U-Statistics and Their Asymptotic Results for Some Inequality and Poverty Measures

U-statistics form a general class of statistics that have certain important features in common. This class arises as a generalization of the sample mean and the sample variance, and typically members of the class are asymptotically normal with good consistency properties. The class encompasses some widely used income inequality and poverty measures, in particular the variance, the Gini index, the poverty rate, the average poverty gap ratios, the Foster-Greer-Thorbecke index, and the Sen index and its modified form. This paper illustrates how these measures come together within the class of U-statistics, and thereby why U-statistics are useful in econometrics.     

Estimating the error variance in nonparametric regression by a covariate-matched u-statistic

For nonparametric regression models with fixed and random design, two classes of estimators for the error variance have been introduced: second sample moments based on residuals from a nonparametric fit, and difference-based estimators. The former are asymptotically optimal but require estimating the regression function; the latter are simple but have larger asymptotic variance. For nonparametric regression models with random covariates, we introduce a class of estimators for the error variance that are related to difference-based estimators: covariate-matched U-statistics. We give conditions on the random weights involved that lead to asymptotically optimal estimators of the error variance. Our explicit construction of the weights uses a kernel estimator for the covariate density.     

Variance estimation for a low income proportion

Summary. Proportions below a given fraction of a quantile of an income distribution are often estimated from survey data in comparisons of poverty. We consider the estimation of the variance of such a proportion, estimated from Family Expenditure Survey data. We show how a linearization method of variance estimation may be applied to this proportion, allowing for the effects of both a complex sampling design and weighting by a raking method to population controls. We show that, for data for 1998–1999, the estimated variances are always increased when allowance is made for the design and raking weights, the principal effect arising from the design. We also study the properties of a simplified variance estimator and discuss extensions to a wider class of poverty measures.     

DISTRIBUTION-FREE TESTS BASED ON SUB-SAMPLE EXTREMA FOR TESTING AGAINST POSITIVE DEPENDENCE

A class of distribution-free tests based on U-statistics has been proposed for testing the null hypothesis of independence against positive quadrant dependence. The tests are based on U-statistics and the Kendall's-tau test belongs to this class.     

Characterizations and o-statistic representations of lu-statistics

The LU-statistics, that is, the class of all L-statistics that are also U-statistics, are characterised both via conditions on their coefficients and as linear combinations of the 0-statistics, which generalize the usual sample order statistics. The explicit computational formulas provided can be used to decide whether a given L-statistic is an LU-statistic and, if so, to find its kernel and its simplest 0-statistic representation.     

Rates of convergence for U-statistic processes and their bootstrapped versions

U-statistic processes are often used to detect a possible change in the distributions of the observations. We obtain the exact rate of convergence in some limit theorems for U-statistics. We discuss the application of the weighted bootstrap to change-point analysis. We show that the bootstrap approximation for U-statistics is as good as the large sample approximations using Gaussian processes. However, the bootstrap approximation is much better when the limit distributions are extreme values.     

Minimum variance rectangular designs for U-statistics

We consider a certain class of rectangular designs for incomplete U-statistics based on Latin squares and show it to be optimal with respect to the minimal variance criterion. We also show it to be asymptotically efficient when compared with the corresponding complete statistics, as well as uniformly more efficient than the random subset selection. We provide the necessary and sufficient conditions for the existence of our design and give some examples of applications.     

Parameter Estimation of Stable Distributions

ABSTRACT

In finance, economics, statistical physics, signal processing, telecommunications, etc., we frequently meet data sets with outliers that transport important information. α-stable distributions are found more suitable in modeling these kind of data. But the lack of simple and effective methods of estimating their parameters limited their applications to wider variety of fields. In this article we develop an unbiased estimator for the stable index α. With the structure of U-statistic, it inherits all the good statistical properties from U-statistics. A consistent estimator of its asymptotic variance is provided. The asymptotic normality of the given estimator holds when using the estimated variance for standardization. Simulation studies are performed. The results support our theory.     

On the variance estimate of a wilcoxon-mann whitney statistic for group ordered data

Two consistent estimators for the non-null variance of Wil-coxon-Mann-Whitney’s statistic applied to grouped ordered data, are considered. The first is based on U-statistics and the sec-ond is obtained by the Delta method. Some examples are given to demonstrate the extent of error when using a null variance esti-mate for constructing confidence intervals. It appears that the two consistent estimates are very close, but may both be disting-uishably larger or smaller than the null variance estimate.     

Analysing competing risks data with transformation models

We present a flexible class of marginal models for the cumulative incidence function. The semiparametric transformation model is utilized in a decomposition for the marginal failure probabilities which extends previous work on Farewell's cure model. Novel estimation, inference and prediction procedures are developed, with large sample properties derived from the theory of martingales and U-statistics. A small simulation study demonstrates that the methods are appropriate for practical use. The methods are illustrated with a thorough analysis of a prostate cancer clinical trial. Simple graphical displays are used to check for the goodness of fit.     

Invariance principles for jackknifing U-statistics for finite population sampling and some applications

For simple random sampling (without replacement) from a finite population, suitable stochastic processes are constructed from the entire sequence of jackknife estimators based on smooth functions of U-statistics and these are approximated (in distributions) by some Brownian bridge processes. Strong convergence of the Tukey estimator of the variance of a jackknife U-statistic has been interpreted suitably and established. Some applications of these results in sequential analysis relating to finite population sampling are also considered.     

中国农村贫困的动态研究

 基于CHNS的八轮农村家庭收入调查数据,本文使用贫困发生率和平方缺口指数对农村贫困进行了实证分析,并把贫困变动分解为长期贫困效应、脱贫效应和返贫效应。实证结论表明,虽然长期贫困发生率呈现明显下降趋势,但是近期长期贫困的平方缺口指数表现出一定程度上升;脱贫效应是减少贫困的主要因素,返贫效应则是加剧贫困的诱因。此外,匿名性与非匿名性的增长率显示,匿名性低估了低收入阶层的实际收入增长率,从而高估了实际贫困。     

贫困缺口总指数的构造、分解与应用

本文构造了一个新贫困指数 ,通过对 指数的因素分解及衍生指数的构造,提出了一些重要贫困指标的预测和推算办法。基于CHNS1991-2011年跨度20年的8轮调查数据进行实证分析,结果表明,我国农村近20年来总贫困缺口率呈现年均6.39%的缩减率,其贡献全部来自贫困人口总规模递减这一因素的影响,而另外两个因素,即平均贫困缺口水平和贫困差异度则呈现年均小幅上升趋势,表明减贫质量不容乐观;以现有减贫模式对我国农村2020年的平均贫困缺口水平、贫困不平等程度、贫困规模等指标进行预测显示,减贫目标依然任重道远;提出了采用精准减贫,重心在于提升贫困人口获取收入的能力和缩小贫困不平等程度的相关建议。     

A general class of poverty measures

Measurement of poverty in the spirit of Sen essentially consists of two steps, namely (1) the identification of the poor and (2) the aggregation of the income distribution of the poor into an indicator called poverty measure. We contribute to the second step from a mathematical and statistical point of view by defining a class of poverty measures and investigating its properties including sensitivity with respect to changes in the income distribution. Various poverty orderings are defined and their relation to stochastic dominance is investigated. Finally estimation of the class of poverty measures from individual and grouped data is considered.     

贫困测度指标及其评价

一、引言一般来说,研究贫困问题主要涉及三个方面:一是贫困的识别;二是贫困程度测定;三是反贫困效果判定以及相应的战略选择。贫困识别问题主要指贫困的基本范畴及贫困线的合理确定,关系到贫困集合的准确界定;贫困程度的测定主要指贫困指数的定义和计算,关系到贫困程度大小的判     

A note on biases of jackknife estimators of third central moments

In this paper we study the biases of jackknife estimators of central third moments which play an important role in improving the accuracy of the normal approximation. It has been found in simulation studies that the jackknife estimator of the skewness coefficient, into which the jackknife variance and third moment estimators are substituted, have downward biases. For the jackknife variance estimators, their asymptotic properties are precisely studied and their biases are discussed theoretically, Here we study the biases of the jackknife estimators of the central third moments for U-statistics theoretically, The results show that the biases are not always downward.     

A new test for bivariate distributions : exponential vs new-better-than-used alternative

This paper presents a new test for testing bivariate exponentiality against the bivariate new better than used (BNBU) class of non-exponential probability distributions. The test statistic is a function of U-statistics and hence asymptotically normally distributed and consistent.     

Nonparametric bayes estimates of estimable parameters with a dirichlet invarant process and inveriant u-statistics

Tne Bayes estimates of estimable parameters of arbitrary degree in the one sample case are obtained against a Dirichlet invariant. process prior and the squared error loss. We also oive the limits of Bayes estimates, which are related to the in- a- variant U-statistics. For a fixed distribution, the limits of Bayes estimates have the asymptotic normal distribution under certain conditjons.     

SMALL SAMPLE VARIANCE ESTIMATORS FOR U-STATISTICS

New unbiased estimators are presented for the dominant and lower-order terms of the variance expansion for U-statistics. In small samples these provide important corrections to the usual estimate of asymptotic standard error which is based on the leading term in the expansion. The new estimators for the first term cannot be recommended. The ordinary jackknife estimator is found to be more effective than the direct estimates of the separate terms.     

On Berry-Esseen theorem for some functions of U-statistics

The rate of convergence in the central limit theorem and in the random central limit theorem for some functions of U-statistics are established. The theorems refer to the asymptotic behaviour of the sequence {g(U_n),n≥1}, where g belongs to the class of all differentiable functions g such that g′εL(δ) and U_n is a U-statistics.