Evolution of the Fuzzy-Set Approach to Multi-Dimensional Poverty Measurement

Bruno Cheli, Achille Lemmi,

Nicoletta Bannuzi and Andrea Regoli

Multi-Dimensional Poverty Measurement: The-State-of-Art

The spectrum of poverty measurements is wide and varies from purely monetary indicators to more or less sophisticated models based on nonmonetary measures (Alkire and Foster, 2011; UNECE, 2017; Aaberge et ah, 2017; Atkinson, 2017; Atkinson et ah, 2017). Measurement choices often remain implicit even though they can have a profound impact on results and related policies (Haughton and Khandker, 2009; Brucker et ah, 2015; Ravallion, 2016; Jenkins, 2018; Berger et ah, 2018; Atkinson, 2019).

The most traditional approach focuses on measuring monetary poverty, which includes conventional poverty analyses using information on household income or consumption expenditure, and more recently also on wealth (Brandolini et ah, 2010; Meyer and Sullivan, 2011; Azpitarte, 2012; OECD, 2013; Jolliffe and Espen, 2016; Serafino and Tonkin, 2017; Brewer et ah, 2017; Kuypers and Marx, 2018). In this framework, poverty is defined as a lack of strictly economic resources, indeed monetary, under the hypothesis that income or, more generally, monetary resources are able to interpret and capture all the aspects that contribute to determining the conditions of material hardship. Economic resources allow people to satisfy their needs and pursue many other goals that they deem important.

The advantages and disadvantages of this approach are well known (Nelson, 2012; Neckermann et ah, 2016), but - across all - its simplicity and direct policy links (especially with monetary transfer policies) still make it the most common and widespread approach for measuring poverty and monitoring its dynamics.

Measures of poverty based on income or consumption expenditure are still the only ones available, for example, in the United States, Japan or Canada, while in Europe the income-based poverty indicator (the so-called at-risk-of-poverty indicator) has been available since 1995, and in 2004 it was juxtaposed and integrated with the material deprivation and low-work- intensity indicators (Eurostat, 2012). For several decades (starting from Townsend, 1979; Sen, 1980; Atkinson and Bourguignon, 1982), in fact, it has been widely acknowledged that reality is much more complex and that poverty, specifically, is a phenomenon concerning a plurality of dimensions (Ravallion, 2011; Stiglitz et al., 2018; World Bank, 2018). The problem of multi-dimensionality is mainly represented by the identification of the relevant dimensions, the availability of adequate and reliable information on them and the method of putting the information together (Duclos et al., 2006; Ferreira and Lugo, 2013; Alkire and Foster, 2011; Lustig, 2011; Labonte et al., 2011; Maasoumi and Lugo, 2008; Alkire and Foster, 2007). Therefore, for many years, this limit has also led to the predominance of uni-dimensional poverty measures, especially within the official statistics.

Measures of poverty that exceed the exclusive use of monetary indicators have been developed in different conceptual contexts, referring to different terms such as exclusion, inclusion or social cohesion, deprivation or ‘capability’ poverty (Back, 2018; Garner and Short, 2015). The latter approach, which arises from the fundamental contribution of Amartya Sen (1993), is what defines the most complete and robust conceptual reference for the development of multi-dimensional poverty measures (Chiappero-Martinetti, 2000). Nevertheless, the theoretical literature on multi-dimensional measurement is still far from being consolidated.

Finally, it is now commonly accepted that income-poor people do not necessarily have to be deprived in other dimensions. For example, in countries where some public services (such as education or health care services) are provided for free or at subsidised prices by the national social security system, even households with very low levels of income may have good health conditions and attain high levels of education for their children, or, in other words, an acceptable standard of living. There is an emerging consensus that multidimensional measures of poverty should complement monetary ones. Despite the fascination represented by the possibility of calculating a composite indicator that facilitates spatial and temporal comparisons and between population subgroups, the excessive aggregation of dimensions and indicators results in an extremely significant informative and analytical loss, even threatening to frustrate the use of multi-dimensional measures. Keeping the dimensions distinct or considering them as appropriate aggregations can help to identify the most critical or serious aspects of poverty and put in place more targeted and effective intervention policies (see Box at the end of the section).

The consensus on the fact that poverty must be seen and measured as a multi-dimensional phenomenon is also enshrined in the 2030 UN Agenda for Sustainable Development which identifies the reduction of poverty in ‘all its forms and dimensions’ among the objectives to be achieved. In particular, Target 1.2 of the SDGs (Sustainable Development Goals) states that by at least 2030 the number of men, women and children of all ages living in poverty should be halved, considered in all its dimensions and calculated according to national definitions (Loewe and Rippin, 2016; United Nations, 2015).

Several multi-dimensional poverty indicators have been developed over time, and they can be broadly classified into two groups: i) aggregate indicators and ii) composite indicators. In the former case, the information for each item is considered at the individual level, and an individual multidimensional measure of poverty is calculated and subsequently aggregated to provide a collective measure. In the latter case, the information for each item is considered at the aggregate level and then it is combined.

Two main examples are represented by the ‘people at-risk-of-poverty or social exclusion’ indicator, developed at the European level and published by Eurostat as a EU2020 indicator,1 and the Multi-dimensional Poverty Index (MPI), developed by the Oxford Poverty and Human Development Initiative (OPHI). Even if they are conceptually different, both approaches need to define threshold levels (for one or more indicators) in order to classify each unit as poor or non-poor.

To summarise, whatever the approach, the choice of the indicators is intertwined with the definition of the respective deprivation thresholds. In multivariate analyses these problems may be amplified by the consideration of intangible dimensions for which it may be even more contentious to identify minimum thresholds (Thorbecke, 2013).

More complex and sophisticated models are also found in the literature, in order to obtain a comprehensive description of the various facets of a complex phenomenon, through a suitable synthesis of the associated elementary indicators. In this framework, Structural Equation Modelling (SEM) offers a powerful yet little explored platform for measuring poverty (Ningaye et al., 2013; Voth-Gaeddert and Oerther, 2014). Among others, particularly promising is the recent development in partial least square-path modelling (PLS-PM) that could help with building Model Based Composite Indicators, providing a better measure of complex social phenomena (Lauro et al., 2018).

The relevant key points, which are preliminary to any discussion of the methods for the multi-dimensional analysis of poverty, are: the selection of the relevant dimensions; the indicators used to measure people’s achievements in these dimensions, and the related issue of the choice of deprivation thresholds; and the weights assigned to each dimension. However, we do not aim to discuss these in this chapter. We simply want to remark how the binary distinction between a ‘bad state’ and a ‘good state’ is too sharp, since deprivation is likely to occur in degrees. Starting from this consideration, we retrace and update the fuzzy-set approach (Zadeh, 1965) for measuring multi-dimensional poverty, which leads to the IFR (Integrated Fuzzy and Relative) method.

Box - Severe Deprivation and At-Risk-Of-Poverty Indicators Across Europe at the Regional Level

The map on the left displays the severe material deprivation (SMD) rate by geographical region, whereas the map on the right refers to the at-risk-of-poverty (AROP) rate (see Figure 2.1). The maps show how the two indicators - related to different dimensions - draw different geographical profiles of hardship at the territorial level. Looking at the map on the left, the worst situations (highlighted by the darkest areas) are observed in regions of Greece, Bulgaria and Hungary and in North Macedonia, whereas the map on the right shows that the highest AROP rates are found in regions of Italy, Spain and Romania. The incidence of deprivation/poverty highlighted by the two indicators sometimes agree and sometimes do not. In particular, the regions of Southern Italy and the regions of Romania outside Bucharest occupy the worst positions on both rankings. It is evident that using different indicators of poverty (in addition to the income dimension) at a disaggregated territorial level to the maximum extent possible helps to draw a more comprehensive and faithful picture of this phenomenon. Nevertheless, the level of geographical disaggregation is not the same for every country, even though it is extremely useful for dealing with specific local intervention policies. If direct estimations are not reliable, it would be recommended to reconstruct them with small area estimation methods (Guadarrama et ah, 2014; Pratesi, 2016).

Poverty as a Fuzzy Concept

The fuzzy-set approach is a valid instrument to measure multi-dimensional poverty, and it offers the additional advantage of overcoming the use of unavoidably arbitrary poverty thresholds. In such a way, it avoids extreme simplifications and loss of statistical information, deriving from the rigid poor/non-poor dichotomy.

Moreover, a number of contributions to poverty research agree in stating that the fuzzy-set approach can be used to analyse poverty in a consistent way with the Sen’s capability approach (Sen, 1993).

The fuzzy-set approach takes into account the vagueness of the concept of poverty, which does not mean to view it as semantically ambiguous (Chiappero Martinetti, 2008). The notion that poverty is an either/or condition was brought into question some 50 years ago by Watts (1968), who claimed that there is a continuous gradation as one crosses any particular poverty line. As Qizilbash (2006) argued, poverty meets all the following requirements of a vague or fuzzy predicate: i) there are ‘borderline’ situations where individuals cannot be unequivocally classified as poor or non-poor; ii) there is not a sharp boundary to separate the poor from the non-poor; iii) the Sorites paradox holds: an increase of 0.01 euros in the monetary resources of a poor may lift him out of the poverty while his real condition remains the same. Two additional sources of vagueness are represented: first, by the lack a consensus regarding the threshold below which an individual can be treated as definitely poor; and second by the fact that the

Severe material deprivation rate by NUTS region (left) and at-risk-of poverty rate by NUTS region (right). Year 2018. Source

Figure 2.1 Severe material deprivation rate by NUTS region (left) and at-risk-of poverty rate by NUTS region (right). Year 2018. Source: https://ec.europa.eu/eurostat/data/database Data refer to the EU27 countries plus the United Kingdom, Norway, Switzerland and North Macedonia. The Nomenclature of Territorial Units for Statistics (NUTS) is used at its most disaggregated level, which is usually the NUTS2 level. Nevertheless, for some countries (namely Austria, Belgium, France, Germany, Portugal and the United Kingdom) the information is available at the country level (NUTSO) only.

variables used in poverty analyses (income, consumption and other deprivation indicators) are likely to be subject to measurement errors (Chakravarty, 2006; Zheng, 2015). These considerations lead to the identification of a grey area, where households and individuals are neither completely poor nor completely non-poor.

A phenomenon with such characteristics cannot be properly addressed through a crisp classification into poor and non-poor, based on the setting of a poverty threshold acting as a clear cut-off line, which constitutes the current practice of the poverty measurement. For example, based on the World Bank’s $1.90-a-day (2011 PPP prices) International Poverty Line (World Bank, 2018), a person who commands over $1.91 is regarded as being undoubtedly non-poor just like another person whose resources are by far larger than the poverty line.

The fuzzy-set approach rejects the traditional dichotomy between poor and non-poor. The fuzziness is accounted for via a poverty membership function, measured on a scale from 0 to 1 - whereby 1 means full membership to the set of the poor and 0 full non-membership - which allows for a blurry (as opposed to sharp) vision of the concept of poverty. The conventional classification into a rigid dichotomy according to the traditional approach to poverty can then be viewed as a special case of the fuzzy conceptualisation of poverty, where the membership function equals 1 for those below the poverty line and 0 for those above the poverty line.

Moreover, the degree of membership can be considered as the relative risk of becoming poor in the future. Therefore, when the focus is on the people who are more vulnerable, the measures derived from a fuzzy logic lend themselves to being used fruitfully by policies aiming at preventing poverty. In the traditional approach, letting the poverty line change only identifies ‘border areas’ where people near the poverty line are located. For example, switching from a poverty line defined as 60% of the median equivalised income (which corresponds to the official Eurostat definition) to a threshold defined as 50% of the same income summary value can identify people who were classified as poor according to the former threshold and non-poor according to the latter threshold and, in general, to detect whether and to what extent the poverty measurement is sensitive to the choice of threshold (Lemmi et al., 2019).

An early attempt to incorporate the fuzzy-set theory (Zadeh, 1965) into poverty measurements and analyses at the methodological level (and in a multi-dimensional framework) was made by Cerioli and Zani (1990). Their original proposal was later developed by Cheli and Lemmi (1995), creating the so-called totally fuzzy and relative (TFR) approach. This approach avoids the need for poverty thresholds, defining a membership function as logically and economically founded, based on the relative position of every individual in the distribution of a specific indicator in a given context, which is summarised by the distribution function. Drawing on this approach, Betti et al. (2006a, 2006b) included the concentration (inequality) measure in the membership function, as defined by the Gini-Lorenz approach. The Betti et al. contribution, known as the integrated fuzzy and relative (IFR) approach, will be described in the next section.

Besides the specifications used in the TFR and IFR approaches, different membership function definitions have been proposed in several applications (Cerioli and Zani, 1990; Dombi, 1990; Chakravarty, 2006; Belhadj, 2011; Abdullah, 2011; Betti and Lemmi, 2013a; Zheng, 2015; Betti, 2017). These include: i) a simple monotone linear function; ii) a trapezoidal-shaped function, which incorporates the choice of two threshold values, so that any individual with a value below the lower threshold is counted as poor while any individual with a value above the upper threshold is counted as nonpoor; iii) a nonlinear function, such as an inverted S-shaped function (sigmoid or logistic curve), which also requires identifying the crossover point of whose membership value equals 0.5; iv) other nonlinear curves, such as a Gaussian or an exponential function; v) and functions based on the experts’ judgement.

Interest in the study of multi-dimensional poverty and deprivation using a fuzzy method was further supported by a recent analysis that aimed to search for a fuzzy counterpart to Alkire and Foster’s Multi-dimensional Poverty Index (Kobus, 2018). The greatest contribution of this study was the removal of the dual cut-off of the MPI methodology, which relies strongly on the definition of a fixed threshold for every single indicator as well as for the final deprivation score.

The Integrated Fuzzy and Relative Method

Fuzzy Approach to the Measurement of Monetary Poverty (‘Fuzzy Monetary’)

The conventional approach based on a poverty line can be viewed as a special case of the fuzzy approach, where the membership function (MF) equals 1 for those below the poverty line and 0 for those above the poverty line. In fact, if we consider a poverty measure based on income, the membership function (p) may be seen as:

where yt is the equivalised income of individual i, and г is the poverty line.

In what has been called the TFR approach, Cheli and Lemmi (1995) define the MF in the fuzzy set of the poor as:

where F(y;) is the cumulated distribution function of income calculated for individual i. In this way, the degree of poverty measured by /t equals 1 for the poorest and 0 for the richest person in the population and simply depends on each individual’s position in the income distribution (which justifies the term ‘totally relative’), therefore implying an ordinal approach to poverty measurement. Moreover, this approach can also be seen as ‘totally fuzzy’ as it defines the degree of poverty in fuzzy terms for every level of income.

The degree of poverty of generic individual i as specified in Equation (2.1) has a precise economic meaning. In particular, it represents the proportion of individuals whose level of income is higher than yr

Equation (2.1) provides the individual measure of deprivation of each statistical unit. By aggregating all these values, we obtain a collective index referring to the overall population, which is given by:

Equation (2.2) represents the fuzzy proportion of poor people according to the monetary variable У, where the symbol FM stands for fuzzy monetary?

It must be noticed that the FM index specified in Equation (2.2) is always equal to approximately O.5.3 In fact it represents the sample estimate of E[1 - F(y(.)] = 1 - £[Е(у()], where F( ) is uniformly distributed in the interval [0,1] so that its mean is 0.5. At first glance, this fact might appear to be a drawback, as when we compute the FM index over different samples or populations we always get the same value. However, since we work in a purely relative context, we cannot (and we don’t aim to) obtain a cardinal poverty measure for the surveyed population. We just evaluate which subgroups are better off and which ones are worse off by comparing the values of the FM index computed for each one.

In the case of international comparisons, we should consider the union of the various national samples and compute the individual memberships as specified in Equation (2.1) on this pooled sample. Hence we would compare the values of the FM index computed for each country based on the corresponding national sample.

In order to facilitate the comparison with the conventional poverty rate, Cheli (1995) proposed to raise the MF as specified in Equation (2.1) to some power a > 1:

Increasing the value of exponent a implies giving more weight to the poorer end of income distribution. In fact, being in general 0 < 1 - F(yf) < 1, as a. increases /t decreases more for those who are less poor compared to those who are poorer. The value of a in Equation (2.3) is arbitrary and implies a political choice. Its introduction essentially aims at facilitating comparisons between the fuzzy measures and the conventional ones; for this reason the value of parameter a can be determined so that the FM index computed for the overall sample is equal to the proportion of the population below the official poverty line.

Betti and Verma (1999) define the fuzzy monetary indicator (FM), using a somewhat refined version of Equation (2.1), namely:

where L(y) represents the ordinate of the Lorenz curve of income for the ith individual. Hence, 1 - L(y) represents the share of the total equivalised income received by all individuals less poor than individual i and it varies from 1 for the poorest to 0 for the richest individual. 1 - L(y) can be expected to be a more sensitive indicator of the actual disparities in income, compared to 1 - Fly). This is illustrated in Figure 2.2, where the two MF definitions, represented in Equations (2.3) and (2.4), are compared (for a = 1) by means of the Lorenz diagram.

Moreover, it may be noted that while the mean of 1 - F(y) values is always 0.5, as we already pointed out, the mean of 1 - L(y) equals (1 + G)/2, where G is the Gini index of the distribution. The more concentrated the income the greater 1 - L(y) is when compared to 1 - F(y).

However, recognising that both definitions of the MF are meaningful, effective and theoretically founded, Betti et al. (2006a) proposed to combine them as follows:4

Henceforth we denote the income-based membership function by FM: in order to distinguish it from the one calculated from non-monetary indicators (FS), which we describe in the next section.

Membership functions as specified in Equations (2.1) and (2.4) in comparison

Figure 2.2 Membership functions as specified in Equations (2.1) and (2.4) in comparison.

Equation (2.5) integrates the original TFR approach, as specified in Equation (2.1), with the information on income shares [1 - L(y,)] which is directly connected to the Gini inequality index. This is what we refer to as the IFR approach.

The FM index obtained by averaging the /г values in Equation (2.5) is expressible as a function of the generalised Gini index Ga:

where Ga is defined (in the continuous case) as:

Ga corresponds to the standard Gini index when a = 1, and it weights the distance [F(y) - L(y)] between the line of perfect equality and the Lorenz curve through the individual’s position in the income distribution, giving more weight to its poorer end.

Fuzzy Approach to the Measurement of Non-Monetary Poverty (‘Fuzzy Supplementary’)

The living standard of the population can also be measured by means of a variety of non-monetary indicators referring to different domains (or dimensions), such as housing conditions, possession of durable goods, general financial situation, perception of hardship, expectations, etc. Quantification and combination of a large set of indicators of this kind involves a number of steps, models and assumptions.

First, from all the available indicators, a selection has to be made of those which are substantively meaningful and useful for the analysis of deprivation. This is a substantive as well as a statistical question.

Second, it is useful to identify the underlying dimensions and to group the indicators accordingly. Taking into account the manner in which different indicators may be grouped together adds to the richness of the analysis; ignoring such dimensionality can result in misleading conclusions (Whelan et al., 2001).

Betti et al. (2006a, 2006b) proposed to measure non-monetary deprivation in a way entirely analogous to the fuzzy monetary method, as described in the previous section.5 On the basis of this approach, the membership function corresponding to Equation (2.5) would be:

where FS stands for fuzzy supplementary, si represents the endowment of the overall supplementary items evaluated for individual i, E(s;) and L(s() are the corresponding values of the distribution function and the Lorenz curve (of s), respectively, whereas a is a parameter to be determined so as to make the overall non-monetary deprivation rate numerically identical to the official head-count ratio of the poor.

In order to construct the fuzzy supplementary index defined in Equation (2.8) one has to begin by grouping the items into dimensions as mentioned at the beginning of this section. Individual items indicating non-monetary deprivation often take the form of simple ‘yes/no’ dichotomies (such as the presence or absence of an enforced lack of certain goods or facilities). However, some items may involve three or more ordered categories, reflecting different degrees of deprivation. Consider the case of C ordered categories (c=l,2,...,C) of some deprivation indicator, with с = 1 representing the most deprived and с = C the least deprived situation. Let ci be the category to which individual i belongs and let dki represent the deprivation score of individual i for what concerns item k, belonging to dimension S. The dki deprivation score can be calculated in two different ways, represented by Equations (2.9) and (2.10):6

Equation (2.9) is based on the assumption that the rank of the categories represents an equally-spaced metric variable, whereas in Equation (2.10) the ranks are replaced by the distribution function of the &th item calculated for the /th individual.

Note that in the case of a dichotomous indicator (C = 2), which is by far the most common situation, dki is dichotomous too and takes a value of 1 (in cases of deprivation) or 0 (otherwise) for both Equations.

Now notice that if dkj measures the specific deprivation for item k, (1 - dkj) reflects the endowment for the same item.

The score for any individual i - that reflects her global endowment (or non-deprivation) - for what concerns dimension S (sSi) is computed as a weighted mean of the specific endowments for any item k belonging to S [S = 1,...,Д):

the upweights are defined as a product of two factors:

where depends only on the distribution of item k in the population and is defined as a decreasing function of the proportion of deprived in this item; on the other side, considers the correlation between k and the other items (in the dimension concerned) in order to reduce the redundancy produced by highly correlated indicators.7

By averaging s5i over dimensions we can obtain the overall deprivation score for the ith individual:

The sj scores are used in Equation (2.8) to derive the fuzzy supplementary measure FS for each individual.

I

Finally, by averaging all the FSp we obtain a collective index of nonmonetary deprivation that is called Fuzzy Supplementary (FS).

Monetary and Non-Monetary Poverty in Combination:

Manifest and Latent Deprivation

In the IFR approach the two measures - FM; concerning income (or monetary) poverty (Equation (2.5)), and FS concerning the overall non-monetary deprivation (Equation (2.8)) - may be combined to construct composite measures indicating the extent to which income poverty and non-monetary deprivation overlap. This is done by using the fuzzy-set union and intersection that we briefly describe below, together with the complement that is used for measuring longitudinal poverty (see the next section).

Given a fuzzy set A with membership function jiv its standard complement A is defined by the membership function ЦА = 1 - pA which can be interpreted as the degree to which a given element does not belong to A.

There is more than one way in which the fuzzy-set union and intersection can be formulated, each representing an equally valid generalisation of the corresponding crisp set operations. The IFR approach uses composite measures of monetary and non-monetary poverty that are based on the min and max operators,8 as we describe below.

Given two fuzzy sets, A and B, with membership functions and цв, respectively, their union defines a new fuzzy set whose membership function can be specified as the maximum of the two membership functions, namely:

On the other hand, the membership function in the intersection of A and В can be defined as the minimum of/<4 and цв, that is:

Considering the two fuzzy sets of the income-poor and the non-monetary deprived whose membership functions are FM: and FS;, respectively, they can be combined to define two different measures, called manifest and latent deprivation.

For any individual i, the measure of manifest deprivation (AT) is specified as follows:

On the other side, the measure of latent deprivation (L.) is defined as the maximum of the two memberships FM( and FSj as follows:

The corresponding collective measures (M and L) for the whole population and for specific sub-populations can be obtained by averaging Equations (2.16) and (2.17).

In this context, it is useful to measure the overlapping degree of the two indices, by means of M to L ratio, which ranges from 0 to 1. When there is no overlap (i.e. when the subpopulation subject to income poverty is entirely different from the subpopulation subject to non-monetary deprivation), the above-mentioned ratio equals 0. On the contrary, when there is complete overlap, i.e. when each individual is subject to exactly the same degree of income poverty and of non-monetary deprivation (FAT = FSi), the M to L ratio equals 1.

Fuzzy Longitudinal Measures of Deprivation

Pt represents the fuzzy set of the poor at time t with membership function pt, and Pt represents its complement, which is the non-poor set, with membership function jlt =1- p, (t = 1,2,...,T). Fuzzy longitudinal measures of deprivation across time require the specification of joint membership functions of the type:

and

where the first expression is the membership in the intersection of T cross- sectional fuzzy sets, whereas the second expression refers to their union. Using the min and max operators for intersection and union, respectively, the above longitudinal measures can be calculated as follows:

In other words, IT represents the individual’s propensity to be poor at all T periods, whereas UT is the propensity to be poor in at least one of the T periods. Furthermore, the propensity to be never poor over all T periods is the complement of Ur, that is UT = 1 - UT . The same result is obtained by considering the intersection of non-poor sets:

Furthermore, a measure that quantifies the ‘exiting from poverty’ event at time t after being poor until then requires the definition of the following membership function:

whereas the ‘re-entering into poverty’ at time t following poverty experiences up to t - 2 and a non-poverty experience at t - 1 is based on the following membership function:

The corresponding rates for the population are computed by simply averaging the above individual contributions.

Recent Developments in Investigating the Accuracy of Fuzzy Poverty Estimates

The most recent studies on fuzzy measures focus on their sampling errors and also when referring to complex sample designs. In particular, the contributions mentioned below converge in finding that fuzzy measures present a smaller sampling error than conventional measures of poverty.

The method proposed by Verma et al. (2013) and Betti and Lemmi (2013 b) aims to improve the sampling accuracy of fuzzy poverty indicators for sub-national areas by accumulating the information in a rotational panel design. Longitudinal poverty and multi-dimensionality are combined with estimation at the local level through small area estimation techniques to produce a cumulative 3D measure of fuzzy poverty.

The estimation of standard errors for fuzzy poverty measures from sample surveys with complex designs drove the study by Betti et al. (2018), the first paper that addressed the computation of accurate standard errors for fuzzy measures. They presented a robust computational methodology through the jack-knife repeated replication (JRR) procedure for variance estimation for multi-dimensional measures of poverty and deprivation through the IFR approach. The substantive finding of their paper was that fuzzy measures tend to be subjected to a smaller sampling error than conventional measures of poverty for a given sample size and design.

Through Monte Carlo simulation, Kim (2015) examined the sampling distribution of three fuzzy measures of poverty: TFR and IFR, as well as the measure derived from Cerioli and Zani’s (1990) approach. Fie found that the three measures have very small confidence intervals and are quite robust to measurement errors. The IFR measure has some advantages over the other measures because, for relatively small sample sizes, it produces more accurate estimates than the other measures. Furthermore, as for the identification of the poor, the IFR measure has strong consistency, whereas the other measures considerably underestimate the number of people with very high membership function values.

Final Remarks

One challenge of poverty measurement consists in identifying who is poor. Such operation is traditionally accomplished by using poverty lines, both in the unidimensional and multi-dimensional framework, thus losing the ambiguity underlying the complex nature of poverty.

In contrast, fuzzy-set theory is able to capture the dimension of uncertainty that classical logic and crisp sets are unable to grasp, without competing with probability. They are, in fact, alternative tools for measuring two different types of uncertainty, that can complement each other.

A significant literature now applies the fuzzy-set approach to poverty measurement, and it also includes policy applications. Among others, Chiappero Martinetti (2006) emphasised several advantages of the fuzzy approach, namely: the possibility of using both quantitative and qualitative variables within the same approach; the opportunity of using multidimensional information for different wellbeing domains; and the chance to make poverty and wellbeing comparisons in a plurality of evaluative spaces. The fuzzy-set method also allows to aggregate across domains by making use of the union, intersection and average operators which satisfy a specific set of properties. Moreover, there are also recent fuzzy reformulations of a variety of multi-dimensional poverty indexes, including the Foster-Greer- Thorbecke (FGT) class of poverty measures (Chakravarty, 2019).

Flowever, some complications remain due to the fact that, in every phase of the research conducted according to these methodologies, it is necessary to be aware of the impact that certain methodological choices may have on the results. These choices may refer, for example, to the shape of the membership functions or the aggregation operations. Therefore, a high interpretative capacity is necessarily required at each step, in order to better fill the gap between the richness and intrinsic vagueness of the theoretical multidimensional poverty concept on the one side, and their formal and empirical representation on the other side. In particular, the choice of a given membership function should be supported by sensitivity analyses or robustness tests also in terms of policy-relevant results (Alkire et al., 2015).

To summarise, a fuzzy approach to poverty may complement and encompass the traditional approach, handling vagueness and complexity, and, mostly, strengthening the connection between theory and data analysis, considering that 'in social investigation and measurement, it is undoubtedly more important to be vaguely right than to be precisely wrong’ (Sen, 1985).

Notes

  • 1 An alternative indicator has been proposed by Guio, Gordon and Marlier, who suggested a list of 13 items (six of them already included in the previous indicator) and also a separate indicator for children aged less than 16 years old based on 18 items (Guio et ah, 2016). For a critical evaluation of the EU 2020 Poverty and Social Exclusion indicators see Maitre et ah (2013).
  • 2 The name ‘Fuzzy Monetary’ was introduced later by Betti and Verma (1999) and Cheli and Betti (1999).
  • 3 This happens when we average the и values over the whole sample. By contrast, when we do it for specific subgroups (such as different geographical regions) the FM index will take values greater or less than 0.5.
  • 4 Again, parameter a is chosen by the authors so that the mean of the MF (Equation 2.5) is equal to the official head count ratio of the poor.
  • 5 This idea had been originally proposed by Betti and Verma (1999) who applied it to Equation (2.4).
  • 6 Equation (2.9) was used in the first papers on the IFR approach (Betti et al., 2006a; 2006b; 2008) whereas Equation (2.10) was introduced later (see for instance Betti et al., 2015).
  • 7 For a complete description of this weighting system see Betti and Verma (1999) or Betti et al. (2008).
  • 8 The reasons that justify this choice are discussed in Betti et al. (2006a, 2006b).

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