Ričardas Zitikis

A flexible Weibull extension

We propose a new two-parameter ageing distribution which is a generalization of the Weibull and study its properties. It has a simple failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model various ageing classes of life distributions including IFR, IFRA and modified bathtub (MBT). The ranges of the two parameters are clearly demarcated to separate these classes. It thus provides an alternative to many existing life distributions. Details of parameter estimation are provided through a Weibull-type probability plot and maximum likelihood. We also derive explicit formulas for the turning points of the failure rate function in terms of its parameters. This, combined with the parameter estimation procedures, will allow empirical estimation of the turning points for real data sets, which provides useful information for reliability policies.

The Asymptotic Distribution of the S-Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S–Gini index. It shows that the theory of L–statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S–Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S–Gini index can be estimated with a prescribed margin of error.

Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses

The conditional tail expectation (CTE) is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

The Vervaat process

Publisher Summary This chapter surveys of asymptotic and other properties of the Vervaat process Vn. In particular, it discusses strong and weak asymptotic negligibility, as well as the law of the iterated logarithm and convergence in distribution of Vn. It also presents a number of examples which dearly indicate the importance of the Vervaat process in various statistical applications. The present chapter surveys the developments in this area and finishing with some new, yet unpublished results.

TESTING HYPOTHESES ABOUT ABSOLUTE CONCENTRATION CURVES AND MARGINAL CONDITIONAL STOCHASTIC DOMINANCE

We consider statistical tests concerning various relationships between two absolute concentration curves (ACC). In particular, we consider tests for determining if the two ACCs coincide, one is above another in a specified order, or do not intersect without specifying which one is above/below the other one. These problems are of interest in the context of marginal conditional stochastic dominance (MCSD). Constructing statistical tests for the MCSD rely on ideas as well as on their modifications developed by Whang, Linton and Maasoumi (2005) in the context of stochastic dominance for distribution functions. Our theoretical considerations are supplemented with a simulation study.

Modelling Deceleration in Senescent Mortality

Mortality deceleration is the observed but yet to be understood phenomenon that the increase in the late-life death rate slows down after a certain species-related advanced age. Various definitions of onsets of mortality deceleration are examined. A new distribution based on the Strehler-Mildvan theory of aging takes on the required shapes. The application is done on mortality data from the 1892 cohort of Swedish women and on Mediterranean fruit flies.

Zenga's New Index of Economic Inequality, Its Estimation, and an Analysis of Incomes in Italy

For at least a century academics and governmental researchers have been developing measures that would aid them in understanding income distributions, their differences with respect to geographic regions, and changes over time periods. It is a fascinating area due to a number of reasons, one of them being the fact that different measures, or indices, are needed to reveal different features of income distributions. Keeping also in mind that the notions of poor and rich are relative to each other, Zenga (2007) proposed a new index of economic inequality. The index is remarkably insightful and useful, but deriving statistical inferential results has been a challenge. For example, unlike many other indices, Zengas new index does not fall into the classes of -, -, and -statistics. In this paper we derive desired statistical inferential results, explore their performance in a simulation study, and then use the results to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW).

Asymptotic Estimation Of The E-Gini Index

Under minimal assumptions on the distribution of income, we demonstrate that Chakravartys empirical (1988, International Economic Review 29, 147–156) E-Gini index is consistent and asymptotically normal. We also derive an explicit formula for the asymptotic variance of the index and then construct a consistent and computationally straightforward estimator for it.Sincere thanks are due to the co-editor Oliver B. Linton and two anonymous referees whose constructive criticism, advice, and queries helped me in reshaping the paper considerably. As advised by a referee, I had the great pleasure of communicating with Garry F. Barrett and Stephen G. Donald and learning about their interesting and closely related results. The starting point of my work on the project was correspondence with Joseph L. Gastwirth in the spring of 2000 that resulted in our joint work on the S-Gini index. I am grateful to Joseph L. Gastwirth for his time, his advice, and his numerous suggestions that followed. The help, in addition to interest in the project, by Ying Zhang of the Statistical Laboratory at the University of Western Ontario is greatly appreciated; the analysis of the dependence of IƒF,I±2 on parameters presented in Table 1 is due to her. This research was partially supported by an NSERC of Canada individual research grant at the University of Western Ontario.

Asymptotic confidence bands for the Lorenz and Bonferroni curves based on the empirical Lorenz curve

Abstract We construct asymptotic confidence bands for the Lorenz and Bonferroni curves that are fundamental tools for analyzing data arising in economics and reliability. The width of the obtained confidence bands is regulated by weight functions depending on the available information about the underlying distribution function. We show that, in some instances, on deleting the smallest and largest observations, the empirical Lorenz and Bonferroni curves become better estimators of the corresponding theoretical ones, and also provide a complete description of such instances. In the process of constructing confidence bands, we prove weighted weak approximation results for the Lorenz and Bonferroni processes, as well as for the Vervaat process that plays a fundamental role in obtaining the main results. We also present examples that indicate the optimality of results.

The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data

Although failure data are usually treated as being continuous, they may have been collected in a discrete manner, or in fact be discrete in nature. Reliability models with bathtub-shaped hazard rate are fundamental to the concepts of burn-in and maintenance, but how well do they incorporate discrete data? We explore discrete versions of the additive Weibull distribution, which has the twin virtues of mathematical tractability and the ability to produce bathtub-shaped hazard rate functions. We derive conditions on the parameters for the hazard rate function to be increasing, decreasing, or bathtub shaped. While discrete versions may have the same shaped hazard rate for the same parameter values, we find that when fitted to data the fitted hazard rate shapes can vary between versions. Our results are illustrated using several real-life data sets, and the implications of using continuous models for discrete data discussed.