The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

Relations for reliability measures of weighted distributions

Relibility measures of weighted distribution of alifeistribution have been derived Sufficientconditions on the weight function have been obtained for the weighted distribution of an IFR distribution to be IFR. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context.     

On collapsing categories in two-way contingency tables

In this paper, we examine the relationships between log odds rate and various reliability measures such as hazard rate and reversed hazard rate in the context of repairable systems. We also prove characterization theorems for some families of distributions viz. Burr, Pearson and log exponential models. We discuss the properties and applications of log odds rate in weighted models. Further we extend the concept to the bivariate set up and study its properties.     

The Lamé class of Lorenz curves

In this paper, the class of Lamé Lorenz curves is studied. This family has the advantage of modeling inequality with a single parameter. The family has a double motivation: it can be obtained from an economic model and from simple transformations of classical Lorenz curves. The underlying cumulative distribution functions have a simple closed form, and correspond to the Singh–Maddala and Dagum distributions, which are well known in the economic literature. The Lorenz order is studied and several inequality and polarization measures are obtained, including Gini, Donaldson–Weymark–Kakwani, Pietra, and Wolfson indices. Some extensions of the Lamé family are obtained. Fitting and estimation methods under two different data configurations are proposed. Empirical applications with real data are given. Finally, some relationships with other curves are included.     

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.     

On the Bayesian estimation of the weighted Lindley distribution

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.     

Lorenz ranking of income distributions

Based on the stochastic comparison of the Lorenz curves of income distributions, five partial orderings of income distributions are obtained. Three of these orderings are the well known star shaped, stochastic and the Lorenz orderings. The other two are new and are studied in some detail. The weakest ordering which is called the Lorenz area ordering is of special importance since it enables us to compare interesting Lorenz curves. This latter ordering leads to a class of income inequality measures which are identical with the linear inequality measures considered by Mehran (1976). A discussion of these measures is presented together with an application to part of Kunzet's (1963) data.     

The Lorenz Curve for Model Assessment in Exponential Order Statistic Models

A goodness-of-fit technique for random samples from the exponential distribution based on the sample Lorenz curve is adapted for use in the exponential order statistic (EOS) model. In the EOS model, only those observations in a random sample from the exponential distribution of unknown size N that are less than some known stopping time T are observable. The model is known as the Jelinski-Moranda model in software reliability, where it is used to estimate the number of bugs in software during development. Distributional results are derived for the distance between the sample Lorenz curve and the population Lorenz curve so that it can be used as a goodness-of-fit test statistic. Simulations show that the test has good power against several alternative distributions. Simulations also indicate that in some cases, model misspecification leads to poor parameter estimation. A plotting procedure provides a means of graphical assessment of fit.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.     

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.     

Zenga Distribution and Inequality Ordering

The aim of this article is to establish an ordering related to the inequality for the recently introduced Zenga distribution. In addition to the well-known order based on the Lorenz curve, the order based on I(p) curve is considered. Since the Zenga 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. This investigation shows that for the Zenga distribution, two of the three parameters are inequality indicators.     

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.     

Chernoff distance for doubly truncated distributions

In a recent paper, Nair et al. [Stat Pap 52:893–909, 2011] proposed Chernoff distance measure for left/right-truncated random variables and studied their properties in the context of reliability analysis. Here we extend the definition of Chernoff distance for doubly truncated distributions. This measure may help the information theorists and reliability analysts to study the various characteristics of a system/component when it fails between two time points. We study some properties of this measure and obtain its upper and lower bounds. We also study the interval Chernoff distance between the original and weighted distributions. These results generalize and enhance the related existing results that are developed based on Chernoff distance for one-sided truncated random variables.     

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.     

An extended Gompertz-Makeham distribution with application to lifetime data

In this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.     

Probability inequalities for weighted distributions

In this paper, properties of weighted distributions for general weight functions are investigated. We establish the subadditivity [superadditivity] property of weighted distributions for log-concave [log-convex] weight functions in the sense of the usual stochastic order. The main result generalizes Lemma 2.3 in Brown (2006). Several interesting moment inequalities are presented.     

A new generalized Lindley distribution

A new generalized Lindley distribution, based on weighted mixture of two gamma distributions, is proposed. This model includes the Lindley, gamma and exponential distributions as and other forms of Lindley distributions as special cases. Lindley distribution based on two gamma with two consecutive shape parameter is investigated in some details. Statistical and reliability properties of this model are derived. The size-biased, the length-biased and Lorenze curve are established. Estimation of the underlying parameters via the moment method and maximum likelihood has been investigated and their values are simulated. Finally, fitting this model to a set of real-life data is discussed.     

Interdistributional Income Inequality

Incomplete moments are used to characterize income inequality and provide the basis for interdistributional Lorenz curves. Four measures of interdistributional inequality are considered and seen to be related to an interdistributional welfare interpretation. Based upon these measures, there has been a significant secular decline in interdistributional inequality between blacks and whites over the past 30 years.     

The Kumaraswamy Burr XII distribution: theory and practice

For the first time, a five-parameter distribution, called the Kumaraswamy Burr XII (KwBXII) distribution, is defined and studied. The new distribution contains as special models some well-known distributions discussed in lifetime literature, such as the logistic, Weibull and Burr XII distributions, among several others. We obtain the complete moments, incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and reliability of the KwBXII distribution. We provide two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. For different parameter settings and sample sizes, various simulation studies are performed and compared to the performance of the KwBXII distribution. Three applications to real data sets demonstrate the usefulness of the proposed distribution and that it may attract wider applications in lifetime data analysis.