliers, also lee-zone multipliers for New Zealand through the analysis of historical wind data recorded at meteorological stations and also utilising a high-resolution convection-resolving numerical weather prediction model (New Zealand Convective-Scale Model (NZCSM)) analyses. New Zealand’s historical wind data have not been analysed in the past two decades for design wind-load purposes. In addition, no attempt has been made to thoroughly homogenise the mean and gust wind speed data recorded prior to the 1990s and to convert them to equivalent Automatic Weather Stations (AWS) records. Furthermore, lee zones, areas affected by the wind speed-up due to the presence of mountains, can significantly influence the design wind loads, thus, it is crucial to estimate the spatial extent and magnitude of the lee multiplier accurately. In this study, the wind data were initially subjected to a robust homogenisation algorithm and then, they were separated into synoptic and non-synoptic events to gain a better understanding of New Zealand’s gust climatology and sources of extreme events. It was demonstrated that synoptic events dominate the design wind speeds at most locations in New Zealand. For extreme value analysis, three different extreme value distributions were used, namely Type I (using Gumbel, Gringorten and BLUE fitting methods), Type III (using maximum likelihood and probability weighted moments methods), and Peaks-Over-Threshold (POT) approach. In addition, the predictions of NZCSM along with historical wind speeds were used to identify the lee zones, which confirms existing zones and provides evidence to support introducing new zones, and obtain estimates of the lee-multipliers. Substantial changes have been proposed for the next version of the Australian/New Zealand wind-loading standard (AS/NZS 1170.2) based on the results of this study. The changes include adding a new wind region to New Zealand, refinements of wind zone boundaries, revising all regional wind speeds and directional multipliers, and modifying the lee-zone regions and multipliers.

Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior

Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.

Estimating the expected number of crashes with traffic conflicts and the Lomax Distribution – A theoretical and numerical exploration

This paper justifies the Lomax distribution for counterfactual modeling of the probability of crash given a traffic conflict. The pre-crash process leading to a conflict or a crash as the result of a failure is discussed as this conceptualization is the basis for proposing a simple model of the probability of a crash at the moment when a conflict is still progressing. Then, a model applicable to heterogeneous conditions is derived; and the model’s relevance, useful properties, and limitations are discussed. The published concepts and study results that support the derived model are provided in the paper.The existing Maximum Likelihood Estimate (MLE) method and the Probability-Weighted Moments (PWM) method of estimating the probability of crash and the expected number of crashes based on the proposed theory are presented. Then, a new Single Parameter Estimation (SPE) method is proposed and evaluated with extensive Monte Carlo experiments. The performance of the MLE, PWM, and SPE methods are compared. The SPE method is found more accurate and efficient than the other two methods. Unlike the benchmark methods, the proposed method produces real estimates in each case.The most important outcome of the presented study is confirmation that traffic conflicts claimed based on sufficiently small threshold separation (such as Time to Collision) allow unbiased estimation of the expected number of crashes during the conflicts observation period. A practical procedure of estimating safety is proposed that identifies the longest suitable threshold separation for each case based on the trends in the estimation results.

Higher moments method for generalized Pareto distribution in flood frequency analysis

The generalized Pareto distribution (GPD) has proven to be the ideal distribution in fitting with the peak over threshold series in flood frequency analysis. Several moments-based estimators are applied to estimating the parameters of GPD. Higher linear moments (LH moments) and higher probability weighted moments (HPWM) are the linear combinations of Probability Weighted Moments (PWM). In this study, the relationship between them will be explored. A series of statistical experiments and a case study are used to compare their performances. The results show that if the same PWM are used in LH moments and HPWM methods, the parameter estimated by these two methods is unbiased. Particularly, when the same PWM are used, the PWM method (or the HPWM method when the order equals 0) shows identical results in parameter estimation with the linear Moments (L-Moments) method. Additionally, this phenomenon is significant when r ≥ 1 that the same order PWM are used in HPWM and LH moments method.

Development of probability wind speed map based on frequency analysis

This study performed a wind speed frequency analysis in order to develop a probability wind speed map. A total of 70 available stations, having a 10-min mean wind speed over a 20-year historical record were selected. Subsequently, the appropriate probability distribution and method of parameter estimation were determined. The application of the Generalized Extreme Value, Gumbel, Weibull, 2-Parameter Gamma, 3-Parameter Gamma and Log-Pearson Type III distribution were used to determine the appropriate distributions for Korea and were found to be useful for frequency analysis of extreme value. The Method of Moment and Probability Weighted Moment Method were used for the parameter estimation of the different distributions. Hence, results show that GEV distribution coupled with Probability Weighted Moment Method is the best fit for the wind speed frequency analysis in Korea. Also, from the results of a 100-year wind velocity frequency analysis, wind speed of Uljin station was calculated to be the highest at 44.8 m/s and Bonghwa station the lowest at 12.9 m/s. Lastly, probability wind speed map was also developed using inverse distance weighted method using ArcGIS.

Skew generalized extreme value distribution: Probability-weighted moments estimation and application to block maxima procedure

ABSTRACTFollowing the work of Azzalini (1985 and 1986) on the skew-normal distribution, we propose an extension of the generalized extreme value (GEV) distribution, the SGEV. This new distribution allows for a better fit of maxima and can be interpreted as both the distribution of maxima when maxima are taken on dependent data and when maxima are taken over a random block size. We propose to estimate the parameters of the SGEV distribution via the probability-weighted moment method. A simulation study is presented to provide an application of the SGEV on block maxima procedure and return level estimation. The proposed method is also implemented on a real-life data.

Comparison of precipitation extremes estimation using parametric and nonparametric methods

ABSTRACTDue to recent occurrences of extreme hydrological events in Central Europe, there is an increasing interest in more accurate prediction of return levels of such events. The precipitation records from six ombrographic stations operated by the Czech Hydrometeorological Institute were analysed in order to estimate the intensity-duration-frequency. Although the longest rainfall series consists of more than 40 years of measurements, the data set also contains records from newly established stations with only short-time series available. The impact of the series length on the estimation quality is part of this study. Parametric and nonparametric approaches to drawing samples are assumed. In the first case, we consider a threshold model and we estimate the unknown parameters using maximum likelihood and probability weighted moments methods. In the latter case, largest order statistics are considered and the bootstrap methodology is applied as a resampling technique together with the moment estimator of extreme value index.Editor D. Koutsoyiannis; Associate editor E. Volpi

An innovative estimation method regarding Weibull parameters for wind energy applications

Two-parameter Weibull distribution is widely-used and accepted as the most popular distribution for the evaluation of wind energy potential. However, it is well-known that parameter estimates have significant effects on the success of Weibull distribution for wind energy applications. Thus, in this paper, an innovative method, PWMBP (probability weighted moments based on the power density method), is developed and proposed for estimating the Weibull parameters in wind energy applications. The salient features of PWMBP compared to other common methods are presented and discussed. In particular, this method is expressed in analytical form and does not need extensive numerical calculations or any iterative procedure. Moreover, the performance of PWMBP is compared with six other commonly-used methods; the maximum likelihood, modified maximum likelihood, graphical, moment, power density and probability weighted moments methods for actual wind data, based on different time periods and regions according to various goodness of fit criteria, moreover energy output performance of these methods are examined. The obtained results indicate that PWMBP provides more accurate and efficient estimation than other methods in estimating the parameters of Weibull distribution. As a result, PWMBP can be used as an improved method to estimate the Weibull parameters for wind energy applications.

The effect of the generalized extreme value distribution parameter estimation methods in extreme wind speed prediction

The modeling and prediction of extreme values of geophysical variables, such as wind, ocean surface waves, sea level, temperature and river flow, has always been a field of main concern for engineers and scientists. The analysis of extreme wind speed particularly plays an important role in natural disasters’ preparedness, prevention, mitigation and management and in various ocean, environmental and civil engineering applications, such as the design of offshore platforms and coastal marine structures, coastal management, wind climate analysis and structural safety. The block maxima (BM) approach is fundamental for extreme value analysis. BM method is closely related to the generalized extreme value (GEV) distribution, which unifies the three asymptotic extreme value distributions into a single one. The most common methods used for the estimation of the GEV parameters are maximum likelihood (ML) and probability weighted moments methods. In this work, several very common as well some less known estimation methods are firstly assessed through a simulation analysis. The results of the analysis showed that the maximum product of spacings (MPS), the elemental percentile (EP), the ordinary entropy method and, in a lesser degree, the ML methods seem to be, in general, superior to the other examined methods with respect to bias, mean squared error and variance of the estimated parameters. The effects of the estimation methods have been also assessed with respect to the n-year design values of real wind speed measurements. The obtained results suggest that the MPS and EP methods, which are rather unknown to the engineering community, describe adequately well the extreme quantiles of the wind speed data samples.

우리나라 빈도별 신적설심 산정 및 적정 확률분포형 선정

본 연구에서는 우리나라의 빈도별 신적설심 지도를 작성하기 위해 빈도분석을 수행하였다. 빈도분석을 위해 우리나라 90개 기상관측소중 20년치 이상의 신적설심 자료를 보유한 66개 기상관측소를 선정하였고 연최대치 신적설심을 수집하였다. 우리나라에 가장 적합한 확률분포형과 매개변수추정방법을 선정하기 위해 극치빈도분석에 널리 사용되는 6가지 확률분포형인 Generalized Extreme Value(GEV), Gumbel, Weibull, 2변수Gamma, 3변수Gamma, Log-Pearson Type III가 적용되었으며 매개변수 추정방법으로는 모멘트법, 확률가중 모멘트법이 적용되었다. 적용결과, 우리나라에는 확률가중모멘트법에 의한 Weibull 분포가 가장 적합한 것으로 나타났다. 66개 기상관측소별 빈도분석 결과를 기반으로 ArcGIS의 크리깅(Kriging) 기법을 통해 결과 값을 공간분포하여 우리나라의 빈도별 신적설심 지도를 작성하였다. 신적설심 지도는 적설심이 관측되지 않는 지역을 포함한 우리나라 전역의 폭설에 대비한 건축구조물 설계에 유용하게 활용될 것으로 판단된다. In this study, snowfall frequency analysis was performed to determine the best fit distribution applicable for Korean in order to make a snowfall frequency map. The available 66 stations having a snowfall over 20 years were selected and the annual maximum snowfall were collected. Generalized Extreme Value, Gumbel, Weibull, 2-Parameter Gamma, 3-Parameter Gamma and Log-Pearson Type III distribution which are useful to extreme value frequency analysis were applied to determine appropriate distribution for Korea. The Method of Moment and the Probability Weighted Moment Method were applied as parameters of the distributions. The results show that Weibull distribution by Probability Weighted Moment Method is the best fit for snowfall frequency analysis. Mapping snowfall frequency were accomplished using snowfall frequency according to the available 66 stations through Kriging method by ArcGIS. Snowfall frequency map is expected to be useful in the design of building structures against heavy snow loading throughout Korea most especially in ungauged areas.

Frequency Analysis of Nonidentically Distributed Hydrometeorological Extremes Associated with Large-Scale Climate Variability Applied to South Korea

AbstractIn the frequency analyses of extreme hydrometeorological events, the restriction of statistical independence and identical distribution (iid) from year to year ensures that all observations are from the same population. In recent decades, the iid assumption for extreme events has been shown to be invalid in many cases because long-term climate variability resulting from phenomena such as the Pacific decadal variability and El Niño–Southern Oscillation may induce varying meteorological systems such as persistent wet years and dry years. Therefore, the objective of the current study is to propose a new parameter estimation method for probability distribution models to more accurately predict the magnitude of future extreme events when the iid assumption of probability distributions for large-scale climate variability is not adequate. The proposed parameter estimation is based on a metaheuristic approach and is derived from the objective function of the rth power probability-weighted sum of observations in increasing order. The combination of two distributions, gamma and generalized extreme value (GEV), was fitted to the GEV distribution in a simulation study. In addition, a case study examining the annual hourly maximum precipitation of all stations in South Korea was performed to evaluate the performance of the proposed approach. The results of the simulation study and case study indicate that the proposed metaheuristic parameter estimation method is an effective alternative for accurately selecting the rth power when the iid assumption of extreme hydrometeorological events is not valid for large-scale climate variability. The maximum likelihood estimate is more accurate with a low mixing probability, and the probability-weighted moment method is a moderately effective option.

RCP4.5 시나리오에 따른 IDF 곡선 분석: 우리나라 30개 저수지를 대상으로

본 연구의 목적은 우리나라 30개 저수지의 미래 IDF (Intensity-Duration-Frequency) 곡선을 산정하여 기후변화로 인한 영향을 분석하는 것이다. RCP (Representative Concentration Pathways) 시나리오를 기반으로 하는 미래 강수자료에 내재하는 계통적 오차를 분위사상법을 이용하여 보정하였고, 미래를 3기간(2025s: 2011~2040, 2055s: 2041~2070, and 2085s: 2071~2100)으로 나누어 미래 강수량의 변화를 분석하였다. 강수자료의 최적 분포형과 매개변수 추정 방법으로 Gumbel 분포와 확률가중모멘트법을 선정하였다. 한강유역, 금강유역, 영산강유역의 강우강도는 현재 1990s 기간에 비해 미래 2085s 기간에 크게 증가하였고, 낙동강유역의 강우강도는 현재 1990s 기간에 비해 미래 2025s 기간에 크게 증가하는 것으로 분석되었다. 본 연구에서 새롭게 도출한 IDF 곡선은 기후변화를 구조물 설계에 반영함으로써 농업용 저수지의 안정성을 증진시킬 수 있을 것이라 사료된다. The main objective of this study is to evaluate the impact of climate change on Intensity-Duration-Frequency (IDF) curves for the selected 30-Reservoirs in South Korea. A bias inherent in future precipitation data based on Representative Concentration Pathways (RCP) scenario was corrected by a quantile mapping method. The trends in future precipitation changes were analyzed for the three different time periods (correspond to 2025s: 2011~2040, 2055s: 2041~2070, and 2085s: 2071~2100). Gumbel distribution was selected as the probability distribution and parameters were estimated by probability weighted moment method. Rainfall intensity of Han, Geum and Yeongsan river watersheds was greatly increased in 2085s period than 1990s period. Rainfall intensity of Nakdong river watershed appeared to increase sharply in 2025s period than 1990s period. The newly developed IDF curve might enhance the stability of agricultural reservoirs through considering climate change impact into structure design process.

Stochastic analyses of maximum daily rainfall series recorded at two stations across the Mediterranean Sea

Independence, stationarity, homogeneity, trend, and periodicity tests are applied on 48-year-long complete and 79-year-long incomplete maximum daily rainfall series recorded at Alexandria, Egypt, and on 61-year-long maximum daily rainfall series recorded at Antalya, Turkey, which are located at the southeastern and northeastern shores of the Mediterranean Sea. The results indicate no significant trend and no periodicity in mean, and both series are independent and homogeneous. Linear regression trend test applied to the 10 % highest part of the Alexandria series indicated a significant increasing trend. Next, frequency analysis is applied on each of these series by the probability distributions of Gumbel, general extreme-values, three-parameter log-normal, Pearson-3, log-Pearson-3, log-logistic, generalized Pareto, and Wakeby. The distributions, except for the generalized Pareto and Wakeby, pass the χ2 and Kolmogorov–Smirnov goodness-of-fit tests at 90 % probability. By visual inspection of the plots of histograms together with the probability density functions, and by the results of the χ2, Kolmogorov–Smirnov, and probability plot correlation coefficient tests, the general extreme-value distribution whose parameters are computed by the method of probability-weighted moments is deemed to be suitable for these two maximum daily rainfall series.

회귀분석 방법을 이용한 서울지점 극치 수문량 산정

본 연구에서는 서울지점의 목표연도별(2020, 2030, 2040년) 재현기간에 따른 확률강수량을 산정하기 위하여 1973년부터 2009년까지 지속시간 1시간 및 24시간에 대한 연 최대 강수량 자료를 사용하였다. Gumbel 분포를 이용해 빈도해석을 실시하였으며, 확률가중모멘트법을 사용하여 분포형의 위치매개변수와 축척매개변수를 산정하였다. 서울지점의 누적평균 극치강수량에 대하여 선형회귀분석을 실시하는 방법과 한강유역에 속하는 14개 지점들의 누적평균 극치강수량에 대하여 Logistic 회귀분석을 실시하는 방법을 이용해 각 방법에 따른 확률강수량을 정상성임을 가정하여 산정된 재현기간에 따른 확률강수량과 비교하였다. 선형회귀분석에 비해 Logistic 회귀분석의 결정계수가 더욱 높은 것으로 나타났고, 확률강수량의 경우, 선형회귀분석을 통한 확률강수량은 정상성에 비하여 매우 높게 나타남을 보였으나, Logistic 회귀분석에서 최대편차를 이용해 산정된 확률강수량의 경우 선형회귀분석에 비해 비교적 안정적인 증가를 보였다. 확률강수량 산정에 있어 정상성 가정에 기초한 전형적인 빈도해석과 비교하였을 때, Logistic 회귀분석은 극치의 선형 증가 거동을 반영하면서도 선형회귀분석을 사용한 목표연도별 확률강우량 산정시 나타나는 지속적 선형 증가의 문제를 보완할 수 있을 것으로 판단된다. In this study, we estimated probable precipitation amounts at Seoul for the target year(2020, 2030, 2040 year) using the 1 and 24 hour annual maximum precipitation data from 1973 through 2009. The Gumbel distribution was chosen and the probability weighted moment method was used to estimate model parameters. The behavior of the mean of extreme precipitation data, scale parameter, and location parameter amount using the linear regression and the logistic regression methods. The probable precipitation amount for the target years using the linear regression methods is much higher than that using a stationary frequency analysis. The probable precipitation amount using the logistic regression showed stable increase but probable precipitation amounts was almost same for different target years since the behavior of logistic curve converges before 2040 year. In estimating probable precipitation, the logistic regression is able to reflect the increase behavior of hydrologic extreme reasonably while probable precipitation amounts for target years using a linear regression method have continuous increase.

Estimation of second order parameters using probability weighted moments

The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ , which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator of ρ looks especially competitive when |ρ | is small.

연최대강우량의 대표확률분포형 결정을 위한 Jackknife기법의 적용

본 연구에서는 전국의 30년 이상의 강우관측기록을 보유하고 있는 기상청 산하 56개 강우관측소의 연 최대치 강우자료들로부터 확률분포형에 대하여 모멘트법, 최우추정법, 확률가중모멘트법을 이용하여 모수를 추정하고, 그 모수의 범위와 확률변수의 범위에 대한 적정성을 알아보았다. 적정성이 있는 모수를 대상으로 적합도 검정법인 x<TEX>$^2$</TEX>-검정, K-S검정, Cramer von Mises (CVM)검정, Probability Plot Correlation Coefficient (PPCC) 검정을 실시한 결과 중, 최근 연구에서 많이 이용되고 있고 표본자료의 크기가 작거나 왜곡된 자료일 경우에도 비교적 안정적인 결과를 얻을 수 있는 확률가중모멘트법과 상관계수에 의한 검정인 PPCC검정을 통과한 분포형을 우선적으로 적합도 평가 대상 분포형으로 선정하였다. 선정된 분포형을 대상으로 적합도 평가기준인 SLSC, MLL, AIC를 적용하여 적합도 평가를 실시하여 대표확률분포형 후보군을 추출하였다. 대표확률분포형 후보군으로 선정된 확률분포형에 대하여 resampling방법인 Jackknife기법을 적용하여 변동성을 파악하고, 변동성이 가장 작게 나타난 분포형을 그 지점의 대표확률분포형으로 결정하였다. 본 논문에서는 분석 결과의 분량을 감안하여 대표적으로 서울, 강릉, 대구, 전주, 부산 지점에 대해 작성하였으며, 확률강우량의 변동성이 가장 작은 확률분포형을 56개 지점의 각 지점 대표확률분포형으로 제시하였으며, Gumbel 분포(GUM)의 선정 비율이 지속기간 12시간, 24시간에 대해 각각 41 %, 32 %로 가장 높게 나타났다. 본 연구에서는 적합도 평가를 함에 있어서 객관적 정량화가 가능한 세 가지 기준과 Jackknife기법을 이용한 새로운 확률분포형 선정의 가능성을 제시하였다. In this study, basic data is consisted annual maximum rainfall at 56 stations that has the rainfall records more than 30years in Korea. The 14 probability distributions which has been widely used in hydrologic frequency analysis are applied to the basic data. The method of moments, method of maximum likelihood and probability weighted moments method are used to estimate the parameters. And 4-tests (chi-square test, Kolmogorov-Smirnov test, Cramer von Mises test, probability plot correlation coefficient (PPCC) test) are used to determine the goodness of fit of probability distributions. This study emphasizes the necessity for considering the variability of the estimate of T-year event in hydrologic frequency analysis and proposes a framework for evaluating probability distribution models. The variability (or estimation error) of T-year event is used as a criterion for model evaluation as well as three goodness of fit criteria (SLSC, MLL, and AIC) in the framework. The Jackknife method plays a important role in estimating the variability. For the annual maxima of rainfall at 56 stations, the Gumble distribution is regarded as the best one among probability distribution models with two or three parameters.

Estimation of asymptotic variances of quantiles for the generalized logistic distribution

In this study, the parameter estimations for the 3-parameter generalized logistic (GL) distribution are presented based on the methods of moments (MOM), maximum likelihood (ML), and probability weighted moments (PWM). The asymptotic variances of the MOM, ML, and PWM quantile estimators for the GL distribution are expressed as functions of the sample size, return period, and parameters. A Monte Carlo simulation was performed to verify the derived expressions for variances and covariances between parameters and to evaluate the applicability of the derived asymptotic variances of quantiles for the MOM, ML and PWM methods. The simulation results generally show good agreement with the analytical results estimated from the asymptotic variances of parameters and quantiles when the shape parameter (β) of the GL distribution is between −0.10 and 0.10 for the MOM method and between −0.25 and 0.45 for the ML and PWM methods, respectively. In addition, the actual sample variances and the root mean square error (RMSE) of asymptotic variances of quantiles for various sample sizes, return periods, and shape parameters were presented. In order to evaluate the applicability of the estimation methods to real data and to compare the values of estimated parameter, quantiles, and confidence intervals based on each parameter estimation method, the GL distribution was fitted to the 24-h annual maximum rainfall data at Pohang, Korea.