Decomposable Measures Research Articles

The three I's of downward income mobility: A directional subgroup decomposable measure

AbstractThe Foster‐Greer‐Thorbecke class of indices and the TIP curve have become classical tools to measure poverty. In this paper we adapt these indices and curve to the income mobility literature and measure downward income mobility considering its three dimensions of incidence, intensity, and inequality. This strategy allows us to incorporate reference dependence, loss aversion and diminishing sensitivity, features emphasized as welfare determinants in Prospect Theory. We propose the use of a class of measures and a Three I's of Downward Mobility curve as a useful graphical device in the income growth framework. Based on this curve we can evaluate the degree of regressivity (or progressivity) of income losses. We provide an empirical illustration for Spain in two recent recession periods. Results show that even if for most households the pandemic shock was not as severe as the Great Recession, some sociodemographic groups had a more similar downward mobility experience in both crises.

Fixed Point Results in Generalized Menger Probabilistic Metric Spaces with Applications to Decomposable Measures

The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our results are meaningful and valuable.

Stochastic dominance and decomposable measures of inequality and poverty

AbstractIn this paper, we characterize some new links between stochastic dominance and the measurement of inequality and poverty. We show that: for second‐degree normalized stochastic dominance (NSD), the weighted area between the NSD curve of a distribution and that of the equalized distribution is a decomposable inequality measure; for first‐degree and second‐degree censored stochastic dominance (CSD), the weighted area between the CSD curve of a distribution and that of the zero‐poverty distribution is a decomposable poverty measure. These characterizations provide graphical representations for decomposable inequality and poverty measures in the same manner as Lorenz curve does for the Gini index. The extensions of the links to higher degrees of stochastic dominance are also investigated.

Segregação ocupacional por sexo no mercado de trabalho brasileiro: uma análise de decomposição para o período 2004-2015

Este artigo analisa a evolução da segregação ocupacional por sexo no mercado de trabalho brasileiro de 2004 a 2015. Além da mensuração da segregação por meio de indicadores tradicionais, como o índice de dissimilaridade e o de Gini, utiliza-se a classe de medidas aditivamente decomponíveis proposta por Hutchens (2004), que permite decompor a segregação em duas parcelas: uma relacionada à segregação entre grandes grupos ocupacionais; e outra que corresponde a uma soma ponderada dos níveis de segregação dentro desses grupamentos. Os resultados mostram que a segregação aumentou entre 2004 e 2015, revertendo a tendência de queda que vinha sendo relatada na literatura referente a anos anteriores a esse período. A análise de decomposição indica que tal crescimento foi impulsionado pela segregação entre os grandes grupos ocupacionais, portanto, entre grupos com ocupações mais heterogêneas entre si. Conclui-se que, à medida que o nível da participação feminina no mercado de trabalho se estabiliza, o processo de integração das ocupações segundo o sexo também apresenta seus primeiros sinais de esgotamento.

Characterizing Horizontal and Vertical Perspectives of Spatial Equity for Various Urban Green Spaces: A Case Study of Wuhan, China.

Equity has been a major concern of urban green space provision. Whether the urban green spaces are equitably provided for socially disadvantaged groups is an important issue in the field of social and environmental justice. This topic is particularly significant in fast-growing Asian countries like China experiencing widening income disparity. This paper examines whether and to what extent the different green spaces (including public parks and urban vegetation) are equitable for all populations (referring to horizontal equity) and also for different social groups (referring to vertical equity) in this typical inland city—Wuhan, China. A novel indicator combining proximity and quality is presented to assess the supply of public parks. The Theil index provides a decomposable measure of overall equity across different regions and vulnerable groups. Both horizontal and vertical perspectives are compared to characterize the spatial equity of urban green spaces (including public parks and urban vegetation) across all population and across different social groups. The empirical analysis of the inland city showed that the overall supply of public parks is far more unequal than mixed or woody vegetation. The distribution of public parks is more inequitable in the outer area, whereas the distribution of mixed or woody vegetation is more inequitable in the inner area. Furthermore, the geographic detector analysis is employed to investigate the spatial relation between socioeconomic contexts and urban green spaces. The spatial heterogeneity of education and age groups is statistically significant for explaining the distribution of public parks. Meanwhile, population density clearly plays a role in the distribution of both public parks and urban vegetation. Per capita income can explain 26% of the distribution of public parks but is not significantly associated with mixed or woody vegetation. Finally, the vertical equity of urban green space is also examined in this paper that the vulnerable groups in the inner area, such as females, residents with low education, children, and the elder suffer from highly unequal accessibility to parks, whereas the vulnerable group in the outer area, such as the migrants gets unequal access to parks.

BMO and the John-Nirenberg Inequality on Measure Spaces

Abstract We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.

Convexity and Closure in Optimal Allocations Determined by Decomposable Measures

A general optimal allocation problem is considered, where the decision-maker controls the distribution of acting agents, by choosing a probability measure on the space of agents. The notion of a decomposable family of probability measures is introduced, in the spirit of a decomposable family of functions. It provides a sufficient condition for the convexity of the feasible set, and the concavity of the value function. Together with additional conditions, closure properties also follow. The notion of a decomposable family of measures covers, both the case of set-valued integrals and the case of convexity in the space of probability measures.

Measuring gender segregation

This paper aims to fill some gaps in the literature concerning the sensitivity of segregation measures. We examine the definitions of regressive and progressive movement, and formally describe the requirements for these movements. As a result of this analysis, we relax a strong assumption established in the literature regarding these movements. Since these measures increase with regressive movement, we are interested in analyzing how the measures’ sensitivities vary as a function of the position of the strata. This analysis allows us to establish how the segregation measures behave with an increase in the number of people in one category in a stratum. We analyze these concepts in terms of the Gini index and the class of additively decomposable measures and analyze the sensitivity of the index of dissimilarity to regressive movements. Data from a national household survey are used to illustrate the results found in the paper.

Continuous m-dimensional distorted probabilities

Fuzzy measures, also known as non-additive measures, monotonic games, and capacities, have been used in many contexts. For example, in economics, risk analysis, in computer science, computer vision and machine learning and, in general, in mathematics.However, when looking at applications, one of the problems that still needs to be solved is how the measure should be defined in an easy and intuitive way. When the reference set is finite, a few families of measures have been established, e.g. distorted probabilities, k-additive and decomposable measures. But, when the reference set is infinite, the only family is distorted probabilities.In this paper we give a definition for m-dimensional distorted probabilities in the case that the reference set is not finite, and we study some properties of this family. We also give a definition for hierarchically decomposable m-dimensional distorted probabilities that relates to another family of measures defined for the finite case.

Controlling Spatial Inhomogeneity in Prototypical Multiphase Microstructures

A wide variety of real random composites can be studied by means of prototypes of multiphase microstructures with a controllable spatial inhomogeneity. To create them, we propose a versatile model of randomly overlapping super-spheres of a given radius and deformed in their shape by the parameter p. With the help of the so-called decomposable entropic measure, a clear dependence of the phase inhomogeneity degree on the values of the parameter p is found. Thus, a leading trend in changes of the phase inhomogeneity can be forecast. It makes searching for possible structure/property relations easier. For the chosen values of p, examples of two and three-phase prototypical microstructures show how the phase inhomogeneity degree evolves at different length scales. The approach can also be applied to preparing the optimal starting configurations in reconstructing real materials.

Extension of a class of decomposable measures via generalized pseudo-metrics

This study considers the application of generalized pseudo-metrics to the extension of decomposable measures. We prove that the extension of a non-strict Archimedean t-conorm-based σ-decomposable measure can be formulated as the closure of a subset of a certain generalized pseudo-metric space. We show that the extension via generalized pseudo-metrics is equivalent to the completion of t-conorm-based σ-decomposable measures and the well-known Carathéodory extension.

The Rising Tide of Absolute Global Income Inequality During 1850–2010: Is It Driven by Inequality Within or Between Countries?

This paper is the first to decompose absolute global income inequality into its within-country and between-country component. The results show a continuous increase of absolute global inequality during 1850–2010, which can be separated into three distinct phases: (1) between 1850 and 1929, within-country inequality explained up to 76 % of absolute global inequality, yet the growth rates of within- and between-country inequality were very similar; (2) a sharp increase in the importance of between-country inequality occurred during the 1929–1950 period, which was followed by a period during which the within- and between-country components were approximately equally relevant; and (3) after 1985, the growth in absolute global inequality was driven primarily by the accelerated growth of within-country income differences. Currently, within-country inequality explains 70 % of absolute global market inequality, a figure close to that of the year 1850. Additional findings include that absolute income convergence between countries took place after 2005, that it is possible to reduce absolute inequality and to grow simultaneously, and that recently within-country net inequality has grown faster than market inequality. The main findings are preserved when different absolute decomposable inequality measures, sample sizes, and purchasing power parity exchange rates are used.

Data-Informed Fuzzy Measures for Fuzzy Integration of Intervals and Fuzzy Numbers

The fuzzy integral (FI) with respect to a fuzzy measure (FM) is a powerful means of aggregating information. The most popular FIs are the Choquet and Sugeno, and most research focuses on these two variants. The arena of the FM is much more populated, including numerically derived FMs such as the Sugeno $\lambda$ -measure and decomposable measure, expert-defined FMs, and data-informed FMs. The drawback of numerically derived and expert-defined FMs is that one must know something about the relative values of the input sources. However, there are many problems where this information is unavailable, such as crowdsourcing. This paper focuses on data-informed FMs, or those FMs that are computed by an algorithm that analyzes some property of the input data itself, gleaning the importance of each input source by the data they provide. The original instantiation of a data-informed FM is the agreement FM, which assigns high confidence to combinations of sources that numerically agree with one another. This paper extends upon our previous work in data-informed FMs by proposing the uniqueness measure and additive measure of agreement for interval-valued evidence. We then extend data-informed FMs to fuzzy number (FN)-valued inputs. We demonstrate the proposed FMs by aggregating interval and FN evidence with the Choquet and Sugeno FIs for both synthetic and real-world data.

A note on the pseudo Stolarsky type inequality for the [formula omitted]-integral

Daraby (2012) proved the pseudo Stolarsky type inequality for pseudo-integrals and Mesiar and Pap (1999) defined the pseudo-integrals with respect to a sup-or int- decomposable measure. In this paper, we defined the interval-valued pseudo-integral(for short, g¯-integral) in the case of a generated semiring with an interval-valued generator g¯ by using interval-representable pseudo-operations and prove the pseudo Stolarsky type inequality for the g¯-integrals.

Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P

Abstract Some results on coherence in probabilistic and in possibilistic frameworks are presented in order to deal with nonmonotonic reasoning. Moreover, we extend these results to conditional decomposable measures. We deal with entailment and prove that it satisfies the axiomatization of System P by referring to conditional necessities or to specific conditional decomposable measures (which include conditional probability). Finally, we study some aspects concerning a notion of irrelevance.

On pseudo-metric spaces induced by σ-⊥-decomposable measures

In this paper, we study the relationship between the pseudo-metric and σ-decomposable measures with respect to a triangular conorm (σ-⊥-decomposable measures, for short). We first construct a pseudo-metric on the measurable sets of a given σ-⊥-decomposable measure, and then discuss several properties such as completeness and continuity of the constructed pseudo-metric space. Finally, we show that the μ-separability and nonatom of the σ-⊥-decomposable measure can be characterized in the constructed pseudo-metric space. Our results suggest that the standard approach for obtaining a metric from a given probability measure can be generalized to the setting of σ-⊥-decomposable measure.

An integral with respect to probabilistic-valued decomposable measures

Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)additivity condition of the considered (sub)measure. It is demonstrated that the introduced integral brings a new tool in approximate reasoning and uncertainty processing with possible applications in several areas.

On Minimal and Maximal p-operator Space Structures

AbstractWe show that L∞(µ), in its capacity as multiplication operators on Lp(µ), is minimal as a p-operator space for a decomposable measure μ. We conclude that L1(μ) has a certain maximal type p-operator space structure that facilitates computations with L1(μ) and the projective tensor product.

Operator Decomposable Measures and Stochastic Difference Equations

We consider the following convolution equation (or equivalently stochastic difference equation) 1 $$\begin{aligned} \lambda _k = \mu _k*\phi (\lambda _{k-1}),\quad k \in {\mathbb Z}\end{aligned}$$ for a given bi-sequence $$(\mu _k)$$ of probability measures on $${\mathbb R}^d$$ and a linear map $$\phi $$ on $${\mathbb R}^d$$ . We study the solutions of Eq. (1) by realizing the process $$(\mu _k)$$ as a measure on $$({\mathbb R}^d)^{\mathbb Z}$$ and rewriting the stochastic difference equation as $$\lambda = \mu *\tau (\lambda )$$ -any such measure $$\lambda $$ on $$({\mathbb R}^d)^{\mathbb Z}$$ is known as $$\tau $$ -decomposable measure with co-factor $$\mu $$ where $$\tau $$ is a suitable weighted shift operator on $$({\mathbb R}^d)^{\mathbb Z}$$ . This enables one to study the solutions of (1) in the settings of $$\tau $$ -decomposable measures. A solution $$(\lambda _k)$$ of (1) will be called a fundamental solution if any solution of (1) can be written as $$\lambda _k*\phi ^k(\rho )$$ for some probability measure $$\rho $$ on $${\mathbb R}^d$$ . Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one–one correspondence between fundamental solutions of (1) and strongly $$\tau $$ -decomposable measures on $$({\mathbb R}^d)^{\mathbb Z}$$ with co-factor $$\mu $$ . We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent $$\ell _p$$ -paths.

On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

We mainly discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, we give the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. Furthermore, we got several important properties of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure.