Modelling of Wage and Income Distributions Using the Method of L-Moments

Commonly used statistical procedure to describe the observed statistical sets is to use their conventional moments or cumulants. When choosing an appropriate parametric distribution for the data set is typically that parameters of a parametric distribution are estimated using the moment method of creating a system of equations in which the sample conventional moments lay in the equality of the corresponding moments of the theoretical distribution. However, the moment method of parameter estimation is not always convenient, especially for small samples. An alternative approach is based on the use of other characteristics, which the author calls L-moments. L-moments are analogous to conventional moments, but they are based on linear combinations of order statistics, i.e., L-statistics. Using L-moments is theoretically preferable to the conventional moments and consists in the fact that L-moments characterize a wider range of distribution. When estimating from sample L-moments, L-moments are more robust to the presence of outliers in the data. Experience also shows that, compared to conventional moments, L-moments are less prone to bias of estimation. Parameter estimates obtained using L-moments are mainly in the case of small samples often even more accurate than estimates of parameters made by maximum likelihood method. Using the method of L-moments in the case of small data sets from the meteorology is primarily known in statistical literature. This paper deals with the use of L-moments in the case for large data sets of income distribution （individual data） and wage distribution （data are ordered to form of interval frequency distribution of extreme open intervals）. This paper also presents a comparison of the accuracy of the method of L-moments with an accuracy of other methods of point estimation of parameters of parametric probability distribution in the case of large data sets of individual data and data ordered to form of interval frequency distribution.

L-Moments and TL-Moments as an Alternative Tool of Statistical Data Analysis

Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics.

A New Class of L-Moments Based Calibration Variance Estimators

Variance is one of themost important measures of descriptive statistics and commonly used for statistical analysis.The traditional second-order central moment based variance estimation is a widely utilized methodology.However,traditional variance estimator is highly affected in the presence of extreme values.So this paper initially,proposes two classes of calibration estimators based on an adaptation of the estimators recently proposed by Koyuncu and then presents a new class of L-Moments based calibration variance estimators utilizing L-Moments characteristics(L-location,Lscale,L-CV)and auxiliary information.It is demonstrated that the proposed L-Moments based calibration variance estimators are more efficient than adapted ones.Artificial data is considered for assessing the performance of the proposed estimators.We also demonstrated an application related to apple fruit for purposes of the article.Using artificial and real data sets,percentage relative efficiency(PRE)of the proposed class of estimators with respect to adapted ones are calculated.The PRE results indicate to the superiority of the proposed class over adapted ones in the presence of extreme values.In this manner,the proposed class of estimators could be applied over an expansive range of survey sampling whenever auxiliary information is available in the presence of extreme values.

Regional Frequency Analysis of Significant Wave Heights Based onL-moments

L-moments are defined as linear combinations of probability-weighted moments, They are, virtually unbiased for small samples, and perform well in parameter estimation, choice of the distribution type and regional analysis. The traditional methods of determining the design wave heights for planning marine structures use data only from the site of interest. Regional frequency analysis gives a new approach to estimate quantile by use of the homogeneous neighborhood informatian. A regional frequency analysis based on L-moments with a case study of the California coast is presented. The significant wave height data for the California coast is offered by NDBC. A 6-site region without 46023 is considered to be a homogeneous region, whose optimal regional distribution is Pearson Ⅲ. The test is conducted by a simulation process. The regional quantile is compared with the at-site quantile, and it is shown that efficient neighborhood information can be used via regional frequency analysis to give a reasonable estimation of the site without enough historical data.

L-Moments Based Calibrated Variance Estimators Using Double Stratified Sampling

Variance is one of the most vital measures of dispersion widely employed in practical aspects.A commonly used approach for variance estimation is the traditional method of moments that is strongly influenced by the presence of extreme values,and thus its results cannot be relied on.Finding momentum from Koyuncu’s recent work,the present paper focuses first on proposing two classes of variance estimators based on linear moments(L-moments),and then employing them with auxiliary data under double stratified sampling to introduce a new class of calibration variance estimators using important properties of L-moments(L-location,L-cv,L-variance).Three populations are taken into account to assess the efficiency of the new estimators.The first and second populations are concerned with artificial data,and the third populations is concerned with real data.The percentage relative efficiency of the proposed estimators over existing ones is evaluated.In the presence of extreme values,our findings depict the superiority and high efficiency of the proposed classes over traditional classes.Hence,when auxiliary data is available along with extreme values,the proposed classes of estimators may be implemented in an extensive variety of sampling surveys.

Forecasting the Rainfall Pattern on Upstream of Hirakud Reservoir Using L-Moment for Accessing the Inflow

Changes in the rainfall pattern are a challenge for filling schedule of reservoir, when it is fulfilling various demands. In monsoon fed reservoirs, the target remains for attaining full reservoir capacity in order to meet various demands during non-monsoon period and the flood control. The planners always eye towards the inflow trend and perspective frequency of rainfall in order to counter the extreme events. In this study, the case of Hirakud reservoir of Mahanadi basin of India is considered as this reservoir meets various demands as well as controls devastating floods. The inflow trend has been detected by using Mann Kendall test. The frequency analysis of monthly rainfall is calculated using L-moment program for finalizing a regional distribution. The falling trend in inflow to reservoir is visualized in the month of July and August. The Wakeby distribution is found suitable for the monthly rainfall of July, September and October, where as in June and August, General Extreme Value (GEV), General Normal (GN) and Pearson Type-III (PT-III) distributions are found suitable. The regional growth factors for the 20, 40, 50 and 100-year return period rain-falls along with inflow to reservoir observed between 1958-2010 are calculated in this study as a referral for reservoir operation policy.

Prediction of Drought with Artificial Neural Networks and Genetic Algorithms Using Precipitation by Remote Perception

The aim of the study is to determine the performance of the regional agricultural drought prediction by the model of ANN(Artificial Neural Networks)type NARX,using the SPI(Standardized Precipitation Index),SPEI(Precipitation Index Standardized Evapotranspiration),VCI(Vegetation Condition Index)and GCI(Global Climate Indexes).There have been determined 10 homogeneous regions through RAF(regional frequency analysis)and L-moments,defining the most arid region and the index representing their respective time scale(SPEI 6 months)which responds to the growth and development of vegetation in the basin correlation Pearson equal to 0.58.Monthly rainfall and temperatures correspond to PISCO data prepared by SENAMHI-Peru,with space resolution of 0.05 degrees.For prediction,they have determined two groups,the first to build the model with 80% of the registration and validation of the model and the hypothesis with the remaining 20%.The results have been satisfactory prediction accepting the null hypothesis.

A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand

In this study, the four-parameter kappa distribution with L-Moments estimation has been used to fit the distribution of weekly rainfall data at Lampao in the Chi River Basin, Thailand. The weekly precipitations with probabilities 0.75 were estimated, and the extreme rainfall estimates obtained can be used for water and agriculture management.

Analysis of Flow Estimation Methods for Small Hydropower Schemes in Bua River

Any hydropower project requires an ample availability of stream flow data. Unfortunately, most of the hydropower projects especially small hydropower projects are conducted on ungauged river and consequently hydrologists have for a longtime used stream flow estimation methods using the mean annual flows to gauge rivers. Unfortunately flow estimation methods which include the runoff data method, area ratio method and the correlation flow methods employ a lot of assumptions which affect their uncertainty. This study was conducted on Bua River in Malawi to unveil the uncertainties of these flow estimation methods. The study was done on a well gauged catchment in order to highlight the variations between the observed, true stream flows and the estimated stream flows for uncertainty analysis. After regionalizing the homogenous sites, catchments using L-moments, an uncertainty analysis was done which showed that the area method is better followed by the correlating flow method and lastly the runoff data method in terms of bias, accuracy and uncertainty.

Spatial Variability and Cartography of Maximum Annual Daily Rainfall under Different Return Periods in Northern Algeria

The estimation of precipitation quantiles has always been an area of great importance to meteorologists, hydrologists, planners and managers of hydrotechnical infrastructures. In many cases, it is necessary to estimate the values relating to extreme events for the sites where there is little or no measurement, as well as their return periods. A statistical approach is the most used in such cases. It aims to find the probability distribution that best fits the maximum daily rainfall values. In our study, 231 rainfall stations were used to regionalize and find the best distribution for modeling the maximum daily rainfall in Northern Algeria. The L-moments method was used to perform a regionalization based on discordance criteria and homogeneity test. It gave rise to twelve homogeneous regions in terms of LCoefficient of variation(L-CV), L-Skewness(L-CS) and L-Kurtosis(L-CK). This same technique allowed us to select the regional probability distribution for each group using the Z statistic. The generalized extreme values distribution(GEV) was selected to model the maximum daily rainfall of 10 groups located in the north of the steppe region and the generalized logistic distribution(GLO) for groups representing the steppes of Central and Western Algeria. The study of uncertainty by the bias and RMSE showed that the regional approach is acceptable. We have also developed maximum daily rainfall maps for 2, 5, 10, 20, 50 and 100 years return periods. We relied on a network of 255 rainfall stations. The spatial variability of quantiles was evaluated by semi-variograms. All rainfall frequency models have a spatial dependence with an exponential model adjusted to the experimental semi-variograms. The parameters of the fitted semi-variogram for different return periods are similar, throughout, while the nugget is more important for high return periods. Maximum daily rainfall increases from South to North and from West to East, and is more significant in the coastal areas of eastern Algeria where it exceeds 170 mm for a return period of 100 years. However, it does not exceed 50 mm in the highlands of the west.

Accounting for the Effects of Climate Variability in Regional Flood Frequency Estimates in Western Nigeria

Extreme flood events are becoming more frequent and intense in recent times, owing to climate change and other anthropogenic factors. Nigeria, the case-study for this research experiences recurrent flooding, with the most disastrous being the 2012 flood event that resulted in unprecedented damage to infrastructure, displacement of people, socio-economic disruption, and loss of lives. To mitigate and minimize the impact of such floods now and in the future, effective planning is required, underpinned by analytics based on reliable data and information. Such data are seldom available in many developing regions, owing to financial, technical, and organizational drawbacks that result in short-length and inadequate historical data that are prone to uncertainties if directly applied for flood frequency estimation. This study applies regional Flood Frequency Analysis (FFA) to curtail deficiencies in historical data, by agglomerating data from various sites with similar hydro-geomorphological characteristics and is governed by a similar probability distribution, differing only by an “index-flood”;as well as accounting for climate variability effect. Data from 17 gauging stations within the Ogun-Osun River Basin in Western Nigeria were analysed, resulting in the delineation of 3 sub-regions, of which 2 were homogeneous and 1 heterogeneous. The Generalized Logistic distribution was fitted to the annual maximum flood series for the 2 homogeneous regions to estimate flood magnitudes and the probability of occurrence while accounting for climate variability. The influence of climate variability on flood estimates in the region was linked to the Madden-Julian Oscillation (MJO) climate indices and resulted in increased flood magnitude for regional and direct flood frequency estimates varying from 0% - 35% and demonstrate that multi-decadal changes in atmospheric conditions influence both small and large floods. The results reveal the value of considering climate variability for flood frequency analysis, especially when non-stationarity is established by homogeneity analysis.

Development of Wages and Salaries of Czech Employees During the Last Period by Educational Attainment and Forecasts for 2011 and 2012

This paper deals with the development of sample characteristics of wage distribution in recent years in the Czech Republic by the highest educational attainment. Gross monthly wage is the variable investigated. We distinguish the following scale of the highest educational attainment： primary and incomplete education, secondary education without GCSE, secondary education with GCSE, higher vocational and bachelor education and tertiary education. Forecasts of wage distribution have been developed for the next two years for all of these categories. Three-parametric lognormal curve formed the basis of the theoretical probability distribution. Parameter values of relevant three-parametric lognormal curves were then estimated using the method of L-moments of parameter estimation. Forecasts of sample values of L-moments were calculated using trend analysis of their past development and the parameters of three-parametric Iognormal curves for forecasts of wage distribution were calculated using the predicted values of the first three sample L-moments. We have obtained the forecasts of wage distribution by the highest educational attainment on the basis of these probability density functions.

基于非参数与L-Moment估计的股市动态极值ES风险测度研究

通过运用带宽非参数方法、AR-GARCH模型对时间序列的条件均值、条件波动性进行建模估计出标准残差序列,再运用L-Moment与MLE(maximum Likelihood estimation)估计标准残差的尾部的GPD参数,进而运用实验方法测度出风险VaR(value at Risk)及ES(ExpectedShortfall),最后运用Back-Testing方法检验测度准确性。结果表明,基于带宽的非参数估计模型比GARCH簇模型在测度ES上具有更高的可靠性;基于非参数模型与L-Moment的风险测度模型能够有效测度沪深股市的动态VaR与ES。

Applications of Burr Ⅲ-Weibull quantile function in reliability analysis

This paper introduces a new family of distributions defined in terms of quantile function.The quantile function introduced here is the sum of quantile functions of life time distributions called Burr Ⅲ and Weibull.Different distributional characteristics and reliability properties are included in the study.Method of Least Square and Method of L-moments are applied to estimate the parameters of the model.Two real life data sets are used to illustrate the performance of the model.

Spatio-temporal trend and statistical distribution of extreme precipitation events in Huaihe River Basin during 1960-2009

Based on the daily precipitation data of 27 meteorological stations from 1960 to 2009 in the Huaihe River Basin, spatio-temporal trend and statistical distribution of extreme precipitation events in this area are analyzed. Annual maximum series （AM） and peak over threshold series （POT） are selected to simulate the probability distribution of extreme pre- cipitation. The results show that positive trend of annual maximum precipitation is detected at most of used stations, only a small number of stations are found to depict a negative trend during the past five decades, and none of the positive or negative trend is significant. The maximum precipitation event almost occurred in the flooding period during the 1960s and 1970s. By the L-moments method, the parameters of three extreme distributions, i.e., Gen- eralized extreme value distribution （GEV）, Generalized Pareto distribution （GP） and Gamma distribution are estimated. From the results of goodness of fit test and Kolmogorov-Smirnov （K-S） test, AM series can be better fitted by GEV model and POT series can be better fitted by GP model. By the comparison of the precipitation amounts under different return levels, it can be found that the values obtained from POT series are a little larger than the values from AM series, and they can better simulate the observed values in the Huaihe River Basin.