Emmanuel Fleurbaix
email: emmanuel.fleurbaix@univ-lorraine.fr

Marc Parant
email: marc.parant@univ-lorraine.fr

Armand Maul
email: armand.maul@univ-lorraine.fr

Carole Cossu-Leguille
email: carole.leguille@univ-lorraine.fr

Toxicity of lanthanides on various fish cell lines

Keywords: Aquatic ecotoxicology, Lanthanide, Mixture effect, Fish cell line, Cytotoxicity

published or not. The documents may come

Introduction

Lanthanides are a group of fifteen elements from Lanthanum (Z = 57) to Lutetium (Z = 71) Fourteen of them are stable and one of them, i.e. Promethium (Z=61) is a radioactive one. These elements may be divided into three groups: light (Lanthanum to Neodymium), medium (Samarium to Gadolinium), and heavy (Terbium to Lutetium). Due to their chemical and physical properties (Yoshida et al., 1997), Lanthanides are widely used in various fields such as industry (Celik et al., 2015), medical [START_REF] Cabella | Cellular labeling with Gd(III) chelates: only high thermodynamic stabilities[END_REF], and agriculture [START_REF] Tai | Biological toxicity of lanthanide elements on algae[END_REF].

In recent decades, the growing use of Lanthanides has contributed to an increasing release of these compounds into the aquatic environment [START_REF] Alonso | Evaluating Rare Earth Element Availability: A Case with Revolutionary Demand from Clean Technologies[END_REF][START_REF] Zhang | An economics strategy for criticality -Application to rare earth element Yttrium in new lighting technology and its sustainable availability[END_REF]. It is generally accepted that surface waters are an important pathway in the transport and distribution of Lanthanides [START_REF] Verplanck | Evaluating the Behavior of Gadolinium and Other Rare Earth Elements through Large Metropolitan Sewage Treatment Plants[END_REF]. As a result of these anthropogenic contaminations, the presence of high level Lanthanide concentrations has been detected worldwide from ng to µg per liter among different aquatic compartments [START_REF] Bau | Anthropogenic origin of positive gadolinium anomalies in river waters[END_REF]Kulaksiz and Bau, 2013;[START_REF] Perrat | Bioaccumulation of gadolinium in freshwater bivalves[END_REF]. Up to now, except for the Netherlands, no other country has set regulatory environmental discharge thresholds of Lanthanides [START_REF] Sneller | Maximum Permissible Concentrations and Negligible Concentrations for Rare Earth Elements (REEs)[END_REF].

The current literature on the ecotoxicity of Lanthanides is mostly confined to Lanthanum, Cerium, and Gadolinium, and studies of Lanthanide mixtures remain limited [START_REF] Herrmann | Aquatic ecotoxicity of lanthanum -A review and an attempt to derive water and sediment quality criteria[END_REF][START_REF] Romero-Freire | Assessment of the toxic effects of mixtures of three lanthanides (Ce, Gd, Lu) to aquatic biota[END_REF][START_REF] Hanana | Toxicity of representative mixture of five rare earth elements in juvenile rainbow trout (Oncorhynchus mykiss) juveniles[END_REF]. Until now, the question of a common toxicity profile for Lanthanides is not solved, some authors agree with this assumption [START_REF] Blinova | Evaluation of the potential hazard of lanthanides to freshwater microcrustaceans[END_REF] while others mentioned a toxicity related to the atomic number (Manusadzianas et al., 2020). Due to the lack of information, no consensus about the Lanthanide ecotoxicity may be established. This lack of consensus emphasized the emerging need to investigate the toxicity of Lanthanides tested alone and in mixtures [START_REF] Pagano | Health effects and toxicity mechanisms of rare earth elements-Knowledge gaps and research prospects[END_REF].

Cells are one of the first levels of organization impacted by pollutants [START_REF] Olabarrieta | In vitro effects of cadmium on two different animal cell models[END_REF], and cell culturing is recognized as a sensitive tool for (eco)toxicity assessment. Numerous studies have used in vitro approaches for human health risk assessment of Lanthanides, but few of them have dealt with aquatic ecotoxicological investigations [START_REF] Pagano | Health effects and toxicity mechanisms of rare earth elements-Knowledge gaps and research prospects[END_REF].

In this context, the aim of our study was to assess the cellular ecotoxicity of Lanthanides. This evaluation was performed by using three fish cell lines, i.e., ZF4 (ATCC ® CRL-2050 ™ ) and ZFL (ATCC ® CRL-2643 ™ ) cells from Danio rerio and RTgill-W1 (ATCC ® CRL-2523 ™ ) cells from Oncorhynchus mykiss.

Firstly, the individual toxic effects of all stable Lanthanides were determined on ZF4 cells.

Then, three of them, i.e., Neodymium (Nd 3+ ), Gadolinium (Gd 3+ ), and Ytterbium (Yb 3+ ) were tested in mixtures on ZF4 cells and two of them i.e., Neodymium (Nd 3+ ) and Ytterbium (Yb 3+ ) were tested alone and in mixtures on ZFL and RT-W1 cells. Differences of toxicity between Lanthanides alone, in mixtures, and on the different cell lines are discussed.

Materials and methods

Chemicals

Stock solutions (5 g/L as salt concentration) were prepared from the following Lanthanide salts: LaCl3-7H2O, CeCl3-7H2O, Pr(NO3)3-6H2O, Nd(NO3)3-6H2O, Sm(NO3)3-6H2O, EuCl3-6H2O, GdCl3-6H2O, TbCl3-6H2O, Dy(NO3)3-6H2O, HoCl3-6H2O, Er(NO3)3-5H2O, Tm(NO3)3-5H2O, YbCl3-6H2O, and LuCl3-6H2O, purchased from Sigma Aldrich, St. Louis, Missouri U.S.A. The purity of all salts was about 99.9 %. Two days prior to experiments, exposure concentrations were obtained by diluting the stock solutions in ultrapure water and stored at 4 °C.

Cell culture

Cell culture was performed as previously reported by [START_REF] Parant | Impact of gadolinium-based contrast agents on the growth of fish cells lines[END_REF]. Briefly: (a) Danio rerio fibroblast-like cells (ZF4; ATCC ® CRL-2050 ™ ) were maintained in a humidified chamber

Toxicity assessment of Lanthanides

Toxicity assessment of individual Lanthanides

ZF4 cells were exposed to the fourteen stable Lanthanides (from Lanthanum to Lutetium except Promethium) at nominal concentrations of 0, 0.5, 5, 50, 500, 1000, 1500, 2000, and 3000 µM during 24, 48, 72, and 96 h, with 4 replicates per condition. For ZFL and RTgill-W1 cells, similar exposure conditions were conducted in the presence of 2 Lanthanides (Nd 3+ and Yb 3+ ).

Toxicity assessment of Lanthanides mixtures

Frequently described as representative of light, medium and heavy Lanthanides, 3 Lanthanides (Nd 3+ , Gd 3+ , and Yb 3+ ) were chosen for the toxicity assessment of Lanthanides mixtures on ZF4 cells. Mixtures of Nd 3+ and Yb 3+ were tested on ZFL and RTgill-W1 cells. Nominal Lanthanides concentrations were selected according to EC values (i.e., EC5, EC10, EC50) determined on ZF4 cells. Respectively, these concentrations were: 164, 235, and 672 µM for Nd 3+ ; 220, 290, and 849 µM for Gd 3+ ; 149, 209, and 576 µM for Yb 3+ . ZF4 cells were exposed to multiple binary mixtures while ZFL and RTgill-W1 cells were exposed to some of these conditions (Table S1).

Experiments were performed for 96 h, repeated three times with six replicates per condition.

2.4.Cytotoxicity assay

According to the Mosmann procedure [START_REF] Mosmann | Rapid colorimetric assay for cellular growth and survival: Application to proliferation and cytotoxicity assays[END_REF], cell viability was evaluated using the MTT reduction assay. Fish cell lines were cultured under standard conditions for 6 days prior to experiments. Sub-confluent cells were trypsinized and counted according to the trypan blue exclusion method (0.4 %) (Gibco ® by Life Technologie ™ , ref: 15250061). ZF4, ZFL, and RTgill-W1 cells were seeded on 96-well plate at initial densities of 2 x 10 4 , 1 x 10 4 , and 3.5 x 10 4 cells/well in 200 µL of culture medium, respectively. After 96 h of exposure, 20 µL of 5 mg/mL MTT (3-(4,5-dimethylthiazol-2-yl) 2,5-diphenyl tetrazolium bromide) was added in each well for control and exposed cells. After 3.5 h of incubation, the exposure medium was removed, and formazan salts were dissolved in 100 µL DMSO under agitation for 4 min.

Absorbance was measured at 570 nm using a microplate spectrophotometer (Asys Expert Plus).

Cell viability was calculated according to the formula: cell viability (%) = (absorbance of exposed cells/absorbance of control) x 100.

Measurement of Lanthanide concentrations

Lanthanide concentrations were measured by Inductively Coupled Plasma Mass Spectrometry (ICP-MS) at the Laboratory Mer, Molécules, Santé (MMS, EA 2160) in Nantes, France.

Measurements were performed in the culture media of ZF4, ZFL, and RTgill-W1 cells. Each media was supplemented with the individual and mixture concentrations of Lanthanide reported in Table S1. All solutions were prepared both with (biotic) and without (abiotic) cells.

Immediately after preparation (T0) and after 96 h (T96), half of the medium (representing the total concentration) was acidified to 1 % (v/v) with concentrated HNO3 65 %. The remaining medium was filtered on a 0.45 µm nitrocellulose membrane (Millipore, ref: HAWPO4700).

The filtrate obtained, representing the dissolved Lanthanides, was acidified to 1 % (v/v) with HNO3 65 % and the filter, representing the undissolved Lanthanides, was stored in a tube. All acidified solutions and the filters were stored at 4 °C prior to ICP-MS analyses.

Statistical methods

Analysis of the individual toxicity of Lanthanides

For each Lanthanide, a dose-response curve was fitted to the viability data using the cumulative distribution function (CDF) of the log-logistic probability distribution. Effective concentrations (EC) inducing 5 % (EC5), 10 % (EC10), and 50 % (EC50) effect on the cell viability with associated 95 % confidence intervals were calculated from these models.

A three-way ANOVA was performed to investigate the effects of Lanthanide, cell type, and time on toxicity, expressed as EC50, after log transformation. To this end, the analysis was carried out on the complete factorial design, associated with the factors "Lanthanide" (2 levels: Nd +3 , Yb 3+ ) "Cell" (3 levels: ZF4, ZFL, RTgill-W1), and "Time" (4 levels: 24h, 48h, 72h and 96h), respectively.

Analyses were performed using R software (version 4.0.0, DRC package, R Development Core Team, 2020).

Analysis of the toxicity of Lanthanides mixtures

The toxicity of Lanthanide mixtures was analyzed using parametric statistical methods. The hypothesis of homogeneity of variances within replicate groups, associated with the different combinations of concentrations considered, was preliminarily tested using the modified Levene test (i.e., the Brown-Forsythe test for the homogeneity of variances).

A weighted log-linear regression model (WLR) was used to describe the variation of cell viability, as a function of two Lanthanide concentrations in each mixture. The model also considered an interaction term, denoted by the sign * in Equation 1:

Log10 (cell viability) = β0 + β1 Lanthanide 1 + β2 Lanthanide 2 + β12 Lanthanide 1 * Lanthanide 2 Eq.1
The use of the WLR model is motivated, on the one hand, by its ability to better meet the requirements of parametric statistical methods (i.e., normality and variance homogeneity) and, on the other hand, to consider the multiplicative nature of the toxic effects on cell viability of the Lanthanides in the mixtures. The weight assigned to each data point was the mean cell viability of the assay (i.e., the condition considered) to which it belonged.

The testing of the regression coefficients (i.e., β0, β1, β2, and β12), and the subsequent construction of the fitted regression models, was carried out following a backward stepwise procedure with a significance threshold set at α = 0.01 for all statistical tests. The predicted response in each mixture with the WLR model was graphically processed using the response surface methodology (RSM approach). The viability rates calculated from each fitted model were thus represented in both a 3D surface plot and a 2D contour plot. Statistical analyses were performed using R software (version 4.0.0, plot3D package, R Development Core Team, 2020).

Results

Lanthanide concentrations in culture media

As described in Materials and Methods (Table S1) a large panel of concentrations was measured for three Lanthanides (Yb 3+ , Nd 3+ and Gd 3+ ) after supplementation of the medium (T0) and after 96h (T96) of exposure. Fig. 1A shows results obtained for one concentration of Yb 3+ on ZF4 cell medium. These results are representative of those obtained with other Lanthanides for all tested concentrations on ZF4 cells (data not shown). Similar trends were observed for the three Lanthanides in ZFL and RTgills-W1 culture media (Fig. S1).

For all tested conditions regardless of Lanthanides, Lanthanides concentration alone or in mixtures and cell lines, total concentrations measured at T0 and after T96 were between 85 and 99 % of the nominal ones (Fig. 1B). In the remainder of the paper, Lanthanide concentrations will therefore be expressed as the corresponding nominal values for all results. Regarding quantification in abiotic conditions, the concentration of Lanthanides measured in the unfiltered medium (i.e. "measured total concentration") was of the same order of magnitude than the one measured in the filtrate, just after supplementation (T0) and after 96 h (T96h) of exposure for all cell culture media. These results suggested that most of the Lanthanides were dissolved in the culture media and that undissolved fraction was negligible (less than 5% of the total measured concentration).

The same trend was observed in the presence of cells (biotic condition) after 96 h of exposure.

(Fig. 1).

Impact of Lanthanides on ZF4 cells

Individual toxicity of 14 Lanthanides on ZF4 cells

The toxicity of the fourteen Lanthanides (from Lanthanum to Lutetium) was assessed on ZF4 cells. Dose-response curves obtained for Neodymium, Gadolinium and Ytterbium for 24 to 96 h of exposure are given in Fig. 2 (dose-response curves for the other Lanthanides are presented in Fig. S2).

ZF4 cell viability was not impacted by the lowest concentrations (from 0.5 to 50 µM), but decreased for concentrations ranging from 500 to 3000 µM, and 100 % mortality was observed at 3000 µM, from 48h of exposure. According to EC50 values, Lanthanides were significantly more toxic (1.3 -2 times) after 48, 72, and 96 h of exposure (Table 1 andTable S2) when compared to 24 h of exposure. After 96 h of exposure, Er 3+ , Tm 3+ , Yb 3+ , and Lu 3+ were the most toxic elements toward ZF4 cells with EC50 values < 600 µM, whereas Eu 3+ and Tb 3+ presented the lowest toxicity with EC50 values > 1000 µM. As shown in Table 2, the cell viability was reduced in the presence of Lanthanides for all tested conditions. To understand this toxicity, the combined effects of Lanthanides in mixtures were studied. The assumption of homogeneity of the variance of the data within the various conditions was not rejected for any mixture examined (Levene test, p > 0.05). Thus, a WLR analysis was performed using all available viability data. The fitted models presented in Table S3 express the variation of cell viability for each mixture studied according to the regression model in Eq.1. The values resulting from the fitting procedure (i.e., F-test, p < 10 -4 ; r² > 0.88; residual coefficient of variation < 15 %) suggested both high explanatory power and good predictive capacity of the fitted models. Moreover, all regression coefficients in these models were statistically significant (p < 0.01) and negative. This indicates a toxic effect of each of the two Lanthanides, as well as the existence of a synergistic effect for each of the binary mixtures considered.

The predicted combined effect of each mixture was explored using the RSM approach. Thus, the effects for the Nd 3+ and Yb 3+ mixture were represented on the 3D plot of the regression surface (Fig. 3A), as well as by the 2D survival level contours, i.e., the isobologram in Fig. 3B.

The synergistic effect, which means that the combination of two lanthanides shows a higher toxicity than expected from the individual components, can be seen from the isobologram in Fig. 3B. Indeed, in the presence of an interaction effect, the isoboles are curved, while they would be straight lines in the absence of interaction. Furthermore, the curvature of the lines (i.e., concavity or convexity) helps to decide between antagonism and synergy, respectively.

Here, the convexity of the isoboles in Fig. 3B reflects a synergistic effect. The same phenomenon was observed for the other mixtures examined in the present study (Fig. S3).

-Impact of Lanthanides on ZFL and RTgill-W1 cells

-Individual Toxicity of Nd 3+ and Yb 3+ on ZFL and RTgill-W1 cells

Cells were exposed to Nd 3+ and Yb 3+ under the same conditions as ZF4 cells. The responses of ZFL (Fig. S4) and RTgill-W1 (Fig. S5) cells were like those observed for ZF4 cells. Indeed, results showed no effect of the lowest Lanthanide concentrations (from 0.5 to 50 µM) and a gradual decrease in cell viability for Lanthanide concentrations ranging from 500 to 3000 µM.

The three-way ANOVA, performed on the data in Table 3, leads to the following conclusions:

(i) there is no statistically significant difference between the mean EC50 of the two Lanthanides considered (p>0.30); (ii) there are differences between the three cell lines: specifically, the mean EC50 value observed for RTgill-W1 cells is significantly higher than the corresponding values of ZF4 and ZFL cells (p<10 -3 ), which are not statistically different from each other; and

(iii) the mean EC50s observed for all factors decrease with time (p<10 -3 ).

Thus, based on EC50 values, the toxicity of Lanthanides increased from 24 to 96 h for all three cell lines (Table 3) RTgill-W1 cells seemed to be less sensitive than ZF4 and ZFL cells, from 48h and especially after 96h of exposure. Based on EC50 values, Nd 3+ was clearly more toxic for ZFL cells compared to ZF4 and RTgill-W1 cells after 96h of exposure.

-Toxicity of Nd 3+ and Yb 3+ mixtures on ZFL and RTgill-W1 cells

To determine whether the toxicity induced by the Lanthanide mixtures on ZF4 cells varied among cell lines, ZFL and RTgill-W1 cells were exposed for 96 h to different Nd 3+ and Yb 3+ concentrations, tested alone and in mixtures. The tested concentrations are EC5, EC10 and EC50 values obtained on ZF4 cells.

As shown in Table 4, the viability of ZFL and RTgill-W1 cells was reduced in the presence of Lanthanides. For these experiments, it may be also noted that the RTgill-W1 cells were less sensitive, with higher viability of cells exposed to Lanthanides tested individually or in binary mixtures, compared to ZFL cells (two-sided Wilcoxon signed-rank test; p ≤ 0.05).

Table 4. ZFL and RTgill-W1 cell viability calculated after 96 h of exposure to Neodymium (Nd 3+ ) and Ytterbium (Yb 3+ ), tested alone and in mixtures. Experiments were repeated three times with six replicates per condition. The percentage of cell viability corresponding to each condition was expressed as the mean ± one standard deviation in parentheses.

Furthermore, the combined effect of Nd 3+ and Yb 3+ in the mixture was assessed both on ZFL and RTgill-W1 cells by performing a WLR analysis, as previously described for ZF4 cells.

The values obtained from the fitting procedure (i.e., F-test, p < 10 -4 ; r² > 0.88, residual coefficient of variation < 15 %) indicated a good fit of the models to the experimental data, as well as high explanatory power and good predictive ability. As shown in Table S4, all regression coefficients of the fitted models were statistically significant (p < 0.01) and negative.

This result led to the same conclusion as for ZF4 cells, namely that a synergistic effect existed in addition to the direct effects of Nd 3+ and Yb 3+ on both ZFL and RTgill-W1 cells.

As observed on ZFL (Fig. S6A) and RTgill-W1 cells (Fig. S6C), the 3D surface plots showed that for a given Nd 3+ concentration, the toxicity increased with the Yb 3+ concentration, and vice versa. However, the toxicity of Lanthanide mixtures was lower for RTgill-W1 cells, with cell viability ranging from 45 to 100 % (Fig. S6C) than for ZFL cells (Fig. 6A).

Furthermore, in addition to the fitted models in Table S4, which highlighted the presence of a synergistic effect, the 2D contour plots, i.e., the isobolograms in Fig. S6B and Fig. S6D, allow visualization of this effect for both cell lines. However, the synergistic effect appears to be more pronounced for ZFL cells than for RTgill-W1 cells, as the convexity of the isoboles is more marked in Fig. S6B than in Fig. S6D, respectively

Discussion

In aquatic ecotoxicology, the bioavailability of Lanthanides remains an important issue to understand their toxicity and assess their environmental risk. Because Lanthanides can form precipitates in the presence of carbonates, phosphates, and sulfates in culture medium (Petersen et al., 1974;[START_REF] Ng | Review of aquatic effects of lanthanides and other uncommon elements 42[END_REF], [START_REF] Ng | Review of aquatic effects of lanthanides and other uncommon elements 42[END_REF] suggested that their toxicity could be linked to indirect effects caused by such precipitates, as previously demonstrated by Petersen et al.

( 1974). The analytical results conducted on total medium spiked with lanthanides and on the filtrates showed that more that 95% of the Lanthanides were in solution, suggesting that, in our medium, the precipitation of Lanthanides was negligible. Nevertheless, the mechanical impact of aggregates, even in small amounts, could not be excluded. Then, it would be of great interest to assess precisely the size of Lanthanides aggregates and then to evaluate the impact of inert aggregates with such size on the cell viability. As previously described by Bergna et al. (1994), silicon dioxide can be organized as aggregates with a size slightly higher than 0.45 µm (corresponding of the filtering size of our filters) and could be used for such experiments. A preliminary experiment was conducted in this way (data not shown) and did not show any toxicity on ZF4 cells at concentrations close to the Lanthanides one. Therefore, it might be assumed that the observed toxicity on fish cell lines during our experiments was mainly induced by the direct effects of Lanthanides, even in the presence of tiny amounts of aggregates.

Concentration-dependent effects of Lanthanides were reported in vitro on the cell viability of human (Rucki et al., 2020), rat [START_REF] Yang | Living Target of Ce(III) Action on Horseradish Cells: Proteins on/in Cell Membrane[END_REF], and fish cell lines [START_REF] Laville | Effects of human pharmaceuticals on cytotoxicity, EROD activity and ROS production in fish hepatocytes[END_REF], but also in vivo using bioassays with aquatic organisms (Gonzalez et al., 2015;[START_REF] Romero-Freire | Assessment of the toxic effects of mixtures of three lanthanides (Ce, Gd, Lu) to aquatic biota[END_REF][START_REF] Blinova | Potential Hazard of Lanthanides and Lanthanide-Based Nanoparticles to Aquatic Ecosystems: Data Gaps, Challenges and Future Research Needs Derived from Bibliometric Analysis[END_REF]. [START_REF] Laville | Effects of human pharmaceuticals on cytotoxicity, EROD activity and ROS production in fish hepatocytes[END_REF] used the MTT assay to investigate Gadolinium toxicity on primary hepatocyte cells from Oncorhynchus mykiss and hepatocyte cell line (PLHC-1) from Poeciliopsis lucida. These authors observed a decrease in the cell viability, after 24 h of exposure to Gadolinium concentrations ranging from 7 µM to 1 mM. Our results agree with these observations. For all exposure durations, concentrations-response curves showed Lanthanide toxicity on ZF4 cells, leading to 100 % mortality at the highest tested concentrations.

Although several adverse effects of Lanthanides have been previously reported, no consensus exists as to regular changes in their toxicity along the Lanthanides series. Some authors [START_REF] Blinova | Evaluation of the potential hazard of lanthanides to freshwater microcrustaceans[END_REF] suggested that Lanthanides should be considered as a uniform group with all elements exhibiting similar toxicity, while others mentioned that their toxicity appears to change as a function of atomic number (Manusadzianas et al., 2020). Our results are not in agreement with [START_REF] Blinova | Evaluation of the potential hazard of lanthanides to freshwater microcrustaceans[END_REF] nor with Manusadzianas et al. ( 2020). Indeed, the impact observed on ZF4 cells shows a high toxicity concerning the four heaviest Lanthanides (Er 3+ , Tm 3+ , Yb 3+ , and Lu 3+ ) but no correlation appears by the comparison of EC50 of light and intermediate Lanthanides depending on their atomic number. Two of them (Eu 3+ and Tb 3+ ) are clearly less toxic when the others display a toxicity of the same order. These observations underlined the need to assess not only La 3+ , Ce 3+ , and Gd 3+ as frequently mentioned in the current literature, but also the other Lanthanides, especially the heavy ones, which are the most toxic but the less studied [START_REF] Pagano | Health effects and toxicity mechanisms of rare earth elements-Knowledge gaps and research prospects[END_REF][START_REF] Herrmann | Aquatic ecotoxicity of lanthanum -A review and an attempt to derive water and sediment quality criteria[END_REF].

Our results clearly show that, in our experimental conditions, all Lanthanides do not induce the same toxicity without displaying any clear relation between the toxicity and the order of the elements within the Lanthanides series. From a molecular point of view, the observed differences of toxicity between Lanthanides could be explained by their abilities to compete differently within cells with essential metallic ions such as Calcium [START_REF] Ansoborlo | Actinide speciation in relation to biological processes[END_REF][START_REF] Bresson | Radionuclide speciation: A key point in the field of nuclear toxicology studies[END_REF].

During our experiments, 8 Lanthanides were used as chloride salts (Yb 3+ , Lu 3+ , La 3+ , Ce 3+ , Eu 3+ , Gd 3+ , Tb 3+ and Ho 3+ ) when 6 of them were used as nitrate salts (Er 3+ , Tm 3+ , Pr 3+ , Nd 3+ , Sm 3+

and Dy 3+ ). It is interesting to note that no link was observed between the salt form used and the toxicity observed. For instance, between the four Lanthanides displaying the highest toxicity (Yb 3+ , Lu 3+ , Er 3+ , Tm 3+ ) two of them were used as chloride salts and the two others were used as nitrate salts.

As previously described in the literature, environmental concentrations of Lanthanides in rivers range from 1 to several ng/L with some spot area showing sometimes high concentrations close to several µg/L (Kulaksız andBau, 2013, Parant et al. 2018). EC50 values measured on ZF4 cells range from to 40 to 160 mg/L. The large difference between the range of these values

shows that even if a cellular toxicity of Lanthanides is observed during our experiments, the risk of cellular toxicity in aquatic organisms in the current situation is low.

Mixture studies are qualified as a more realistic point of view to determine the effects of pollutants in aquatic systems [START_REF] Altenburger | Mixture effects in samples of multiple contaminants -An interlaboratory study with manifold bioassays[END_REF]. In environmental toxicology, to predict and understand the biological responses induced by the effects of mixtures (i.e., synergism, additivivity or antagonism) is one of the greatest challenges, and at the same time an essential aspect of ecotoxicological risk assessment [START_REF] Norwood | Effects of Metal Mixtures on Aquatic Biota: A Review of Observations and Methods[END_REF]. To date, few studies were carried out to assess the impacts of Lanthanide mixtures using various organisms such bacteria, algae, rotifers, daphnids, ostracods, and fish [START_REF] Hao | Bioconcentration of Rare Earth Elements lanthanum, gadolinium and yttrium in algae (Chlorella Vulgarize Beijerinck): Influence of chemical species[END_REF][START_REF] Tai | Biological toxicity of lanthanide elements on algae[END_REF][START_REF] Tan | Biotic ligand model explains the effects of competition but not complexation for Sm biouptake by Chlamydomonas reinhardtii[END_REF][START_REF] Blinova | Evaluation of the potential hazard of lanthanides to freshwater microcrustaceans[END_REF][START_REF] Romero-Freire | Assessment of the toxic effects of mixtures of three lanthanides (Ce, Gd, Lu) to aquatic biota[END_REF][START_REF] Hanana | Toxicity of representative mixture of five rare earth elements in juvenile rainbow trout (Oncorhynchus mykiss) juveniles[END_REF]. In these studies, different mixture toxicities were observed, according to Lanthanide compositions. [START_REF] Tai | Biological toxicity of lanthanide elements on algae[END_REF] showed an additive effect for the mixture of fourteen Lanthanides on Skeletetonema costatum, while a synergistic effect was found on Chlamydomonas reinhardtii in the presence of Sm 3+ , mixtures, Romero et al. (2019) showed synergistic effects on bacteria and algae, and antagonistic ones on rotifers, by means on standardized bioassays. Based on these observations, it is a challenge to analyze the effects of Lanthanide mixtures at the organism level due to the diversity of studied endpoints, biological variations, stress factors, and sensitivity of the selected organisms [START_REF] Liess | Predicting the synergy of multiple stress effects[END_REF]. Moreover, possible invalid data interpretations may result due to a lack of statistical analysis of the interaction effects [START_REF] Kerr | Modeling dose response using generalized linear models[END_REF].

The combination of studies at the cellular level with statistical and modelling methods could be an interesting approach to overcome these limitations. The results of the WLR analysis and the RSM approach used in the present study showed no antagonist effects for all mixtures tested whatever the cell line used. Dose-response curves showed concentration-dependent effects of Lanthanides on cell viability, as well as synergistic effects for each of the five mixtures examined mixture, which were more pronounced at higher concentrations after 96 h of exposure.

Our results are in accordance with [START_REF] Jonker | Significance testing of synergistic/antagonistic, dose level-dependent, or dose ratio-dependent effects in mixture dose-response analysis[END_REF], namely synergistic effect becoming more marked at high pollutant concentrations. Although synergistic effect was observed and confirmed by statistical tests for each of the five mixtures examined in this study, we must acknowledge that, at this time, we have no mechanistic or biological explanation to offer for interpreting synergism. Further studies are therefore needed to confirm these observations and eventually clarify this phenomenon. We suggest that statistical and modelling methods could be alternative and more powerful tools to predict the toxicity of Lanthanide mixtures compared with the classical models of concentration addition (Lowe and Muischeck, 1926) and independent action [START_REF] Bliss | The Toxicity of Poisons Applied Jointly1[END_REF].

Three fish cell lines were used to evaluate Nd 3+ , Gd 3+ , and Yb 3+ toxicity. RTgill-W1 appeared as the less sensitive cells, compared to the two others, exposed on Lanthanides alone as well as in mixture. Gills are the first tissues in contact with the external medium and possess potentially more efficient defense mechanisms against external aggressions [START_REF] Perrat | Bioaccumulation of gadolinium in freshwater bivalves[END_REF], which can make them less sensitive than other cellular models. No clear difference of sensitivity in both individual or mixture toxicity of Lanthanides could be observed between ZF4 and ZFL cells.

Our results suggested that the in vitro approaches to assess the biological responses of living organisms exposed to abnormal Lanthanide concentrations in aquatic systems can be useful to compare the sensitivity of cells with various tissue origins.

Conclusion

Lanthanides are frequently employed in new technologies and are considered as emerging pollutants due to their growing release into the aquatic environment. Increased risks of interaction with living organisms can therefore be expected. In an innovative way, the present study is conducted from an aquatic environmental point of view by evaluating the cellular impact of Lanthanides individually and in mixtures. However, it is still uncertain whether all

Lanthanides have the same toxicity, regarding few studies in the literature carried out on the toxicity of some elements. Our results highlighted that Nd 3+ , Gd 3+ , and Yb 3+ have a toxic impact on fish cell lines but could not be considered as representatives of the toxicity of the other light, medium, and heavy Lanthanides, respectively. The evaluation of the toxicity of the fourteen Lanthanides on ZF4 cells showed, on the one hand, that the heavy Lanthanides (i.e., Er 3+ , Tm 3+ , Yb 3+ , and Lu 3+ ) were the most toxic and, on the other hand, that Eu 3+ and Tb 3+ , one medium and one light Lanthanide, were the least toxic elements. More generally, these observations did not show the existence of a relationship between increasing toxicity and atomic mass of lanthanides. Thus, our results support the need to pursue more investigations on the toxicity of all the fourteen Lanthanides to better understand their intracellular behavior (specific abilities to bind with biomolecules and to compete with essential ions) in order to elucidate the reasons of differences in toxicity among individual elements.

The analysis of Lanthanide mixtures highlighted synergistic effects in all tests conducted with the three cell lines from two fish models. Synergistic effects became stronger at high Lanthanide concentrations confirming the need of further research on the toxic potential of their mixtures in aquatic environments. The use of in vitro approaches coupled with statistical and modelling methods appears to be an effective way to assess the ecotoxicological risk of Lanthanides and their mixtures.

(

  28 °C, 5 % CO2) in DMEM: F12 medium (ATCC ® 30-2006 ™ ) supplemented with 10 % (v/v) fetal calf serum (ATCC ® 30-2021 ™ ). The medium was renewed every 3 days. Cells were passed once a week at 1 x 10 6 cells/flask T75; (b) Danio rerio hepatic cells (ZFL; ATCC ® CRL-2643 ™ ) were cultured in mixed medium composed by 50 % (v/v) Leibovitz's L-15 medium (ATCC ® 30-2008 ™ ), 35 % (v/v) DMEM HG (Gibco ® by Life Technologie ™ , ref: 12100), 15 % (v/v) Ham's F12 (Gibco ® by Life Technologie ™ , ref: 21700), 0.15 g/L (v/v) sodium bicarbonate solution (Sigma Aldrich, ref: 58761), 15 mM (v/v) HEPES solution (Sigma Aldrich, ref: H0887), 0.01 mg/mL (v/v) bovine insulin (Sigma Aldrich, ref: E4127), and 5 % (v/v) inactivated fetal calf serum (ATCC ® 30-2020 ™ ). Cells were incubated at 28 °C without CO2 and the medium was renewed every 3 days. Cells were passed twice a week at 1.25 x 10 6 cells/flask T75; (c) Oncorhynchus mykiss epithelial cells (RTgill-W1; ATCC ® CRL-2523 ™ ) were maintained at 15 °C without CO2 in Leibovitz's L-15 medium (Gibco ® by Life Technologie ™ , ref: 11415-064) supplemented with 10 % (v/v) fetal calf serum (Eurobio Abcys , ref: CVFSVF00-01). The medium was renewed every 3 days. Cells were passed once a week at 2.5 x 10 6 cells/flask T75.

Fig. 1 .

 1 Fig. 1. (A) Concentration (mg/L) of Ytterbium (Yb 3+ ) under abiotic and biotic conditions after the supplementation (T0) and after 96 h (T96h) of exposure in the media of ZF4 cells. (B) percentage of nominal concentration considering all tested conditions under abiotic and biotic conditions after the supplementation (T0) and after 96 h (T96h) of exposure in the media regardless of Lanthanides, Lanthanides concentration (alone or in mixtures) and cell lines. Data are expressed as the mean ± one standard deviation (n = 3) per condition.

Fig. 2 .

 2 Fig. 2. Dose-response curves for the viability of ZF4 cells exposed for 24 to 96 h to Neodymium (A), Gadolinium (B) and Ytterbium (C) at nominal concentrations ranging from 0.5 to 3000 µM. Data are expressed as the mean ± one standard deviation (n = 3) per condition.

Fig. 3 .

 3 Fig. 3. (A) Three-dimensional response surface plot (3D) showing the combined effect of the Neodymium (Nd 3+ ) and Ytterbium (Yb 3+ ) mixtures on the viability of ZF4 cells after 96 h of exposure; (B) Two-dimensional level contour plots (2D), i.e. isobologram showing the relationship between combinations of Neodymium (Nd 3+ ) and Ytterbium (Yb 3+ ) concentrations to attain predetermined values of ZF4 cell viability after 96 h of exposure (print in color).

Fig. S2 .

 S2 Fig. S2. Dose-response curves for the viability of ZF4 cells exposed for 24 to 96 h to Lanthanium (A), Cerium (B), Praseodymium (C), Samarium (D), Europium (E), Terbium (F), Dysprosium (G), Holmium (H), Erbium (I), Thulium (J) and Lutetium (K) at nominal concentrations ranging from 0.5 to 3000 µM. Data are expressed as the mean ± one standard deviation (n = 3) per condition.

1 2

 1 Fig. S3. Three-dimensional response surface plot (3D) showing the effect of Neodymium (Nd 3+ ) and Gadolinium

  Fig. S4. Dose-response curves fitted on the viability data of ZFL cells exposed to Neodymium (Nd 3+ ) and Ytterbium (Yb 3+ ) at nominal concentrations ranging from 0.5 to 3000 µM for 24 (A), 48 (B), 72 (C), and 96 h (D).Data are expressed as the mean ± the standard deviation (n = 4) per condition.

Fig. S5 .

 S5 Fig. S5. Dose-response curves fitted on the viability data of RTgill-W1 cells exposed to Neodymium (Nd 3+ ) and

Table 1 .

 1 Effective concentrations (with associated 95 % confidence intervals between square brackets) inducing 50 % (EC50) of the effect on ZF4 cell viability. Values were calculated from dose-response curves obtained after 96 h of exposure to fourteen Lanthanides.

			EC50 [95 % CI]
	Lanthanide	Time (h)	Molarity concentration (µM)
		24	1633 [1430 -1836]		1188 [1014 -1363]
		48	1027 [916 -1139]		994 [845 -1143]
	Lanthanum	72	966 [862 -1071]	Terbium	928 [799 -1056]
		96	864 [760 -969]		1069 [865 -1273]
		24	1546 [1304 -1788]		1079 [809 -1349]
		48	907 [792 -1023]		753 [661 -845]
	Cerium	72	835 [700 -971]	Dysprosium	674 [610 -739]
		96	760 [663 -857]		668 [593 -744]
		24	1951 [1318 -2585]		1131 [978 -1283]
		48	800 [675 -924]		936 [778 -1093]
	Praseodymium	72	883 [770 -996]	Holmium	805 [683 -926]
		96	793 [706 -879]		673 [547 -799]
		24	1504 [1287 -1722]		731 [638 -825]
		48	665 [557 -773]		583 [517 -649]
	Neodymium	72	583 [525 -641]	Erbium	570 [530 -609]
		96	672 [567 -776]		480 [449 -512]
		24	1479 [1193 -1766]		1360 [1181 -1539]
		48	858 [646 -1071]		777 [659 -895]
	Samarium	72	841 [700 -982]	Thulium	747 [635 -860]
		96	823 [696 -950]		501 [298 -705]
		24	1500 [1235 -1765]		1342 [1201 -1482]
		48	1003 [834 -1173]		698 [638 -758]
	Europium	72	1053 [911 -1196]	Ytterbium	715 [611 -818]
		96	1047 [917 -1178]		576 [488 -665]
		24	1601 [1319 -1882]		1145 [1001 -1289]
		48	919 [758 -1079]		587 [493 -680]
	Gadolinium	72	890 [747 -1032]	Lutetium	312 [0 -650]
		96	849 [717 -981]		242 [79 -404]

Toxicity of Nd 3+ , Gd 3+ , and Yb 3+ mixtures on ZF4 cells ZF4

  cells were exposed for 96 h to different concentrations of 3 Lanthanides (Nd 3+ , Gd 3+ , and

	Yb 3+ ), tested individually and in binary mixtures. Mixture experiments were conducted using
	the EC5, EC10, and EC50 values calculated on ZF4 cells after 96 h of exposure.

Table 2 .

 2 ZF4 cell viability calculated after 96 h of exposure to Neodymium (Nd 3+ ), Gadolinium (Gd 3+ ), and Ytterbium (Yb 3+ ) tested alone and in mixtures. Experiments were repeated three times with six replicates per condition. The percentage of cell viability corresponding to each condition was expressed as the mean ± one standard deviation in parentheses.

	Lanthanide		Mixture Nd 3+ + Yb 3+
	Condition	Cell viability (%) Condition	Cell viability (%)
	Control	100 (± 1)	EC5 Nd 3+ + EC5 Yb 3+	69 (± 6)
	EC5 Nd 3+ : 164 µM	93 (± 4)	EC5 Nd 3+ + EC10 Yb 3+	58 (± 11)
	EC5 Gd 3+ : 220 µM	89 (± 4)	EC5Nd 3+ + EC50Yb 3+	30 (± 9)
	EC5 Yb 3+ : 149 µM	81 (± 2)	EC10 Nd 3+ + EC5Yb 3+	58 (± 13)
	EC10 Nd 3+ : 235µM	85 (± 2)	EC10 Nd 3+ + EC10 Yb 3+	54 (± 7)
	EC10 Gd 3+ : 290 µM	80 (± 4)	EC50 Nd 3+ + EC5 Yb 3+	44 (± 14)
	EC10 Yb 3+ : 209 µM	70 (± 5)	EC50 Nd 3+ + EC50 Yb 3+	4 (± 3)
	EC50 Nd 3+ : 672 µM	48 (± 11)
	EC50 Gd 3+ : 849 µM	51 (± 7)
	EC50 Yb 3+ : 576 µM	51 (± 5)
	Mixture Nd 3+ + Gd 3+		Mixture Gd 3+ + Yb 3+
	Condition	Cell viability (%) Condition	Cell viability (%)
	EC5 Nd 3+ + EC5 Gd 3+	80 (± 4)	EC5 Gd 3+ + EC5 Yb 3+	61 (± 5)

Table 3 .

 3 Effective concentrations (with associated 95 % confidence intervals in brackets) inducing 50 % (EC50) loss in cell viability in ZF4, ZFL, and RTgill-W1 cell lines exposed to Neodymium and Ytterbium. Values were calculated from dose-response curves after 24, 48, 72, and 96 h of exposure to Neodymium and Ytterbium.

	EC50 [95 % CI]

Table S1 .

 S1 Nominal Lanthanide concentrations tested alone and in mixtures on ZF4 cells. Bold text represents the 39 conditions tested on ZFL and RTgill-W1 cells.

	40
	41
	Individual exposures
		Nd 3+	Gd 3+	Yb 3+
	EC5:	164 µM	220 µM	149 µM
	EC10:	235 µM	290 µM	209 µM
	EC50:	672 µM	849 µM	576 µM
	Mixture exposures
	Nd 3+ + Gd 3+		Nd 3+ + Yb 3+	Gd 3+ + Yb 3+
	EC5 Nd 3+ + EC5 Gd 3+	EC5 Nd 3+ + EC5 Yb 3+	EC5 Gd 3+ + EC5 Yb 3+
	EC5 Nd 3+ + EC10 Gd 3+	EC5 Nd 3+ + EC10 Yb 3+	EC5 Gd 3+ + EC10 Yb 3+
	EC5 Nd 3+ + EC50 Gd 3++	EC5 Nd 3+ + EC50 Yb 3+	EC5 Gd 3+ + EC50 Yb 3+
	EC10 Nd 3+ + EC5 Gd 3+	EC10 Nd 3+ + EC5 Yb 3+	EC10 Gd 3+ + EC5 Yb 3+
	EC10 Nd 3+ + EC10 Gd 3+	EC10 Nd 3+ + EC10 Yb 3+	EC10 Gd 3+ + EC10 Yb 3+
	EC50 Nd 3+ + EC5 Gd 3+	EC50 Nd 3+ + EC5 Yb 3+	EC50 Gd 3+ + EC5 Yb 3+
	EC50 Nd 3+ + EC50 Gd 3+	EC50 Nd 3+ + EC50 Yb 3+	EC50 Gd 3+ + EC50 Yb 3+
	42
	43
	44

Table S2 .

 S2 Effective concentrations inducing 5 % (EC5) and 10 % (EC10) effect on ZF4 cell viability with associated

% confidence intervals between square brackets. Values are calculated from dose-response curves fitted 47 obtained for24, 48, 72, and

h of exposure to fourteen Lanthanides.

Acknowledgments

We thank Clément Bojic and Justine Flayac for technical support in cell culture, and Danièle Pauly for the preparation of the stock solutions of Lanthanides. We warmly thank Davide Vignati for his proofreading and advice in English improvement.

Funding

This work was supported by the Lorraine region and French National Research Agency through the "Ecotoxicology of Rare Earth Elements in Aquatic Systems" project with the reference ANR-16-CE34-0012/ECOTREE.

Ethics approval

The cellular models ZF4 (ATCC ® CRL-2050 ™ ) and ZFL (ATCC ® CRL-2643 ™ ) cells from Danio rerio and RTgill-W1 (ATCC ® CRL-2523 ™ ) cells from Oncorhynchus mykiss do not benefit from any particular protection status. No legal professional qualification is currently required for their experimental use in the laboratory. Table S3. WLR analysis on each binary mixture data (n = 42) from ZF4 cells. A backward stepwise procedure was performed with α = 0.01. Nd 3+ + Gd 3+ Log10 (cell viability) = -3.902 x 10 -4 Nd 3+ -2.252 x 10 -4 Gd 3+ -1.106 x 10 -6 Nd 3+ * Gd 3+ < 10 -4 0.885

CRediT authorship contribution statement

Nd 3+ + Yb 3+ Log10 (cell viability) = -3.648 x 10 -4 Nd 3+ -5.664 x 10 -4 Yb 3+ -1.432 x 10 -6 Nd 3+ * Yb 3+ < 10 -4 0.927

Gd 3+ + Yb 3+ Log10 (cell viability) = -2.922 x 10 -4 Gd 3+ -4.350 x 10 -4 Yb 3+ -6.754 x 10 -6 Gd 3+ * Yb 3+ < 10 -4 0.931


Frank A Cowell

Emmanuel Flachaire

Sanghamitra Bandyopadhyay

Reference Distributions and Inequality Measurement

Keywords: generalised entropy measures, income distribution, inequality measurement, JEL Classication: D63, C10

We investigate the general problem of comparing pairs of distribution which includes approaches to inequality measurement, the evaluation of unfair income inequality, evaluation of inequality relative to norm incomes, and goodness of t. We show how information theory can be used to provide insights on the problem and characterise a class of divergence measures using a parsimonious set of axioms. The problems of appropriate statistical implementation are discussed and empirical illustrations of the technique are provided using a variety of reference distributions.

Introduction

There is a broad class of problems in distributional analysis that involve comparing two distributions. This may involve judging whether a functional form is a good t to an empirical distribution; it may involve computation of the divergence of an empirical distribution from a theoretical economic model; it may involve the ethical evaluation of an empirical distribution with reference to some norm or ideal distribution. This paper shows how the class of problems can be characterised in a way that has a natural interpretation in terms of familiar analytical tools.

4

Our approach to the problem is based on standard results in information theory that allow one to construct a distance concept that is appropriate for characterising the divergence between the empirical distribution function and a proposed reference distribution. The connection between information theory and the economic interpretation of distributions is established by exploiting the close relationship between entropy measures (based on probability distributions) and measures of inequality and distributional change (based on distributions of income shares). The approach is adaptable to other elds in economics that make use of models of distributions. The paper is structured as follows. In Section 2 we explain the connection be-tween information theory and the analysis of income distributions. Section 3 introduces dierent concepts of reference distribution that are relevant for dierent versions of the generic problem under consideration. Section 4 sets out a set of principles for distributional comparisons in terms of aggregate divergence and show how these characterise a class of measures. Section 5 performs a set of experiments and applications using the proposed measures and UK income data. Section 6 concludes.

Information and income distribution

Comparisons of distributions using information-theoretic approaches has involved comparing entropy-based measures which quantify the discrepancies between the probability distributions. This concept was rst introduced by [START_REF] Shannon | A mathematical theory of communication[END_REF] and then further developed into a relative measure of entropy by [START_REF] Kullback | On information and suciency[END_REF]. In this section, we show that generalised entropy inequality measures are obtained by little more than a change of variables from these entropy measures. We will then use this approach to discrepancies between distributions in order to formulate appropriate inequality measures.

2.1

Entropy: basic concept Take a variable y distributed on support Y . Although it is not necessary for much of the discussion, it is often convenient to suppose that the distribution has a well-dened density function f (•) so that, by denition, ´Y f (y)dy = 1. Now consider the information conveyed by the observation that an event y ∈ Y has occurred when it is known that the density function was f . [START_REF] Shannon | A mathematical theory of communication[END_REF] suggested a simple formulation for the information function g: the information content from an observation y when the density is f is g (f (y)) =

-log f (y). The entropy is the expected information H(f ) := -E log f (y) = -ˆY log f (y) f (y)dy.

(1)

In the case of a discrete distribution, where Y is nite with index set K and the probability of event k ∈ K occurring is p k , the entropy will be -k∈K p k log p k .

Clearly g(p k ) decreases with p k capturing the idea that larger is the probability of event k the smaller is the information value of an observation that k has actually occurred; if event k is known to be certain (p k = 1)

the observation that it has occurred conveys no information and we have

g(p k ) = -log(p k ) = 0.
It is also clear that this denition implies that if k and k are two independent events then g (p

k p k ) = g (p k ) + g (p k )
It is not self-evident that the additivity property of independent events is essential and so it may be appropriate to take a generalisation of the Shannon (1948) approach 5 where g is any convex function with g (1) = 0 [START_REF] Khinchin | Mathematical Foundations of Information Theory[END_REF]). An important special case is given by g

(f ) = 1 α-1 [1 -f α-1 ]
where α > 0 is a parameter. From this we get a generalisation of (1), the α-class

entropy H α (f ) := Eg (f (y)) = 1 α -1 1 -E(f (y) α-1 ) , α > 0.
(2)

Entropy and inequality

To transfer these ideas to the analysis of income distributions it is useful to perform a transformation similar to that outlined in [START_REF] Theil | Economics and Information Theory[END_REF]. Suppose we specialise the model of Section 2.1 to the case of univariate probability distributions: instead of y ∈ Y , with Y as general, take x ∈ R + where x can be thought of as income. Let the distribution function be F so that a proportion q = F (x)

of the population has an income less than or equal to x. Given that the population size is normalised to 1, we may dene the income share function

s : [0, 1] → [0, 1] as s (q) := F -1 (q) ´1 0 F -1 (t) dt = x µ (3)
where F -1 (•) is the inverse of the function F and µ is the mean of the income distribution. One way of reading (3) is that those located in a small neighbourhood around the q-th quantile have a share s (q) dq in total income. It is clear that the function s (•) has the same properties as the regular density function f (•):

s (q) ≥ 0, for all q and ˆ1 0 s (q) dq = 1.

(4)

5 Using l'Hôpital's rule we can see that when α = 0 H α takes the form (1). For discussion of H α see [START_REF] Havrda | Quantication method in classication processes: concept of structural α-entropy[END_REF], [START_REF] Ullah | Entropy, divergence and distance measures with econometric applications[END_REF].

We may thus use s (•) rather than f (•) to characterise the income distribution. Replacing f by s in (1), we obtain

H(s) = - ˆ1 0 s (q) log[s (q)]dq = - ˆ∞ 0 x µ log x µ dF (x) (5)
The Theil inequality index is dened by

I 1 := ˆ∞ 0 x µ log x µ dF (x) (6)
and thus we have I 1 = -H(s).

I α = ˆ∞ 0 1 α(α -1) x µ α -1 dF (x) (7)
and thus, replacing f by s in (2), it is clear that

I α = -α -1 H α (s), α > 0.
One of the attractions of the form ( 7) is that the parameter α has a natural interpretation in terms of economic welfare: for α > 0 the measure I α is top-sensitive in that it gives higher importance to changes in the top of the income distribution; α < 0 it is particularly sensitive to changes at the bottom of the distribution; [START_REF] Atkinson | On the measurement of inequality[END_REF]'s index of relative inequality aversion is identical to 1 -α for α < 1.

Divergence entropy

It is clear that there is a close analogy between the α-class of entropy measures (2) and the generalised entropy inequality measure (7). Eectively it requires little more than a change of variables. We will now develop an approach to the problem of characterising changes in distributions using a similar type of argument.

Let the divergence between two densities f 2 and f 1 be λ := f 1 /f 2 ; clearly the dierence in the distributions is large when λ is far from 1. Using an entropy formulation of a divergence measure, one can measure the amount of information in λ using some convex function, g (λ), such that g(1) = 0.

The expected information content in f 2 with respect to f 1 , or the divergence of f 2 with respect to f 1 , is given by

H(f 1 , f 2 ) = ˆY g f 1 f 2 f 1 dy (8)
which is nonnegative (by Jensen's inequality) and is zero if and only if f 2 = f 1 .

Corresponding to (2), we have the class of divergence measures

H α (f 1 , f 2 ) = 1 α -1 ˆY 1 -f 1 f 1 f 2 α-1 dy, α > 0 (9)
In the case α = 1 we obtain the [START_REF] Kullback | On information and suciency[END_REF] generalisation of the Shannon entropy (1)

H 1 (f 1 , f 2 ) = ˆY f 1 log f 2 f 1 dy = -E f 1 log f 1 f 2 , ( 10
)
known as the relative entropy or divergence measure of f 2 from f 1 . When f 2 is the uniform density, (10) becomes (1).

Discrepancy and distributional change

The transformation used to derive the Theil inequality measure from the entropy measure may also be applied to the case of divergence entropy measures. Consider a pair (x, y) jointly distributed on R 2 + : for example x and y could represent two dierent denitions of income. Given that the population size is normalised to 1, we may dene the income share functions s 1 and s

2 : [0, 1] → [0, 1] as s 1 (q) = F -1 1 (q) ´1 0 F -1 1 (t) dt = x µ 1 and s 2 (q) = F -1 2 (q) ´1 0 F -1 2 (t) dt = y µ 2 (11)
where F -1 1 is the inverse of the marginal distribution of x, F -1 2 is the inverse of the marginal distribution of y and µ 1 , µ 2 are the means of the marginal distributions of x and y.

We may now use the concept of relative entropy to characterise the transformed distribution. Instead of considering a pair of density functions f 1 , f 2 , we consider a pair of income-share functions s 1 , s 2 . Replacing f 1 and f 2 by s 1 and s 2 in (10) we obtain

H 1 (s 1 , s 2 ) = - ˆ1 0 s 1 (q) log s 2 (q) s 1 (q) dq (12)
A normalised version of the measure of distributional change, proposed by [START_REF] Cowell | Generalized entropy and the measurement of distributional change[END_REF], for two n-vectors of income x and y can be written:

J 1 (x, y) := 1 n n i=1 x i µ 1 log x i µ 1 / y i µ 2 . ( 13
)
In the case of a discrete distribution with n point masses it is clear that we have J 1 (x, y) = -H 1 (s 1 , s 2 ).

Replacing f 1 and f 2 by s 1 and s 2 in equation ( 9), and rearranging, we obtain

H α (s 1 , s 2 ) = 1 α -1 ˆ1 0 1 -s 1 (q) α s 2 (q) 1-α dq (14)
The J class of distributional-change measure, proposed by [START_REF] Cowell | Generalized entropy and the measurement of distributional change[END_REF] for two n-vectors of income x and y is

J α (x, y) := 1 nα(α -1) n i=1 x i µ 1 α y i µ 2 1-α -1 , ( 15
)
where α takes any real value; the limiting form for α = 0 is given by

J 0 (x, y) = - 1 n n i=1 y i µ 2 log x i µ 1 / y i µ 2 (16)
and for α = 1 is given by ( 13); note that J α (x, y) ≥ 0 for arbitrary x and y. 6 The family (15) represents an aggregate measure of discrepancy between two distributions. Again, for a discrete distribution with n point masses, it is clear that J α (x, y) = -α -1 H α (s 1 , s 2 ). The analogy between the α-class of divergence measures and the measure of discrepancy ( 15) is evident and requires no more than a change of variables. Once again the parameter α has the natural welfare interpretation pointed out in section 2.2.

Reference distributions

The analysis in section 2 provides a natural lead into a discussion of the divergence between an Empirical Distribution Function (EDF) and a theoretical reference distribution F * . In order to do it, for xed q values, we need to compare the corresponding q-quantiles given by the EDF and F * .

For instance, Figure 1 presents an EDF (red dots) and a theoretical reference distribution (blue line) on a reverse graph, with the q values in the x-axis and the income quantiles in the y-axis.

6 To see this write (15) as

n i=1 y i nµ 2 [ψ (q i ) -ψ (1)] , where q i := x i µ 2 y i µ 1 , ψ (q) := q α α [α -1]
Because ψ us a convex function we have, for any (q 1 , ..., q n ) and any set of non negative weights (w 1 , ..., w n ) that sum to 1,

n i=1 w i ψ (q i ) ≥ ψ ( n i=1 w i q i ). Letting w i = y i / [nµ 2 ]
and using the denition of q i we can see that w i q i = x i / [nµ 1 ] so we have n i=1 w i ψ (q i ) ≥ ψ (1) and the result follows. In other words, we would replace x i and y i in (15), respectively, by EDF -1 (q i ) and F -1 (q i ).

The standard approach in the statistics literature is based upon the empirical distribution function (EDF)

F (x) = 1 n + 1 n i=1 ι (x i ≤ x) ,
where ι is an indicator function such that ι (S) = 1 if statement S is true and ι (S) = 0 otherwise. 7 Let us denote {x (1) , x (2) , ..., x (n) } the members of the sample in increasing order. The corresponding values given by the EDF are the adjusted sample proportion q = { 1 n+1 , 2 n+1 , . . . , n n+1 } and, for each q, the corresponding value for the reference distribution is equal to

y i = F -1 * i n + 1 (17)
7 Note that we use 1 n+1 rather than 1 n to avoid an obvious problem where i = n. Had we used i n in (17) then y n would automatically be set to sup (X) where X is the support of F * .

A divergence measure between the EDF and a theoretical reference distribution F * would be given by replacing x i and y i in (15), respectively, by

EDF -1 * i n+1 = x (i) and F -1 * i n+1
. Thus, we would have

J α = 1 nα(α -1) n i=1   x (i) μ α F -1 * i n+1 µ (F * ) 1-α -1   , α = 0, 1 (18)
The limiting forms for α = 0, 1, dened in ( 16) and ( 13) become

J 1 = 1 n n i=1
x (i) μ log

x (i) μ / F -1 * i n+1 µ (F * ) , (19)
J 0 = - 1 n n i=1 F -1 * i n+1 µ (F * ) log x (i) μ / F -1 * i n+1 µ (F * )
.

(

) 20
This index can be used to measure the divergence between an empirical income distribution, given by a sample of individual incomes, and any theoretical reference distribution.

This index would require the choice of a specic value or values for the parameter α according to the judgment that one wants to make about the relative importance of dierent types of discrepancy: choosing a large positive value for α would put a lot of weight on parts of the distribution where the observed incomes x i greatly exceed the corresponding values y i in the reference distribution ; choosing a substantial negative value would put a lot of weight on cases where the opposite type of discrepancy arises.

The most equal reference distribution

Let us assume that the most equal income distribution is when the same amount is given to each individuals:

F -1 * i n + 1 = μ for i = 1, . . . , n (21)
If we use this (egalitarian) distribution as the reference distribution in ( 18), then we have

J α = 1 nα(α -1) n i=1 x i μ α -1 , α = 0, 1 (22)
Figure 2: Quantile approach with the most equal reference distribution and the limiting forms, for α = 0, 1, are equal to

J 0 = - 1 n n i=1 log x i μ and J 1 = 1 n n i=1 x i μ log x i μ (23)
These measures are nothing but the standard Generalised Entropy inequality measure. In other words, the standard GE inequality measures are divergence measures between the EDF and the most equal distribution, where everybody gets the same income. They tell us how far a distribution is from the most equal distribution. A sample with a smaller index has a more equal distribution.

Figure 2 presents the quantile approach for this case. We can see that the EDF is always above (below) the reference distribution for large (small) values of incomes. It makes clear that large (small) values of α would be more sensitive to changes in high (small) incomes.

The most unequal reference distribution

Rather than selecting the most equal distribution as a reference distribution, we can reverse the standard approach by using the most unequal distribution as a reference distribution. It requires one to measure how far a sample is from the most unequal distribution, rather than how far it is from the most equal distribution.

The most unequal income distribution is when one person gets all the income and the others zero:

F -1 * i n + 1 = 0 for i = 1, . . . , n -1 nμ for i = n (24)
If we use this distribution as the reference distribution in ( 18), then we have:

J α = 1 α(α -1) max x i nμ α -1 , α < 1, α = 0 (25)
In the limiting case α = 0, we have

J 0 = -log max x i nμ (26)
This follows immediately from l'Hopital's rule. In the limiting case α = 1, the index is undened. This index tells us how far a distribution is from the most unequal distribution. However, two major drawbacks make this index useless in practice:

1. To be comparable for two dierent samples, the index should use the same reference distribution in both samples. Here max x i = x (n) is an estimate of the n/(n + 1)-quantile in F * . It follows that, if n = 100, the reference distribution is when the top 1% gets all the income, whereas if n = 1000 it is when the top 0.1% gets all the income. The reference distribution diers with the sample size.

2. The presence of zero incomes in the reference distribution produce undesirable properties: the index is independent on how the rst n -1 ordered incomes are distributed. The rst n-1 ordered incomes do not appear explicitly in the formula, the index depends on them through the mean only. It follows that, the mean being constant, the distribution of the n -1 rst ordered incomes does not matter. For instance, the two samples {8, 8, 8, 8, 8, 8, 8, 8, 20} and {1, 1, 1, 1, 15, 15, 15, 15, 20}

produce the same value of the index.

These two drawbacks lead us to consider the following reference distribution, where the top 100k% richest gets 100p% of the total income:

F -1 * i n + 1 = (1 -p)μ/(1 -k) for i = 1, . . . , n(1 -k) pμ/k for i = n(1 -k) + 1, . . . , n (27)
with 0 ≤ k ≤ 1, 0 ≤ p ≤ 1 and z denotes the smallest integer not less than z. Small values of k and large values of p produce very unequal distributions, where a few people get nearly all the income, and the rest get nearly zero.

For instance, in setting k = 0.01 and p = 0.99 we take the case where the top 1% richest gets 99% of the total income. If we use this distribution as the reference distribution in (18), we obtain:

J α,k,p = 1 nα(α -1) n i=1 x (i) μ α c 1-α i -1 , α = 0, 1, (28)
where

c i = (1 -p)/(1 -k) if i ≤ n(1 -k) p/k if i > n(1 -k) (29)
The limiting forms for α = 0, 1, dened in ( 19) and (20) become

J 1,k,p = 1 n n i=1
x (i) μ log

x (i) c i μ and J 0,k,p = - 1 n n i=1 c i log x (i) c i μ . (30)
There is two interesting special cases. If p = k, everybody gets the same income value, μ, and the reference distribution is the most equal distribution.

If k = 1/n and p = 1, only one individual gets all the income, nμ, and the reference distribution is the most unequal distribution.

In practice, k and p have to be xed: (1) to avoid the rst drawback, k and p should be independent of the sample size, with k > 1/n and p > 1/n ; (2) to avoid the second drawback, zero incomes are not allowed in the reference distribution, that is, if p = 1 we have k = 1. Finally, to make our index J α,k,p useful in practice, we need to use constant values, such that 1/n < k < 1 and 1/n < p < 1, or p = k = 1.

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In empirical studies, we could use several values of k and p. For instance, k = 1 -p = 0.05, 0.01, 0.005, correspond to the reference distributions with the top 5%, 1% and 0.5% getting, respectively, 95%, 99% and 99.5% of the total income.

Other reference distributions

Clearly, other reference distributions could be used. For instance, if we assume that productive talents are distributed in the population according to a continuous distribution of talents F * and that wages should be related to talent, a situation in which everyone received the same income to everybody might be considered as unfair. In this case one might use F * as the reference distribution and make use of the index (18): any deviation from F * would come from something else than talent. If total income is nite, it makes sense to use a distribution dened on a nite support. For instance, we could use a Uniform distribution or a Beta distribution with two parameters, which can provide a variety of appropriate shapes.

Axiomatic foundation

We may put the informal discussion of the use of distributional-change measures on to a rigorous footing using the representation of the problem in section 4.1 and the principles described in section 4.2.

Representation of the problem

The distributional change problem can be characterised as the relationship between two n-vectors of incomes x and y. An alternative equivalent approach is to work with z : = (z 1 , z 2 , ..., z n ), where each z i is the ordered pair (x i , y i ), i = 1, ..., n and belongs to a set Z, which we will take to be a connected subset of R + × R + . The divergence issue clearly focuses on the discrepancies between the x-values and the y-values. The problem can then be approached in two steps.

1. We represent the problem as one of characterising a weak ordering 8 on

Z n := Z × Z × ... × Z n .
where, for any z, z ∈ Z n the statement z z should be read as the income pairs in z constitute at least as close according to as the income pairs in z . From we may derive the antisymmetric part and symmetric part ∼ of the ordering.9

2. We use the function representing to generate the index J.

In the rst stage of step 1 we introduce some properties for , many of which are standard in choice theory and welfare economics.

Z n . Axiom 2 [Monotonicity] If z, z ∈ Z n dier only in their ith component then d (x i , y i ) < d (x i , y i ) ⇐⇒ z z .
Axiom 3 [Symmetry] For any z, z ∈ Z n such that z is obtained by permuting the components of z: z ∼ z .

In view of Axiom 3 we may without loss of generality impose a simultaneous ordering on the x and y components of z, for example x 1 ≤ x 2 ≤ ... ≤ x n and y 1 ≤ y 2 ≤ ... ≤ y n . 11 For any z ∈ Z n denote by z (ζ, i) the member of Z n formed by replacing the ith component of z by ζ ∈ Z.

Axiom 4 [Independence] For z, z ∈ Z n such that: z ∼ z and z i = z i for some

i then z (ζ, i) ∼ z (ζ, i) for all ζ ∈ [z i-1 , z i+1 ] ∩ z i-1 , z i+1 .
If z and z are equivalent in terms of overall discrepancy and the discrepancy at position i is the same in the two cases then a local variation at i simultaneously in z and z has no overall eect.

Axiom 5 [Zero local discrepancy] Let z, z ∈ Z n be such that, for some i and j, x i = y i , x j = y j , x i = x i + δ, y i = y i + δ, x j = x j -δ, y j = y j -δ and, for all k = i, j, x k = x k , y k = y k . Then z ∼ z .

The principle states that if there is zero local discrepancy at two positions in the distribution then moving x-income and y-income simultaneously from one position to the other has no eect on the overall discrepancy.

Theorem 1 Given Axioms 1 to 5 (a) is representable by the continuous function given by

n i=1 φ i (z i ) , ∀z ∈ Z n (32)
where, for each i, φ i : Z → R is a continuous function that is strictly decreasing in |x i -y i | and (b)

φ i (x, x) = a i + b i x (33)
Proof. Axioms 1 to 5 imply that can be represented by a continuous function Φ :

Z n → R that is increasing in |x i -y i |, i = 1, ...,
φ i (x i + δ, x i + δ) -φ i (x i , x i ) -φ j (x j + δ, x j + δ) + φ j (x j + δ, x j + δ) = 0
which can only be true if

φ i (x i + δ, x i + δ) -φ i (x i , x i ) = f (δ)
for arbitrary x i and δ. This is a standard Pexider equation and its solution implies (33).

Corollary 1 Since is an ordering it is also representable by

φ n i=1 φ i (z i ) (34)
where, φ i is dened as in (32), (33). and φ : R → R continuous and strictly monotonic increasing.

This additive structure means that we can proceed to evaluate overall discrepancy one income-position at a time. The following axiom imposes a very weak structural requirement, namely that the ordering remains unchanged by some uniform scale change to both x-values and y-values simultaneously.

As Theorem 2 shows it is enough to induce a rather specic structure on the function representing .

Axiom 6 [Income scale irrelevance] For any z, z ∈ Z n such that z ∼ z , tz ∼ tz for all t > 0.

Theorem 2 Given Axioms 1 to 6 is representable by

φ n i=1 x i h i x i y i (35)
where h i is a real-valued function.

Proof. Using the function Φ introduced in the proof of Theorem 1 Axiom 6 implies

Φ (z) = Φ (z ) Φ (tz) = Φ (tz )
and so, since this has to be true for arbitrary z, z we have

Φ (tz) Φ (z) = Φ (tz ) Φ (z ) = ψ (t)
where ψ is a continuous function R → R. Hence, using the φ i given in (32), we have for all :

φ i (tz i ) = ψ (t) φ i (z i ) i = 1, ..., n.
or, equivalently φ i (tx i , ty i ) = ψ (t) φ i (x i , y i ) , i = 1, ..., n.

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So, in view of [START_REF] Aczél | Functional Equations in Several Variables[END_REF], page 346 there must exist c ∈ R and a function h i : R + → R such that

φ i (x i , y i ) = x c i h i x i y i . ( 37
)
From ( 33) and ( 37) it is clear that

φ i (x i , x i ) = x c i h i (1) = a i + b i x i , (38)
which, ifφ i (x, x) is non-constant in x, implies c = 1. Putting (37) with c = 1 into (34) gives the result.

This result is important but limited since the function h i is essentially arbitrary: we need to impose more structure.

Income discrepancy

We now focus on the way in which one compares the (x, y) discrepancies in dierent parts of the income distribution. The form of ( 35) suggests that discrepancy should be characterised terms of proportional dierences:

d (z i ) = max x i y i , y i x i .
This is the form for d that we will assume from this point onwards. We also introduce:

Axiom 7 [Discrepancy scale irrelevance] Suppose there are z 0 , z 0 ∈ Z n such that z 0 ∼ z 0 . Then for all t > 0 and z, z such that d (z) = td (z 0 ) and

d (z ) = td (z 0 ): z ∼ z .
The principle states this. Suppose we have two discrepancy proles z 0 and z 0 that are regarded as equivalent under . Then scale up (or down) all the income discrepancies in z 0 and z 0 by the same factor t. The resulting pair of discrepancy proles z and z will also be equivalent. 12 Theorem 3 Given Axioms 1 to 7 is representable by

φ n i=1 x α i y 1-α i ( 39
)
where α = 1 is a constant. 13

Proof. Take the special case where, in distribution z 0 the income discrep- ancy takes the same value r at all n income positions.

If (x i , y i ) represents a typical component in z 0 then z 0 ∼ z 0 implies r = ψ n i=1 x i h i x i y i ( 40
)
where ψ is the solution in r to n i=1

x i h i x i y i = n i=1 x i h i (r) (41)
12 Also note that Axiom 7 can be stated equivalently by requiring that, for a given z 0 , z 0 ∈ Z n such that z 0 ∼ z 0 , either (a) any z and z found by rescaling the x-components will be equivalent or (b) any z and z found by rescaling the y-components will be equivalent.

13 The following proof draws on [START_REF] Ebert | Measurement of inequality: an attempt at unication and generalization[END_REF].

In (41) can take the x i as xed weights. Using Axiom 7 in (40) requires tr = ψ n i=1

x i h i t

x i y i , for all t > 0.

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Using ( 41) we have n i=1

x i h i tψ n i=1 x i h i x i y i = n i=1 x i h i t x i y i (43)
Introduce the following change of variables

u i := x i h i x i y i , i = 1, ..., n (44)
and write the inverse of this relationship as

x i y i = ψ i (u i ) , i = 1, ..., n (45)
Substituting ( 44) and ( 45) into (43) we get n i=1

x i h i tψ n i=1 u i = n i=1 x i h i (tψ i (u i )) . (46)
Also dene the following functions

θ 0 (u, t) := n i=1 x i h i (tψ (u)) (47)
θ i (u, t) := x i h i (tψ i (u)) , i = 1, ..., n. (48)
Substituting ( 47),( 48) into (46) we get the Pexider functional equation

θ 0 n i=1 u i , t = n i=1 θ i (u i , t)
which has as a solution

θ i (u, t) = b i (t) + B (t) u, i = 0, 1, ..., n where b 0 (t) = n i=1 b i (t)
see [START_REF] Aczél | Lectures on Functional Equations and their Applications[END_REF], page 142. Therefore we have

h i t x i y i = b i (t) x i + B (t) h i x i y i , i = 1, ..., n (49)
From [START_REF] Eichhorn | Functional Equations in Economics. Reading Massachusetts[END_REF], Theorem 2.7.3 the solution to ( 49) is of the form

h i (v) = β i v α-1 + γ i , α = 1 β i log v + γ i α = 1 (50)
where β i > 0 is an arbitrary positive number. Substituting for h i (•) from ( 50) into (2) for the case where β i is the same for all i gives the result.

4.4

The J index

For the required index use the natural cardinalisation of the function (39), n i=1 x α i y 1-α i , and normalise with reference to the case where both the observed and the modelled distribution exhibit complete equality, so x i = µ 1 and y i = µ 2 for all i. This gives the following class of measures of divergence (aggregated discrepancy):

J α (x, y) := 1 nα(α -1) n i=1 x i µ 1 α y i µ 2 1-α -1 . (51)
This normalised index can be implemented straightforwardly for a proposed model of an empirical distribution.

14 Of course this would require the choice of a specic value or values for the parameter α in (51). 15

Implementation

We now look at the practicalities of the class of measures J α dened in (18) and J α,k,p dened in (28).

Statistical properties

It is necessary to establish the existence of an asymptotic distribution for J α and J α,k,p in order to justify its use in practice. If the most equal distribution is taken as the reference distribution (k = p = 1), the index J α,1,1 is nothing 14 The form (51) implies that it is valid for mean-normalised distributions.

15 Compare this with the discussion of the interpretation of α in terms of upper-and lower-tail sensitivity in the context of inequality (page 4).

but the standard GE inequality measure, which is asymptotically Normal and has well-known statistical properties.

16 If a continuous distribution is taken as the reference distribution, it can be shown that the limiting distribution of nJ α is that of

1 2µ F * ˆ1 0 B 2 (t)dt F -1 * (t)f 2 * (F -1 * (t)) - 1 µ F * ˆ1 0 B(t)dt f * (F -1 * (t)) 2 (52)
where f * is the density of distribution F * and B(t) is a Brownian bridge. This random variable can have an innite expectation. It is only if F * has a bounded support that the limiting distribution has reasonable properties see [START_REF] Cowell | Goodness of t: an axiomatic approach[END_REF] and [START_REF] Davidson | Statistical inference in the presence of heavy tails[END_REF] for more details. If we use a continuous parametric reference distribution, since total income is nite, it makes sense to use a distribution F * dened on a bounded support only.

For instance, one could use a Uniform distribution or a Beta distribution with two parameters, which can provide many dierent shapes. The same approach can be used for nJ α,k,p , noting that the last statistic is equivalent to the statistic dened in (9) in [START_REF] Cowell | Goodness of t: an axiomatic approach[END_REF], where 2i/(n + 1) is replaced by c i dened in (29).

In the two cases, the limiting distribution of J α and J α,k,p exists, but is not tractable. It is enough to justify the use of bootstrap methods for making inference. To compute a bootstrap condence interval, we generate B bootstrap samples by resampling from the original data, and then, for each resample, we compute the index J.

where c b 0.025 and c b 0.975 are the 2.5 and 97.5 percentiles of the EDF of the bootstrap statistics -for a comprehensive discussion on bootstrap methods, see [START_REF] Davison | Bootstrap Methods[END_REF], [START_REF] Davidson | Bootstrap methods in econometrics[END_REF]. For wellknown reasons see [START_REF] Davison | Bootstrap Methods[END_REF] or [START_REF] Davidson | Bootstrap tests: How many bootstraps?[END_REF] the number B should be chosen so that (B + 1)/100 is an integer: here we set B = 999 unless otherwise stated.

To be used in practice, we need to determine the nite sample properties of J α and J α,k,p . The coverage error rate of a condence interval is the probability that the random interval does not include, or cover, the true value of the parameter. A method of constructing condence intervals with good nite sample properties should generate a coverage error rate close to 16 Among others, see [START_REF] Cowell | Income distribution and inequality measurement: The problem of extreme values[END_REF], [START_REF] Davidson | Asymptotic and bootstrap inference for inequality and poverty measures[END_REF], [START_REF] Schluter | Edgeworth expansions and normalizing transforms for inequality measures[END_REF], [START_REF] Schluter | The econometrics of inequality measurement[END_REF], [START_REF] Davidson | Statistical inference in the presence of heavy tails[END_REF] the nominal rate. For a condence interval at 95%, the nominal coverage error rate is equal to 5%. We use Monte-Carlo simulation to approximate the coverage error rate bootstrap condence intervals in several experimental designs.

In our experiments, samples are drawn from a lognormal distribution. For xed values of α, k, p and n, we draw 10 000 samples. For each sample we compute J α or J α,k,p and its condence interval at 95%. The coverage error rate is computed as the proportion of times the true value of the inequality measure is not included in the condence intervals. 17 Condence intervals perform well in nite samples if the coverage error rate is close to the nominal value, that is, close to 0.05.

Table 1 presents the coverage error rate of bootstrap condence intervals at 95% of J α and J α,k,p for several reference distributions. The standard GE measures use the most equal reference distribution, it corresponds to J α,1,1 .

When the top 1% richest gets 99% of the income is the reference distribution we use the index J α,0.01,0.99 ; when the top 5% richest gets 99% of the income is the reference distribution we use J α,0.05,0.99 . In addition, we examine J α with two continuous (bounded) parametric reference distributions, the Beta(1,1) distribution which is equal to the Uniform(0,1), and the Beta(2,2) which is a symmetric inverted-U-shape distribution. Table 1 shows that the nite sample properties of the indices with alternative reference distributions are not very dierent from those of the standard GE measures, except for J α,0.01,0.99 when n ≤ 500. The coverage error rate is close to 0.05 for very large samples. For small and moderate samples, further investigations are required to improve the nite sample properties, with, for instance, a fast double or triple bootstrap, see [START_REF] Davidson | Improving the reliability of bootstrap tests with the fast double bootstrap[END_REF] and [START_REF] Davidson | The iterated bootstrap[END_REF]. 18

Application

Let us compare the performance of the statistic J α,k,p with that of conventional GE inequality measures using as a case study UK income data from the Family Expenditure Survey, for years 1979 and 1988.

19 In Table 5.2

we present the results of indices J α and J α,k,p estimated with three dierent types of reference distribution, along with bootstrap condence intervals at 95%.

Equality

The top panel of Table 5.2 presents estimates of J α,k,p using an equality reference distribution. Clearly, when we select the most equal distribution as the reference distribution, i.e. k = p = 1, the index J α,k,p is reduced to the standard GE inequality measure. Estimates for standard GE measures, J α,1,1 are tabulated in the rst row, for values of α ranging from -1 to 2. 20 When α = 1, J α,1,1 is the Theil index. For values of α = 0.5, 1, 1.5, 2, J α,1,1 represents the several (transformed) Atkinson indices.

21 All estimates of standard GE measures increase considerably between 1979 and 1988, suggesting a signicant rise in inequality in the 80s.

Extreme inequality

The key point highlighted in earlier sections was that changing the reference distribution from which we measure the distance of the empirical distribution opens up the possibility for researchers to choose the exact distribution from which they wish to measure distance of the empirical distribution. While standard GE indices tell us about the distance of the empirical distribution from an equal reference distribution, one can change the focus to that of its distance from an unequal reference distribution. In the second panel of Table 5.2 we present estimates of J α,k,p using several extreme inequality reference distributions. The interpretation of the size of the J α,k,p index now is the reverse of the interpretation of standard GE measures. For a standard GE inequality measure, a small value of J α,1,1 corresponds to the empirical distribution being close to the equal reference distribution compared to that of a large value of J α,1,1 . However, for an unequal reference distribution a small value of J α,k,p corresponds to the empirical distribution being close to the particular extreme inequality reference distribution that has been specied.

To illustrate we focus on two dierent unequal reference distributions:

one, where the top 1% of the income distribution receive 99% of the income, 21 See [START_REF] Cowell | Measuring Inequality[END_REF] for details.

and second, where the top 5% of the income distribution receive 99% of the income. From Table 5.2 we can see that, with one exception, the values of J α,k,p have dropped between years 1979 and 1988: in other words, it is almost always true that the distance from the extreme inequality reference distribution has decreased. The exception is the case (k = 0.05, p = 0.99, α = 2)

where the movement relative to the reference distribution is not signicant.

The implication is that UK inequality grew during the 1980s whether one interprets this in terms of distance from equality, or as distance from a reference unequal distribution, except for one case. This case concerns top-sensitive inequality where, in terms of distance from maximum inequality, the change in the distribution is inconclusive.

Theoretical distribution

Finally let us consider how inequality changed using a continuous reference distribution F * . The last panel of Table 5.2, tabulates the results for three such F * (introduced in Section 5.1) taken from the Beta distribution family. Did UK income inequality, interpreted as a distance from a Beta-family reference distribution increase? We can see that the values of J α are not statistically dierent between 1979 and 1988 when the Beta(1,1) (uniform) or Beta(2,5) (unimodal, right skewed) is used as the reference distribution distribution, while they are statistically dierent when the Beta(2, 2) (unimodal symmetric) is used as the reference distribution.

The estimates of the standard GE inequality measures J α,1,1 and of those of J α,k,p and J α in Table 5.2 provide us with dierent information about divergence of the empirical distribution from the chosen reference distribution. By varying the values of k and p, one can specify the exact skewness of the reference distribution one would like to measure distance of the empirical distribution from. Likewise, by varying the values of α one can focus on dierent parts of the income distribution. A large value of α implies a greater weight on parts of the distribution where the observed incomes are vastly dierent from the corresponding values in the reference distribution.

Finally, one can choose specic parametric distributions which correspond to the relevant reference distribution that the researcher is interested in.

Conclusion

The problem of comparing pairs of distributions is a widespread one in distributional analysis. It is often treated on an ad-hoc basis by invoking the concept of norm incomes and an arbitrary inequality index.

Our approach to the issue is a natural generalisation of the concept of inequality indices where the implicit reference distribution is the trivial perfectequality distribution. It is also a natural application of information theory to assessment of income distributions. The approach uses the same ingredients as loss functions applied in other economic contexts. Its intuitive appeal is supported by the type of axiomatisation that is common in modern approaches to inequality measurement and other welfare criteria. The axiomatisation yields indices that can be interpreted as measures of discrepancy. They are related to the concept of divergence entropy in the context of information theory. Furthermore, they oer a degree of control to the researcher in that the J α indices form a class of measures that can be calibrated to suit the nature of the economic problem under consideration. Members of the class have a distributional interpretation that is close to members of the well-known generalised-entropy class of inequality indices.

In eect the user of the J α -index is presented with two key questions:

Figure 1 :

 1 Figure 1: Quantile approach

  We obtain B bootstrap statistics, J b α , b = 1, . . . , B. The percentile bootstrap condence interval is equal to CI perc = [c b 0.025 ; c b 0.975 ]

  ported incomes are known to be misrepresented. The years 1979 and 1988 have been chosen to represent the maximum recorded dierence in inequality across the available years, post-1975. 20 A large value of α implies greater weight on parts of the distribution where the observed incomes are vastly dierent from the corresponding values in the reference distribution.

  1. the income discrepancies underlying inequality are with reference to what? 2. to what kind of discrepancies do you want the measure to be particularly sensitive? As our empirical illustration has shown, dierent responses to these two key questions provide dierent interpretations from the same set of facts.

  The analogy between the Shannon entropy measure (1) and the Theil inequality measure (6) is evident and requires no

more than a change of variables. The transformed version due to Theil is more useful in the context of income distribution because it enables a link to be established with several classes of inequality measures. The generalised entropy inequality measure is dened by

See, for example,[START_REF] Bartels | Economic Aspects of Regional Welfare, Income Distribution and Unemployment[END_REF] and[START_REF] Nygård | Measuring Income Inequality[END_REF]. See also[START_REF] Ebert | Measures of distance between income distributions[END_REF]'s characterisation of absolute inequality indices in ternms of distance between income distributions.

2Almås et al. (2011) compare actual and equalizing earnings; their work is related to Paglin's Gini[START_REF] Paglin | The measurement and trend of inequality: a basic revision[END_REF]) and Wertz's Gini[START_REF] Wertz | The measurement of inequality: comment[END_REF]. Also see[START_REF] Jenkins | Inequality measurement using norm incomes -were Garvy and Paglin onto something after all? Review of Income and[END_REF] and[START_REF] Garvy | Inequality of income: Causes and measurement[END_REF]

).3 On the fairness reference distribution see, for exampleAlmås et al. (2011) and De- vooght

(2008).4 The[START_REF] Cowell | Goodness of t: an axiomatic approach[END_REF] approach diers from that developed here in that it deals with the problem of continuous reference distributions on unbounded support.

This implies that it has the minimal properties of completeness, reexivity and tran- sitivity.

9 For any z, z ∈ Z n z z means [z z ] & [z z]; z ∼ z means [z z ] & [z z].

Note that the derivation which follows diers from that provided in[START_REF] Cowell | Measures of distributional change: An axiomatic approach[END_REF] used to establish the class of measures of distributional change using explicit assumptions of dierentiability and additive separability. Here we adopt a minimalist approach that uses neither of these strong assumptions and that focuses directly on the divergence issue.

in the present case, of course, the components of x and y will be in the same order.

The true values are computed replacing x (i) in (18) by F -1 ( i n+1 ), where F is the distribution of x, that is, the lognormal distribution in our experiments.

Such developments are beyond the scope of this paper and are the subject of future research.

The application uses the before housing costs income variable of the Family Expen- diture Survey for years[START_REF] Wertz | The measurement of inequality: comment[END_REF] 1988 (Department of Work and

Pensions 2006), deated and equivalised using the McClement's adult-equivalence scale, excluding households with self-employed individuals. We exclude households with self-employed individuals as re-