Robust Inference for Inverse Stochastic Dominance

The notion of inverse stochastic dominance is gaining increasing support in risk, inequality, and welfare analysis as a relevant criterion for ranking distributions, which is alternative to the standard stochastic dominance approach. Its implementation rests on comparisons of two distributions’ quantile functions, or of their multiple partial integrals, at fixed population proportions. This article develops a novel statistical inference model for inverse stochastic dominance that is based on the influence function approach. The proposed method allows model-free evaluations that are limitedly affected by contamination in the data. Asymptotic normality of the estimators allows to derive tests for the restrictions implied by various forms of inverse stochastic dominance. Monte Carlo experiments and an application promote the qualities of the influence function estimator when compared with alternative dominance criteria.     

Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit

In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small simulation. From the simulation, the performance of the new estimator is roughly comparable with maximum likelihood for positive values of the shape parameter and often much better than maximum likelihood for negative values.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.     

Consistent Nonparametric Tests for Lorenz Dominance

This article proposes consistent nonparametric methods for testing the null hypothesis of Lorenz dominance. The methods are based on a class of statistical functionals defined over the difference between the Lorenz curves for two samples of welfare-related variables. We present two specific test statistics belonging to the general class and derive their asymptotic properties. As the limiting distributions of the test statistics are nonstandard, we propose and justify bootstrap methods of inference. We provide methods appropriate for case where the two samples are independent as well as the case where the two samples represent different measures of welfare for one set of individuals. The small sample performance of the two tests is examined and compared in the context of a Monte Carlo study and an empirical analysis of income and consumption inequality.     

Jackknife empirical likelihood-based inferences for Lorenz curve with kernel smoothing

The Lorenz curve describes the wealth proportion for an income-ordered population. In this paper, we introduce a kernel smoothing estimator for the Lorenz curve and propose a smoothed jackknife empirical likelihood method for constructing confidence intervals of Lorenz ordinates. Extensive simulation studies are conducted to evaluate finite sample performances of the proposed methods. A real dataset of Georgia professor’s income is used to illustrate the proposed methods.     

洛伦兹曲线的半参数估计

洛伦兹曲线与基尼系数是研究社会收入分配差异的重要工具.社会收入分配是一个复杂的过程,用尽可能精确的曲线给出洛伦兹曲线的估计进而给出基尼系数的估计,历来是统计学者和经济学者的工作目标.基于将参数方法与非参数方法相结合的思想给出洛伦兹曲线的半参数估计,进而导出基尼系数的估计,并据此进行了实证分析.     

On Distributions of Exceedances Based on Generalized Order Statistics

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.     

A hierarchy of lorenz curves based on the generalized tukey's lambda distribution

A hierarchy of Lorenz curves based on the generalized Tukey's Lambda distribution is proposed. Representations of the corresponding distribution and density function are also provided, together with popular inequality measures. Estimation methods are suggested. Finally, a comparison with other parametric families of Lorenz curves is established.     

Testing sphericity and intraclass covariance structures under a growth curve model in high dimension

In this article, we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a growth curve model. The maximum likelihood estimator (MLE) for the mean in a growth curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.     

M-Estimator of a Generalized Linear Model with Measurement Errors

In this article, we propose a generalized linear model and estimate the unknown parameters using robust M-estimator. Under suitable conditions and by the strong law of large numbers and central limits theorem, the proposed M-estimators are proved to be consistent and asymptotically normal. We also evaluate the finite sample performance of our estimator through a Monte Carlo study.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

Comparison of Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM) as General Platforms for Distribution Fitting

Distribution fitting is widely practiced in all branches of engineering and applied science, yet only a few studies have examined the relative capability of various parameter-rich families of distributions to represent a wide spectrum of diversely shaped distributions. In this article, two such families of distributions, Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM), are compared. For a sample of some commonly used distributions, each family is fitted to each distribution, using two methods: fitting by minimization of the L ₂ norm (minimizing density function distance) and nonlinear regression applied to a sample of exact quantile values (minimizing quantile function distance). The resultant goodness-of-fit is assessed by four criteria: the optimized value of the L ₂ norm, and three additional criteria, relating to quantile function matching. Results show that RMM is uniformly better than GLD. An additional study includes Shore's quantile function (QF) and again RMM is the best performer, followed by Shore's QF and then GLD.     

Interval Estimations for the Mean of the Generalized Poisson and Generalized Negative Binomial Distributions

In this article, we extend the Wald, score, skewness-corrected score, likelihood ratio, and mid-P intervals for the means of the generalized Poisson and generalized negative binomial distributions. These distributions are the members of the discrete version of the natural exponential family (NEF) with cubic variance function (CVF). Also, the coverage probabilities, the distal and mesial noncoverage probabilities, and the lengths of the proposed confidence intervals are estimated by means of a Monte Carlo simulation study. Finally, some practical examples are provided to show the applicability of the proposed intervals in applied studies.     

Generalized elliptical distributions

A family of distributions generated by an operator acting on generalized normal density is introduced. This family contains as particular cases many known distributions, including the generalized normal, generalized t, and generalized gamma distributions. Several mathematical properties of the family (including expansions, characteristic function, moments, cumulants, and order statistics properties) are derived. Estimation procedures are derived too by the method of moments, method of maximum likelihood, and the method of empirical characteristic function. A real data application is presented. Finally, extensions to the multivariate case are outlined.     

Statistical inference for the difference of two Lorenz curves

Testing the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

A hierarchy of lorenz curves based on the generalized tukey's lambda distribution

A hierarchy of Lorenz curves based on the generalized Tukey's Lambda distribution is proposed. Representations of the corresponding distribution and density function are also provided, together with popular inequality measures. Estimation methods are suggested. Finally, a comparison with other parametric families of Lorenz curves is established.