Quantile Versions of the Lorenz Curve

The classical Lorenz curve is often used to depict inequality in a population of incomes, and the associated Gini coefficient is relied upon to make comparisons between different countries and other groups. The sample estimates of these moment-based concepts are sensitive to outliers and so we investigate the extent to which quantile-based definitions can capture income inequality and lead to more robust procedures. Distribution-free estimates of the corresponding coefficients of inequality are obtained, as well as sample sizes required to estimate them to a given accuracy. Convexity, transference and robustness of the measures are examined and illustrated.


Introduction
The Lorenz curve and the associated Gini coefficient are routinely employed for comparisons of income inequality in various countries. There are also numerous applications of them in the biological, computing, health and social sciences. These concepts have nice mathematical properties, and thus are the subject of numerous theoretical studies; for a recent review see Kleiber (2005). However, when it comes to statistical inference for the Lorenz curve and the Gini coefficient, thorny issues arise. An excellent review of existing methods and new proposals for estimating the standard error of the Gini coefficient are investigated by Davidson (2008). However, as this author notes, such methods will not work when the variance of the income distribution is large or fails to exist, and of course this means that they are undermined when there are outliers in the data. Cowell & Victoria-Feser (1996) show that most of the inequality measures in the econometrics literature are very sensitive to outliers and have unbounded influence functions.
There are methods available for resolving these inferential obstacles. One is to choose a parametric income model and then to find optimal bounded influence estimators for the parameters; for example, Victoria-Feser & Ronchetti (1994) do this for the gamma and Pareto models. And, Victoria-Feser (2000) shows how to robustly choose between parametric models and then find robust estimates of inequality indices based on a single data sample, even if it has been grouped or truncated. In a series of papers Cowell & Victoria-Feser (2002, 2003, 2007 investigate damaging effects of data contamination on transfer properties of various inequality indices, as well as dealing with the effects of truncation of non-positive and/or large data values. They propose semi-parametric models for overcoming these issues. We go one step further here, redefining the basic concept of the Lorenz curve in terms of quantiles instead of moments, and then determining what has been gained and lost in terms of conceptual clarity, inference and resistance to contamination. Examples of this approach are the standardized median in lieu of the standardized mean, and quantile measures of skewness and kurtosis, rather than the classical moment-based measures, Staudte (2013bStaudte ( , 2014Staudte ( , 2015. Ratios of quantiles based on one sample are often presented as measures of inequality, and inferential procedures for them are in Prendergast & Staudte (2015a,b).
The role of quantiles in inequality measures is long-standing. Gastwirth (1971) observed that the definition of the Lorenz curve could be extended to all distributions having a finite mean µ by expressing the cumulative income as an integral of the quantile function. More recently Gastwirth (2012) showed that the inequality coefficient of Gini (1914) could be made much more sensitive to shifts in income inequality if the mean in its denominator were replaced by the median; this also has the advantage of protecting the coefficient from large outliers. Kampke (2010) compares the effects of means versus medians on poverty indices. It is in this spirit that we begin in Section 2 by introducing three simple quantile versions of the Lorenz curve for distributions on the positive axis, and their associated coefficients of inequality. Numerous examples demonstrate how these curves and coefficients agree or disagree with the moment-based classical version. In particular, the effects of an income transfer function on the inequality coefficients are illustrated for the Type II Pareto model.
In Section 3 we study empirical versions of these inequality curves and their associated estimated coefficients. The latter estimates are found to have predictable distributionfree standard errors, unlike the sensitive Gini coefficient. For an assumed scale model, confidence intervals for the inequality coefficients are given. It is not surprising that these quantile measures of inequality are resistant to outliers, and in Section 4 we show that they have bounded influence functions.
While the quantile versions of the Lorenz curve are not always convex, they are so for common distributions used to model incomes, as explained in Section 5. A summary and further research problems are given in Section 6.
2 Quantile analogues of the Lorenz curve

Definitions and basic properties
Let F be the class of all cumulative distribution functions F with F (0) = 0. Such F will be interpreted as 'income' distributions and p = F (x) as the proportion of incomes less than or equal to x. Define the quantile function associated with F ∈ F at each p ∈ [0, 1] by Q(F ; p) = F −1 (p) ≡ inf{x : F (x) ≥ p}. If the support of F is infinite; that is F (x) < 1 for all x > 0, this infimum does not exist for p = 1, and then we define Q(F ; 1) = +∞. When the meaning of F is clear, we will sometimes write x p or Q(p) for Q(F ; p).
The mean income of those with proportion p of smallest incomes is µ = µ p (F ) = xp 0 x dF (x)/p, and the mean income of the entire population is defined by µ = µ(F ) = lim p→1 µ p . Let F 0 ⊂ F be the set of F for which µ(F ) exists as a finite number. For each F ∈ F 0 the Lorenz curve of F is defined by L 0 (F ; p) ≡ p µ p /µ, for 0 ≤ p ≤ 1. The lowest proportion of incomes p have proportion L 0 (p) of the total wealth.
What we are proposing here is to replace µ p , the mean of the proportion p of those with wealth less than x p , by its median x p/2 = Q(F ; p/2). In addition, we replace the mean µ of the entire population by one of three quantile measures of its size: x 1/2 , x 1−p/2 , or (x p/2 + x 1−p/2 )/2. The robustness merits of this last divisor, a symmetric quantile average, are investigated by Brown (1981).
Definition 1 For F ∈ F and p ∈ [0, 1] let x p = Q(F ; p). The three quantile-based functions whose graphs reveal income inequality are defined for each p by: .
For each p the first measure L 1 (p) compares the typical (median) wealth of the poorest proportion p of incomes with the typical (median) wealth of the entire population. The second measure compares the bottom typical wealth with the top typical wealth; for example L 2 (0.2) corresponds to the popular '20-20 rule', which compares the mean wealth of the lowest 20% of incomes with the largest 20%. For each p the third L 3 (p) gives the typical wealth of the poorest 100 p % incomes, relative to the mid-range wealth of the middle 100(1 − p) % of incomes. In all cases, extreme incomes are down-weighted because of multiplication by the factor p, as it is for the Lorenz curve L 0 (p) = p µ p /µ.
All of these quantile inequality curves {(p, L i (p))} are scale invariant and monotone increasing from L i (0) = 0 to L i (1) = 1, and all satisfy L i (p) ≤ p for 0 ≤ p ≤ 1. Each L i (p) ≡ p when all incomes are equal. None are strictly speaking 'Lorenz' curves, because they are not convex for all F ∈ F 0 , as examples will show. Nevertheless, for most commonly assumed income distributions F , they are convex, see Section 5. Some examples of the quantile curves are depicted in Figures 1-2, which compares their graphs with the Lorenz curve. Note that L 0 (p) ≡ L 1 (p) ≡ L 3 (p) ≡ p 2 for the uniform distribution. And, L 2 (p) ≈ p 3 for the log-normal distribution.

Coefficients of inequality
The relative measure of dispersion, or concentration ratio due to Gini (1914) is defined for F ∈ F 0 by G 0 = E|X 1 − X 2 |/(2µ), where X 1 , X 2 are independent and each distributed as F , and µ is the mean of F . It is known, see Sen (1986) for example, to equal twice the area between the Lorenz curve and the diagonal line; it is an indicator, on the scale of 0 to 1, of 'how far' the inequality graph is from the diagonal line representing equal incomes; the further it is, the larger the Gini coefficient.  Definition 2 For each of the L i given in (1) define the respective coefficients of inequality Specific numerical comparisons of the G i s are given in Table 1. It lists a variety of F ranging from uniform to very long-tailed distributions and the associated values of Gini's index for the four G i s. The rankings of different F s by these four measures of inequality are seen to be very similar and the Spearman rank correlation of G 0 with G i for i = 1, 2 and 3 are respectively 0.84, 0.88 and 0.88, for this list of F s. For more background material on distributions, see Johnson et al. (1994Johnson et al. ( , 1995. Proposition 1 Given F ∈ F let m = F −1 (0.5) denote its median. Choose two incomes Y 1 , Y 2 independently and randomly from those incomes less than the median, and let V = max{Y 1 , Y 2 }. It then follows that G 1 defined by (2) is given by

Proof:
Let Y have the conditional distribution of X given X ≤ m; then its distribution function F Y (y) = 2F (y), for 0 ≤ y ≤ m and the distribution of V is determined by For each of the three integrals in (2), make the change of variable v = F −1 (p/2). The results are then immediate by observing that each integral with respect to the measure dF V equals the corresponding claimed expected value.
Proposition 1 shows that G 1 ≤ G 2 and G 3 ≤ G 2 for all F . It also allows for simple alternative interpretations of the three quantile inequality coefficients defined in (2) which can be compared with Gini's original definition as a relative measure of concentration. Note that the Gini measure has been criticized for placing too much emphasis on the central part of the distribution. As Proposition 1 shows, the quantile versions can also be criticized for the same reason, because the main ingredient is the maximum of two randomly chosen incomes from the lower half of the population. This maximum arises because in the definition (1) all the ratios are multiplied by p, which down-weights ratios involving relatively small and large incomes.

Tranference of income
The effect of income transfers on inequality measures is of great interest to economists, see Kleiber (2005) and Fellman (2012). The basic idea Dalton (1920) is that if one transfers Table 1: Values of G i to 3 decimal places for various F . Also listed are the rankings of F induced by the various G i . were so extended, L 0 (p) would be 0 for 0 < p < 1 and the associated coefficient would be 1.
income from some members of the population having income above the mean to others having income below the mean, then the inequality measure should reflect this by decreasing. In keeping with our preference for quantiles over moments, we suggest replacing the mean by the median in defining the transference principle for inequality measures.
Definition 3 Given X ∼ F ∈ F, and let m ≡ x 0.5 = F −1 (0.5) be the median. We define a median preserving transfer (of income) function Y = t(X) ∼ F Y as one satisfying In words, a median preserving transfer function can only increase income that is less than the median, and only decrease income if it exceeds the median. It follows that The effect on the quantile inequality curves is then easily seen: L 1 (F ; p) = p x p/2 /x 0.5 ≤ p y p/2 /y 0.5 = L 1 (F Y ; p); that is, the transfer function can only increase L 1 (p) at each p. This implies the associated coefficient of inequality (2) satisfies G 1 (F ) ≥ G 1 (F Y ). We say that L 1 preserves the ordering induced by the transfer function. The reader may readily verify that for i = 2, 3 the other quantile inequality curves satisfy For any non-trivial transfer function we will have G i (F ) > G i (F Y ), a positive reduction in the coefficient of inequality. Can we quantify this amount for any specific transfer functions? An example is given in Section 2.4.

Example of transference
Suppose one wants to increase all incomes less than a specific threshold b (say the poverty line) so that they equal b. That is; to be found, say, by transference from those with incomes above the median or some higher thresh-hold c. One possibility is to charge a levy of amount d on those with income exceeding c, leading to the following transfer function In the interest of fairness one could also charge a proportional amount for those with income between c and c + d so that Y = c for c < x < c + d, but this unnecessarily complicates our presentation.
At this point it is convenient to introduce the pth cumulative income by C(F ; p) = xp 0 y dF (y), where x p = Q(F ; p). As Cowell & Victoria-Feser (2002) point out, this function is fundamental to analysis of Lorenz curves, and C(1; F ) = µ and L 0 (F ; p) = C(F ; p)/C(1; F ). We want to determine C(F ; p) for the Type II Pareto distribution having shape parameter a > 1 and scale parameter σ > 0. Now 1 − F a,σ (x) = (1 + x/σ) −a , which has mean µ = σ/(a − 1) and pth quantile Q(F a,σ ; p) = σ{(1 − p) −1/a − 1}. Integrating by parts we obtain This amount can be obtained by a levy d on each income greater than c = x 1−p .
For the Pareto distribution with parameters a = 2, σ = 100, 000 , the median income is 41,421.36 and the mean income is µ = 100, 000. For p = 0.2, say, the quantities of interest are the poverty line b = 11, 803.40, the mean cumulative income µ 0.2 = 5, 572.80 and d = 6, 230.60. All those having income greater than the 0.8 quantile 123, 606.30 would need to pay an impost of d = 6, 230.60.
The absolute and relative effects of such a transfer function are depicted in Figure 3 for two income distributions, Pareto with a = 1.1 and a = 2. For the first distribution, the change in the Gini coefficient G 0 is larger than for the G 2 and G 3 coefficients, but less than that for G 1 ; but the relative effect plot shows that the G 1 coefficient is most sensitive of the four, especially for p 0 near 0.25. For the second distribution both G 0 and G 1 are roughly the same in terms of sensitivity to changes by transference and again preferable to G 2 and G 3 .
Many other transfer functions and income distributions could be considered, but those are applications beyond the scope of this work. It is important, of course, to identify real changes in coefficients of inequality after implementing a transfer of income. Estimation of G i is discussed in Section 3. Another factor that we have not included here are the costs of implementation of a transfer function.  as the graph of the piecewise linear connection of the points (i/n, L 0 (i/n)), i = 0, 1, . . . , n. The empirical distribution function defined for each x by F n (x) = ( x i ≤x i)/n has inverse Q(F n ; p) = F −1 n (p) = x ([np]+1) for 0 ≤ p < 1, and so empirical versions of the quantile curves (1) can be expressed in terms of the n order statistics. Such curves are discontinuous, but there are several continuous quantile estimators available, including kernel density estimators Sheather & Marron (1990) and the linear combinations of two adjacent order statistics studied by Hyndman & Fan (1996). Many of the latter are implemented on the software package R Development Core Team (2008), and here we use the Type 8 version of the quantile command recommended by Hyndman & Fan (1996). It linearly interpolates between the points (p [k] , x (k) ), where p [k] = (k − 1/3)/(n + 1/3) and is a continuous function of p in (0, 1). We also denote this estimatorx p =Q(p).

Empirical quantile inequality curves
Definition 4 All of the L i curves defined by (1) are functions of the quantile function Q(F ; p), so given the estimatorx p =Q(p) one can by substitution obtain estimators of each of the L i (p) for any p in (0, 1); we call these estimatorsL i (p), for i = 1, 2, and 3.

Empirical coefficients of inequality
With few exceptions, such as the uniform distribution, one cannot analytically compute the G i (F )s, but using modern software packages such as R Development Core Team (2008), it is easy to get very good approximations to them for many F of interest as follows.
Definition 5 Given a large integer J define a grid in (0,1) with increments of size 1/J by p j = (j − 1/2)/J, for j = 1, 2, . . . , J. Then evaluate the quantile function Q(p j ) for p j in the grid and find G i (J) ≡ (2/J) j {p j − L i (p j )} for each i = 1, 2 and 3.
Clearly one can make G i (J) as close to G i as desired by choosing J sufficiently large. We will estimate G i (J), and hence G i , as follows. LetL i (p j ) be the estimated inequality curve value at p j , for each p j in the grid. ThenĜ i (J) is defined bŷ In our computations, we used J = 1000. Hereafter we write G i for G i (J) andĜ i for G i (J), but it is understood that these are computed on a grid with increments 1/J.

Simulation studies
It will be seen that despite the fact that the values of the quantile coefficients of inequality G i (F ) vary greatly over the range of F in Table 1, the standard errors of estimation are fairly predictable. By 'standard error' ofĜ i , we mean the square root of the mean squared error. Initial simulations suggested that Bias[Ĝ i ] = o(n −1/2 ) and Var[Ĝ i ] = O(1/n) so in Figure 4 we show some examples of √ n SE[Ĝ i (F )], plotted as a function of ln(n), for n ranging from 20 to 1600. These plots are based on 1000 replications at each of the selected values of n for various F . In all four plots it is seen that the standard errors of G 2 (F ) ≈Ĝ 3 (F ) ≈ 1/(2 √ n) whileĜ 1 (F ) is a little larger. This enables one to choose a sample size which guarantees a desired standard error for each of the three estimators. Attempting to estimate Gini's coefficient of inequality by means of the Lorenz curve areas has no such simple sample size solution.
It is also interesting to plot √ n SE[Ĝ i (F a )] versus a as in Figure 5, where F a denotes the Pareto distribution with shape parameter a ranging from 0.25 : 2.5/0.1. Again all three standard errors of the estimated inequality coefficients derived from the L i -curves are well behaved, but those for the Lorenz curve are quite irregular. For a ≤ 1 the Lorenz curve is not defined because E a [X] = +∞ but if one defines the curve to be 0 in this case the  corresponding measure of inequality is 1 and this can be estimated. Even if one restricts attention to 1 < a < 2, these plots show that for increasing n the standard error is growing at a faster rate than the others, (because for a < 2 the variance of F is infinite).
The results in Table 2 suggest that one can choose the minimum sample size required to obtain SE[Ĝ 1 ] ≤ c; it is n 1 (c) = (0.55/c) 2 . So for example, for standard error c = 0.01, one needs n ≥ n 1 ≈ 3000. Note that this accuracy is achieved for all F in Table 2. Similarly for G 2 , G 3 the required sample size is a little smaller n 2 (c) = (0.43/c) 2 = n 3 (c).

Confidence intervals for the coefficients of inequality
Recall from (6) that for each i = 1, 2, 3 and large fixed J the estimated coefficient of inequality isĜ i = (2/J) j {p j −L i (p j )}. Now the estimateL i (p j ), as a ratio of finite linear combinations of quantile estimates, is consistent for L i (p j ), soĜ i is also consistent for G i . Further, Prendergast & Staudte (2015b) show that n 1/2 {L i (p j ) − L i (p j )} is asymptotically normal with mean 0 and variance depending on certain quantiles and quantile densities of  the underlying F . Beach & Davidson (1983) find the limiting joint normal distribution of estimates of a finite number of Lorenz curve ordinates, based on a finite number of sample quantiles, assuming that F ∈ F 0 ∩ F , where F is specified in Definition 6. In the same way, for F ∈ F , the limiting joint normal distribution of the estimated ordinatesL i (p j ), j = 1, . . . , J can be established. We do not need an analytic expression for the covariance matrix, because we only require the asymptotic normality of the estimated G i , which being an average of the p j −L i (p j ), is immediate. Its asymptotic variance is available from the expected value of the squared influence function see (9) and (10).
Here we present the results of a modest simulation study of confidence intervals for G i of the formĜ i ± 1.96σ i / √ n , with nominal coefficient 95% , with results in Table 3. For the lognormal distribution, the respective σ i found in Table 2  To obtain distribution-free confidence intervals for G i , one needs consistent estimates for the asymptotic variance σ 2 i , a project beyond the scope of this work.

Robustness properties
In this section we show that the quantile inequality curves and their associated coefficients of inequality have bounded influence functions, which guarantees that a small amount of contamination can only have a limited effect on the asymptotic bias of estimators of these quantities. For background material on robustness concepts for functionals, see Hampel et al. (1986), although we attempt to make the presentation self-contained. To this end, we must restrict F ∈ F to the following subclass of smooth distributions: Definition 6 F = {F ∈ F : f = F exists and is strictly positive.} For F ∈ F with inverse x p = Q(p) = F −1 (p), we define the quantile density by The quantile density terminology is due to Parzen (1979), and its importance was earlier recognized by Tukey (1965) who called it the 'sparsity index'.
In order to find the influence function of the L i -curves at any specific p in (0, 1) we also require the mixture distribution which places positive probability the point z (the contamination point) and 1 − on the income distribution F . Formally, it is defined for is the indicator function. The influence function for any functional T is then defined for each z as the IF(z; ). The influence function of the pth quantile functional T (F ) = Q(F ; p), where F ∈ F of Definition 6, is well-known to be (Staudte & Sheather, 1990, p.59) where x p = F −1 (p) and q(p) is given by (7). One can show that E F [IF(Z; Q( · ; p), F ), F )] = 0 and Var F [IF(Z; Q( · , p), F ), F )] = E F [IF 2 (Z; Q( · , p), F ), F )] = p(1 − p) q 2 (p). One reason for calculating this variance is that it arises in the asymptotic variance of the functional applied to the empirical distribution F n , namely Q(F n ; p). That is, n 1/2 [Q(F n ; p) − Q(F ; p)] → N (0, p(1 − p) q 2 (p)) in distribution; and sometimes a simple expression for the asymptotic variance is not otherwise available.  Cowell & Victoria-Feser (2002) show that the influence function of the Lorenz curve at the point p is unbounded, implying that a small amount of contamination can lead to a large bias in estimation of its value; on the other hand the quantile inequality curves proposed here all have bounded influence functions, provided only that F ∈ F . To see this, note that each T i (F ) = L i (F ; p) = px p/2 /d i (p), where d 1 (p) = x 1/2 , d 2 (p) = x 1−p/2 and d 3 (p) = (x p/2 + x 1−p/2 )/2 are all quantile functionals or an average of them.

Influence functions of quantile inequality curves
Proposition 2 The influence function of the functional defined by T i (F ) = L i (F ; p) is a multiple p of the derivative of the ratio of two functionals, so by elementary calculus we have for each p ∈ (0, 1) .
For each case i = 1, 2 and 3 one only requires substitution of the respective quantile influence functions for the d i s found in (8).
While these influence functions are complicated, the are easy to compute and plot using currently available software. Specific examples are shown Figure 6 when the underlying F = F a is the Pareto distribution with shape parameter a = 1 and are plotted as functions of a possible contamination at z. For small p there is very little influence on L i (F ; p) of contamination at any point z. However, as p increases, there is a noticeable increase in influence on L 1 (F a ; p) for z near the median, which equals one in this case. Contamination at z near zero is slightly negative, then rises to a positive relatively large positive peak as z approaches the median, and then drops to a small negative and constant influence again as z increases past the median. This is to be expected, because when the median is pulled to the left by contamination, then L 1 (F ; p) = p x p/2 /x 0.5 is increased, but when the median is pulled to the right, the values of L 1 (F ; p) are decreased.
The other two L i (F ; p) are similarly affected by contamination at z, but to a lesser extent. Plots of the influence functions of the quantile inequality curves for other Pareto(a) distributions (not shown) are similar to those in Figure 6, and again the peak is located at the median F −1 a (0.5) = 2 1/a − 1. Similar influence function plots are obtained for uniform, lognormal and Weibull distributions, again with peaks near their respective medians.

Influence of contamination at z on the graph {p, L i (p)}
We have found, for each fixed 0 < p < 1, the influence functions IF(z; L i (p), F ). Now we consider, for fixed z, the graph {(p, IF(z; L i (p), F ))}, which shows the influence of contamination at z on the respective inequality curves {(p, L i (p))}. Examples are shown in Figure 7, again for F the Pareto (a = 1) distribution, and selected values of z.
First we concentrate on only the solid lines corresponding to L 1 (p). Inspection of (9) shows that the discontinuity points are x 1/2 = 1 and x p/2 . Now z < x p/2 if and only if p > 2F (z). Thus in the upper left plot of Figure 7 where z = 0.5 < x 1/2 there are only two cases of interest: p < 2F (0.5) = 2/3 and p > 2/3; in the first interval (0, 2/3) the influence of contamination at z = 0.5 on the L 1 -curve is positive and increasing in p, but its influence is negative for p in (2/3, 1). For the top right plot 2F (z) = 1 so the influence of contamination z = 1 at the median on the L 1 -curve is positive and increasing for all p. For the other two plots z exceeds the median 2F (z) > 1 and there is only a slight negative influence of z on the L 1 -curve for all p.
The influence of contamination at z on the graphs of L 2 (p), L 3 (p) is also shown in Figure 7 as dashed and dotted lines, respectively. Such influence is similar to that on L 1 (p) in the top two plots where z does not exceed the median. But in the lower plots where z exceeds the median, the contamination is positive and increasing on (0, 2(1 − F (z))) and negative for larger p. For the bottom left plot this interval is (0, 0.952), and for the bottom right it is (0, 0.8). Details are left as an exercise for the reader. Further increasing the values of z only diminishes its effect on the graphs.

Influence functions of quantile coefficients of inequality
The influence functions of the inequality coefficients associated with the L i -curves are easily found, because the functional G i (F ) = 1 − 2 1 0 L i (F ; p) dp, which contains an average of L i (F ; p) values over p ∈ (0, 1).
Proposition 3 For each i = 1, 2 and 3 the influence function of the inequality coefficients G i are given respectively by One only needs to justify taking the derivative G i (F (z) ) with respect to at = 0 under the integral sign. An argument based on the Leibniz Integration Rule is given in the Appendix.

Convexity of the quantile inequality curves
One of the nice mathematical properties of the Lorenz curve {p, L 0 (F ; p)} is that it is convex for all distributions F ∈ F 0 . The quantile-based versions (1) are defined for all F in the larger class F, but need not be convex. In particular, empirical versions are often not convex over (0, 1).
The following examples demonstrate that for the more commonly assumed income distributions, the quantile inequality curves are convex. See Johnson et al. (1994Johnson et al. ( , 1995 for background material on these distributions. Table 4: Examples of distributions F (x) and associated quantile functions and their densities. In general, we denote x p = Q(p) = F −1 (p), but for the normal F = Φ with density ϕ, we write z p = Φ −1 (p). the support of each F is (0, +∞), except for the normal and Type I Pareto, the latter having support on [1, +∞). Figure 9 shows that for the very U-shaped Beta distribution with parameters (0.1, 0.05) only the Lorenz curve is convex. This distribution appears to have a symmetric density, but in fact is quite asymmetric, with mean 2/3, and the quartiles 0.050,0.997, and 1.000, to three decimal places. The inequality coefficients are G 0 = 0.329, G 1 = 0.453, G 2 = 0.455 and G 3 = 0.403. Note that the Gini coefficient G 0 < 1/3, its value for the uniform distribution, a non-intuitive result to us. Other plots, not shown, for parameters (0.05, 0.1), (0.1, 0.1) and (0.05, 0.05) indicate that all four L i curves are convex.
It is 'obvious' from the lower left plot in Figure 1 that all three L i (p) curves are convex on (0,1) for the lognormal distribution. Proving it using the calculus is not as straightforward as one might expect. Note that Q(p) = e zp , q(p) = e zp /ϕ(z p ). Further, observe that L 1 (p) = p exp(z p/2 ) and that exp(z p/2 ) is not convex, so one cannot use the fact that two monotone increasing convex functions is convex. Taking derivatives, Thus L 1 (p) > 0 if and only if 4ϕ(z p/2 ) + p(1 + z p/2 ) > 0 and this again, while obvious from a plot, is not readily verified. Next consider L 2 (p) = p {exp(z p/2 ) exp(−z 1−p/2 )} = p exp(2z p/2 ). The argument is very similar to that for L 1 : Thus L 2 (p) > 0 if and only if 4ϕ(z p/2 ) + p(2 + z p/2 ) > 0, a weaker condition than required for convexity of L 1 .
For the Weibull distribution with shape parameter β > 0, we have The term −p ln(2/(2−p)) is a decreasing function in p with limit equal to -2 as p approaches 0. Consequently, L 1 (p) > 0 so that L 1 (p) is convex. For L 2 (p) and L 3 (p), again we used computational minimization for all β values up to 100. Neither had a negative minimum so both were found to be convex.

Summary and further research
We have shown that quantile versions of the Lorenz curve have most of the properties of the original definition, with two exceptions. The first exception is convexity, which is not satisfied for some very U-shaped distributions and many empirical ones. Nevertheless, for all continuous distributions commonly used to model population incomes, the quantile versions are convex.
The second exception is the first order transference principle, which is mean-centric. When replaced by a median-centric definition, this principle is satisfied for all three quantile versions of the Lorenz curve. We then studied a specific example of a transfer function and showed how it could be measured by the associated inequality coefficients, defined as twice the area between the quantile inequality curve and the equity diagonal. These inequality coefficients can also be interpreted as expected values of certain functions of independent randomly drawn incomes from the population.
These concepts have distinct advantages over the traditional Lorenz curve and Gini index. They are defined for all positive income distributions, and their influence functions are bounded. Distribution-free confidence intervals for the ordinates of inequality curves at fixed points are readily found, since they are just ratios of finite linear combinations of quantiles. In addition, we showed that the standard errors of estimates for the quantile analogues of the Gini coefficient do not appear to depend much on the underlying population model, so that sample sizes can be chosen in advance to obtain desired standard errors. Simulation studies suggest that these sample inequality coefficients approach normality very rapidly, and confidence intervals for them can be constructed when the underlying scale family is known. One way to obtain distribution-free confidence intervals for them would be to find distribution-free estimates of their standard errors, which involves quantile density estimation.
Many other challenges remain. It would be good to have simple necessary and sufficient conditions in terms of the underlying income distribution for convexity of each of the inequality curves. If one is interested in confidence bands for the quantile curves, one could utilize functionals of the quantile process to determine them, starting with the results in Doss & Gill (1992). Finally, applications to other fields which use diversity indices Patil & Taillie (1982) would be of interest, as well as connections to the 'Lorenz dominance' literature, see Aaberge & Mogstad (2011) and references therein.
Proof for i = 2.
Proof for i = 3.