A new generalized Weibull distribution in income economic inequality curves

Researchers have been developing various extensions and modified forms of the Weibull distribution to enhance its capability for modeling and fitting different data sets. In this note, we investigate the potential usefulness of the new modification to the standard Weibull distribution called odd Weibull distribution in income economic inequality studies. Some mathematical and statistical properties of this model are proposed. We obtain explicit expressions for the first incomplete moment, quantile function, Lorenz and Zenga curves and related inequality indices. In addition to the well-known stochastic order based on Lorenz curve, the stochastic order based on Zenga curve is considered. Since the new generalized Weibull distribution seems to be suitable to model wealth, financial, actuarial and especially income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. Also, the estimation of parameters by maximum likelihood and moment methods is discussed. Finally, this distribution has been fitted to United States and Austrian income data sets and has been found to fit remarkably well in compare with the other widely used income models.     

Parametric Versus Nonparametric Inference on Zenga Index of Inequality: Issues and Evidence From Survey Data

Recent growing disparities suggests to move from inequality measures based on comparing the incomes of the less fortunate with the overall mean, as the Gini, to the new Zenga index, which instead contrasts the means of the less and the more wealthy subpopulations. After providing a thorough analysis of the theoretical and practical aspects of obtaining parametric and non-parametric confidence intervals for the Zenga inequality measure, we develop a cross-regional study based on the Swiss Income and Consumption Survey, wave 2005. Results show that coverage accuracy and average length of confidence intervals improve when the parametric model offers a good fit to the data.     

Contrasting the Gini and Zenga indices of economic inequality

The current financial turbulence in Europe inspires and perhaps requires researchers to rethink how to measure incomes, wealth, and other parameters of interest to policy-makers and others. The noticeable increase in disparities between less and more fortunate individuals suggests that measures based upon comparing the incomes of less fortunate with the mean of the entire population may not be adequate. The classical Gini and related indices of economic inequality, however, are based exactly on such comparisons. It is because of this reason that in this paper we explore and contrast the classical Gini index with a new Zenga index, the latter being based on comparisons of the means of less and more fortunate sub-populations, irrespectively of the threshold that might be used to delineate the two sub-populations. The empirical part of the paper is based on the 2001 wave of the European Community Household Panel data set provided by EuroStat. Even though sample sizes appear to be large, we supplement the estimated Gini and Zenga indices with measures of variability in the form of normal, t-bootstrap, and bootstrap bias-corrected and accelerated confidence intervals.     

The New Zenga curve in the context of reliability analysis

ABSTRACT

In this article, we look into the properties and characterizations of the New Zenga curve. The relationship of the curve with other measures of inequality as well as some reliability concepts are examined. Classification of lifetime distributions using the Zenga curve and an illustration for the behaviour of the curve using a survival data are also provided.     

On the decomposition by sources of the Zenga 1984 point and synthetic inequality indexes

In this work we provide a decomposition by sources of the inequality index \(\zeta \) defined by Zenga (Giornale degli Economisti e Annali di economia 43(5–6):301–326, 1984). The source contributions are obtained with the method proposed in Zenga et al. (Stat Appl X(1):3–31, 2012) and Zenga (Stat Appl XI(2):133–161, 2013), that allows to compare different inequality measures. This method is based on the decomposition of inequality curves. To apply this decomposition to the index \(\zeta \) and its inequality curve, we adapt the method to the “cograduation” table. Moreover, we consider the case of linear transformation of sources and analyse the corresponding results.     

On the parameters of Zenga distribution

In 2010 Zenga introduced a new three-parameter model for distributions by size that can be used to represent income, wealth, financial and actuarial variables. This paper proposes a summary of its main properties, followed by a focus on the interpretation of the parameters in terms of inequality. The scale parameter μ is equal to the expectation, and it does not affect the inequality, while the two shape parameters α and θ are inverse and direct inequality indicators respectively. This result is obtained through stochastic orders based on inequality curves. A procedure to generate a random sample from Zenga distribution is also proposed. The second part of this article looks at the parameter estimation. Analytical solution of method of moments is obtained. This result is used as a starting point of numerical procedures to obtain maximum likelihood estimates both on ungrouped and grouped data. In the application, three empirical income distributions are considered and the aforementioned estimates are evaluated. A comparison with other well-known models is provided, by the evaluation of three goodness-of-fit indexes.     

Relationship between the weighted distributions and some inequality measures

In this paper, we study the relationships between the weighted distributions and the parent distributions in the context of Lorenz curve, Lorenz ordering and inequality measures. These relationships depend on the nature of the weight functions and give rise to interesting connections. The properties of weighted distributions for general weight functions are also investigated. It is shown how to derive and to determine characterizations related to Lorenz curve and other inequality measures for the cases weight functions are increasing or decreasing. Some of the results are applied for special cases of the weighted distributions. We represent the reliability measures of weighted distributions by the inequality measures to obtain some results. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context by concentration curves. We also review and extend the problem of stochastic orderings and aging classes under weighting. Finally, the relationships between the weighted distribution and transformations are discussed.     

Asymptotic Confidence Intervals for a New Inequality Measure

This work aims at assessing, by simulation methods, the performance of asymptotic confidence intervals for Zenga's new inequality measure. The results are compared with those obtained on Gini's measure, perhaps the most widely used index for measuring inequality in income and wealth distributions. Our findings show that the coverage accuracy and the size of the confidence intervals for the two measures are very similar in samples from economic size distributions.     

Some remarks on Lorenz ordering-preserving functionals

Basing on two well-known characterization results on stochastic dominance and continuous majorization relation, the ordering-preserving property-with respect to Lorenz ordering-is deduced for a wide class of families of functionals on a class of distributions. As a consequence the isotonicity ofZ Zenga concentration index is deduced as an immediate application of a characterization result, in particular of the first degree stochastic dominance relation. Moreover it is also shown that a classical inequality by Fan and Lorenz is a basic reference for the determination of a wide class of Lorenz ordering-preserving functionals. Isotonicity ofZ could also be seen as a straighforward application of Fan and Lorenz inequality.     

On Stein Identity,Chernoff Inequality,and Orthogonal Polynomials

In this article, we present a general method for deriving Stein-like identity and Chernoff-like inequality based on orthogonal polynomials. In order to illustrate our method, some applications are given with respect to normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution, not only for random variables but also for random vectors, resulting corresponding Stein-like identity and Chernoff-like inequality are obtained consequently. Within our best knowledge, some of our matrix version results are new in the literature. In addition, forward difference formulae of Charlier polynomials, Krawtchouk polynomials and Meixner polynomials, Stein-like identity, and Chernoff-like inequality with respect to Beta distribution, as well as Rodrigues formula of Meixner polynomials are also prepared in the first time within our limited information. Interestingly, as far as normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution are concerned, we found that their Stein-like identity and corresponding Chernoff-like inequality are related closely, by examining their Rodrigues formula.     

Linearization of inequality indices in the design-based framework

Linearization methods are customarily adopted in sampling surveys to obtain approximated variance formulae for estimators of statistical functionals under the design-based approach. In the present paper, following the Deville [Variance estimation for complex statistics and estimators: linearization and residual techniques. Surv Methodol. 1999;25:193–203] approach stemming from the concept of design-based influence function, we provide a general result for linearizing a large family of population functionals which includes many of the inequality measures considered in social, economic and statistical studies, such as the Gini, Amato, Zenga, Atkinson and Generalized Entropy indices. The feasibility of our theoretical results is assessed by some simulation studies involving real and artificial data.     

Relations for m-generalized order statistics via an Opial-type inequality

An Opial-type inequality is applied to obtain relations for expectations of functions of m-generalized order statistics (m-gOSs), their distribution functions, as well as moment-generating functions. Respective inequalities for common order statistics and record values are contained as particular cases.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

Exact Non-parametric Confidence Intervals for Quantiles with Progressive Type-II Censoring

It is shown how various exact non-parametric inferences based on order statistics in one or two random samples can be generalized to situations with progressive type-II censoring, which is a kind of evolutionary right censoring. Ordinary type-II right censoring is a special case of such progressive censoring. These inferences include confidence intervals for a given parent quantile, prediction intervals for a given order statistic of a future sample, and related two-sample inferences based on exceedance probabilities. The proposed inferences are valid for any parent distribution with continuous distribution function. The key result is that each observable uncensored order statistic that becomes available with progressive type-II censoring can be represented as a mixture with known weights of underlying ordinary order statistics. The importance of this mixture representation lies in that various properties of such observable order statistics can be deduced immediately from well-known properties of ordinary order statistics.     

Multivariate Excess Wealth Ordering of Generalized Order Statistics

The concept of generalized order statistics (GOSs) was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this paper is to investigate conditions on the underlying distribution function and the parameters on which GOSs are based, to establish multivariate excess wealth ordering of GOSs from one sample and two samples, respectively.     

Some distributional results through past entropy

Measure of uncertainty in past lifetime distribution plays an important role in the context of Information Theory, Forensic Science and other related fields. In this paper we provide characterizations of quite a few continuous and discrete distributions based on certain functional relationships among past entropy, reversed hazard rate and expected inactivity time. Based on past entropy, a conditional measure of uncertainty has been defined, which has helped in defining a new stochastic order and an ageing class. The properties of the stochastic order and those of the ageing class are also studied here.     

Properties of a randomization test for multifactor comparisons of groups

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F _pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.     

Features and performance of some outlier detection methods

A review of several statistical methods that are currently in use for outlier identification is presented, and their performances are compared theoretically for typical statistical distributions of experimental data, considering values derived from the distribution of extreme order statistics as reference terms. A simple modification of a popular, broadly used method based upon box-plot is introduced, in order to overcome a major limitation concerning sample size. Examples are presented concerning exploitation of methods considered on two data sets: a historical one concerning evaluation of an astronomical constant performed by a number of leading observatories and a substantial database pertaining to an ongoing investigation on absolute measurement of gravity acceleration, exhibiting peculiar aspects concerning outliers. Some problems related to outlier treatment are examined, and the requirement of both statistical analysis and expert opinion for proper outlier management is underlined.