A REVIEW OF STOCHASTIC DOMINANCE METHODS FOR POVERTY ANALYSIS

Stochastic dominance techniques have been mainly employed in poverty analyses to overcome what it is called the multiplicity of poverty indices problem. Moreover, in the multidimensional context, stochastic dominance techniques capture the possible relationships between the dimensions of poverty as they rely upon their joint distribution, unlike most multidimensional poverty indices, which are only based on marginal distributions. In this paper, we first review the general definition of unidimensional stochastic dominance and its relationship with poverty orderings. Then we focus on the conditions of multivariate stochastic dominance and their relationship with multidimensional poverty orderings, highlighting the additional difficulties that the multivariate setting involves. In both cases, we focus our discussion on first‐ and second‐order dominance, though some guidelines on higher order dominance are also mentioned. We also present an overview of some relevant empirical applications of these methods that can be found in the literature in both univariate and multivariate contexts.     

Stochastic comparisons of multivariate reversed frailty model

Multivariate frailty approaches are most commonly used to define distributions of random vectors, which represent lifetimes of individuals or components and stochastically compare them in terms of various multivariate orders. In this paper, we study a multivariate shared reversed frailty model and a general multivariate reversed frailty mixture model, and derive sufficient conditions for some of the stochastic orderings to hold among the random vectors. We also consider a particular case of a general multivariate mixture model in which the baseline distribution function is represented in terms of a copula and study stochastic comparisons (stochastic and lower orthant order) among the two random vectors.     

Stochastic comparisons of coherent systems

The study of stochastic comparisons of coherent systems with different structures is a relevant topic in reliability theory. Several results have been obtained for specific distributions. The present paper is focused on distribution-free comparisons, that is, orderings which do not depend on the component distributions. Different assumptions for the component lifetimes are considered which lead us to different comparison techniques. Thus, if the components are independent and identically distributed (IID) or exchangeable, the orderings are obtained by using signatures. If they are just ID (homogeneous components), then ordering results for distorted distributions are used. In the general case or in the case of independent (heterogeneous) components, a similar technique based on generalized distorted distributions is applied. In these cases, the ordering results may depend on the copula used to model the dependence between the component lifetimes. Some illustrative examples are included in each case.     

Poverty in America 1970–1990: who did gain ground? An application of stochastic dominance criteria employing simultaneous inequality tests in a partial panel

Atkinson ( 1987 ) proposed stochastic dominance criteria for analysing poverty which, under certain conditions, establish orderings of states for any poverty line and any poverty measure within given class, refocusing debate on the nature of the income distribution of the poor. Employing new empirical techniques, these criteria are implemented for the United States from 1970 to 1990 using the Panel Study of Income Dynamics. Results highlight the pivotal role of family size scale economies in consumption, indicate different experiences for white versus non‐white groups and suggest that optimism over the progress of the poor is not warranted. Copyright © 2003 John Wiley & Sons, Ltd.     

International welfare comparisons and nonparametric testing of multivariate stochastic dominance

This paper outlines a class of statistical procedures that permit testing of a broad range of multidimensional stochastic dominance hypotheses and, more generally, welfare hypotheses that rely upon multiple stochastic dominance conditions. We apply the procedures to data on income and leisure hours for individuals in Germany, the UK, and the USA. We find that no country first‐order stochastically dominates the others in both dimensions for all years of comparison. Furthermore, while in general the USA stochastically dominates Germany and the UK with respect to income, in most periods Germany stochastically dominates with respect to leisure hours. Finally, we find evidence that bivariate poverty (which refers, for example, to the working poor, that is, those who have little leisure and low income) is lower in Germany than in either the UK or the USA. On the other hand, poverty comparisons between the UK and the USA are sensitive to the subpopulation of individuals considered. Copyright © 2007 John Wiley & Sons, Ltd.     

CONTINUOUS PREFERENCE ORDERINGS REPRESENTABLE BY UTILITY FUNCTIONS

This paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. We start with a historical perspective on the notions of utility and preferences, continue by defining the mathematical concepts employed in this literature, and then list several key contributions to the topic of representability. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.     

Moment inequalities for order statistics from ordered families of distributions

Summary In this paper some inequalities for the variance and covariance of convex monotone functions of order statistics from ordered families of distributions are presented. The considered order relations in the set of distributions are the stochastic ordering relation and the convex ordering relation. Stochastic comparisons of spacings and their sums are also given. As corollaries the results for IFR and DFR distributions are obtained.     

Stochastic comparisons of distorted distributions,coherent systems and mixtures with ordered components

A distribution function F is a generalized distorted distribution of the distribution functions \(F_1,\ldots ,F_n\) if \(F=Q(F_1,\ldots ,F_n)\) for an increasing continuous distortion function Q such that \(Q(0,\ldots ,0)=0\) and \(Q(1,\ldots ,1)=1\). In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions \(F_1,\ldots ,F_n\) are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1–3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures.     

Distribution-sensitivity of rank-dependent poverty measures

We propose a criterion to rank poverty measures on the basis of distribution-sensitivity. The criterion compares reactions to ‘lossy’ transfers among the poor. We focus on the class of rank-dependent poverty measures and provide distribution-sensitivity rankings of the poverty gap ratio, the first and second Sen measures, the Thon measure, the Shorrocks measure, and the Thon, Kakwani and S-Gini classes of measures. Moreover, we discuss the relationship between the proposed criterion and two alternative distribution-sensitivity criteria based on the Arrow–Pratt theory of risk aversion. Finally, we provide an empirically tractable necessary and sufficient condition for unanimous poverty rankings by all continuous and replication invariant rank-dependent poverty measures that exhibit a predetermined minimum degree of distribution-sensitivity.     

Aggregate Poverty Measures

The way poverty is measured is important for an understanding of what has happened to poverty as well as for anti-poverty policy evaluation. Sen's (1976) pathfinding work has motivated many researchers to focus on the way poverty should be measured. A poverty measure, argued by Sen, should satisfy certain properties or axioms and the desirability of a poverty measure should be evaluated by these axioms. During the last two decades, many researchers have adopted the axiomatic approach pioneered by Sen to propose additional axioms and develop alternative poverty measures. The objective of this survey is to provide a clarification on the extensive literature of aggregate poverty measures. In this survey, we first examine the desirability of each axiom, the properties of each poverty measure, and the interrelationships among axioms. The desirability of an axiom cannot be evaluated in isolation, and some combination of axioms may make it impossible to devise a satisfactory poverty measure; some axioms can be implied by other axioms combined and so are not independent; some others are ad hoc and are disqualified as axioms for poverty measurement. Based on the interactions among axioms, we identify the ‘core’ axioms which together have a strong implication on the functional form of a poverty measure. We then review poverty measures that have appeared in the literature, evaluating the interrelationships among different measures, and examining the properties of each measure. The axioms each measure satisfies/violates are also summarized in a tabular form. Several ‘good’ poverty measures, which have not been documented by previous surveys, are also included.     

Stochastic dominance efficient sets and stochastic spanning

Decisions in Economics and Finance - We derive sufficient conditions for non-emptiness of the efficient sets for stochastic dominance relations, usually employed in economics and finance. We do so...     

Ross risk vulnerability for introductions and changes in background risk

We present two theorems that provide necessary and sufficient conditions for an expected utility maximizer to become more risk averse in the sense of Ross with respect to bearing a foreground risk after the introduction of any independent fair or unfair additive background risk. We call these decision makers Ross risk vulnerable, and show that Ross decreasing absolute risk aversion and Ross decreasing absolute prudence are jointly sufficient for Ross risk vulnerability. Restrictions on utility necessary and sufficient for Ross risk vulnerability with respect to stochastic dominance deteriorations of an existing background risk are also presented. Our analysis concludes with applications of Ross risk vulnerability.     

On rankings and top choices in random utility models with dependent utilities

Different aggregate preference orders based on rankings and top choices have been defined in the literature to describe preferences among items in a fixed set of alternatives. A useful tool in this framework is constituted by random utility models, where the utility of each alternative, or object, is represented by a random variable, indexed by the object, which, for example, can capture the variability of preferences over a population. Applications are derived in diverse research fields, including computer science, management science and reliability. Recently, some stochastic ordering conditions have been provided for comparing alternatives by means of some aggregate preference orders in the case of independent random utility variables by Joe (Math Soc Sci 43:391–404, 2002). In this paper we provide new conditions, based on some joint stochastic orderings, for aggregate preference orders among the alternatives in the case of dependent random utilities. We also provide some examples of application in different research fields.      

IDENTIFYING VULNERABILITY TO POVERTY: A CRITICAL SURVEY

In the economic literature on poverty, various methods have been proposed for measuring a phenomenon known as ‘vulnerability’. However, after more than a quarter century of research, no consensus has been reached on how to identify such vulnerable individuals within a given population. Some misunderstandings have also arisen from the overlapping of other closely related concepts, such as the expectation of being poor, expected poverty, multi‐period poverty and risk exposure. This paper offers a detailed conceptual discussion on vulnerability to poverty and its related elements, reviewing a wide range of identifying criteria provided in the literature. It is found that according to the state of the art in this field of research, two key elements stand out in identifying vulnerable individuals: an expected well‐being below the poverty line and a relevant risk of falling into poverty due to downside deviation from a reference level of well‐being. The traditional classification of vulnerability approaches has been updated into four groups: (i) those that stress the element of exposure to risk; (ii) those that emphasize the element of expected poverty; (iii) those that define vulnerability through a utility gap and (iv) those that are supported by a mean‐risk dominance criterion.     

New Results for additive and multiplicative risk apportionment

We provide new characterizations of the preference for additive and multiplicative risk apportionment when risk ordering relies on stochastic dominance. We then point out a simple property of risk apportionment with additive risks: Quite generally, an observed preference for additive risk apportionment in a specific risk environment is preserved when the decision-maker is confronted to other risk situations, so long as the total order of stochastic dominance relationships among risk couples remains the same. The main objective of this paper is to check whether this simple property also holds for multiplicative risks environments. We explain why this is not the case in general, and then provide a set of conditions under which this property holds. We also show that it holds – and even more strongly – in the case of CRRA utility functions due to a particular feature of this family of utility functions.     

Robust growth-equity decomposition of change in poverty: The case of Iran (2000–2009)

This paper examines a robust nonparametric methodology for decomposition of change in poverty into growth and redistribution components. The decomposition is exact, symmetric and free of residual terms. It is equivalent to the Shapley value decomposition in this two-component case. We avoid parametric assumptions about the underlying distributions and Lorenz functions. All of the currently popular poverty measures may be decomposed as suggested in this paper. We identify the issues that arise with parametric approaches to decomposition. An empirical application is given based on recent data on real consumption in rural and urban areas of Iran in 2000, 2004 and 2009 (covering the country's third and fourth five-year development plans). We find that both ‘pure growth’ and ‘redistribution’ components are present in a striking change in poverty, especially among rural households. It would appear that stochastic dominance rankings of the consumption distributions make poverty analyses and decompositions robust to the choice of a poverty line, or poverty measure.     

Concavifiability and constructions of concave utility functions

Concavifiable convex preference orderings are characterized and minimally concave utilities are constructed, using three different approaches. One involves the intersection of arbitrary lines with the three indifference surfaces, another involves conditions on the normals of two indifference surfaces and is related to the super-gradient map of a possible concave utility. In the third approach it is assumed that the ordering is induced by a twice-differentiable utility and Perror's integral of a certain expression formed from the derivatives is used. A possible economic interpretation of minimally concave utilities is suggested, and it is shown that one cannot select concave utilities so that they depend continuously on the ordering.     

A new family of dispersive orderings

In this paper we introduce a new family of univariate dispersion orderings which, under mild conditions, includes the weakly dispersive ordering as a particular case. Different properties of the new class of dispersion orderings are analyzed. Some connections with other univariate dispersion orderings are stated. Finally an application of the new family of dispersion orders to the field of genetics is developed.     

A new axiomatization of rank-dependent expected utility with tradeoff consistency for equally likely outcomes

This paper provides a new axiomatization of rank-dependent expected utility by relaxing tradeoff consistency to hold only when the involved outcomes are equally likely. The resulting weakened variant of tradeoff consistency is considerably simpler and intuitively more appealing than the original condition. Consequently, an improved tool for empirical studies and the elicitation of utilities is provided. It turns out that in the presence of weak ordering, simple-continuity, and stochastic dominance, the new tradeoff consistency for equally likely outcomes is sufficient to derive rank-dependent expected utility for general lotteries.     

Multiple Comparisons with the Best: Bayesian Precision Measures of Efficiency Rankings

A large literature measures the allocative and technical efficiency of a set of firms using econometric techniques to estimate stochastic production frontiers or distance functions. Typically, researchers compute only the precision of individual efficiency rankings. Recently, Horrace and Schmidt (Journal of Applied Economics 15, 1–26, 2000) have applied sampling theoretic statistical techniques known as multiple comparisons with a control (MCC) and multiple comparisons with the best (MCB) to make statistical comparisons of efficiency rankings. As an alternative, this paper offers a Bayesian multiple comparison procedure that we argue is simpler to implement, gives the researcher increased flexibility over the type of comparison, and provides greater, and more intuitive, information content. For these methods and a parametric bootstrap technique, we carry out multiple comparisons of technical efficiency rankings for a set of U.S. electric generating firms, estimated using a distance function framework. We find that the Bayesian method provides substantially more precise inferences than obtained using the MCB and MCC methods.Jel Classification: C11, C32, D24.