Zenga’s New Index of Economic Inequality, Its Estimation, and an Analysis of Incomes in Italy

For at least a century academics and governmental researchers have been developing measures that would aid them in understanding income distributions, their diﬀerences with respect to geographic regions, and changes over time periods. It is a challenging area due to a number of reasons, one of them being the fact that different measures, or indices, are needed to reveal diﬀerent features of income distributions. Keeping also in mind that the notions of ‘poor’ and ‘rich’ are relative to each other, M. Zenga has recently proposed a new index of economic inequality. The index is remarkably insightful and useful, but deriving statistical inferential results has been a challenge. For example, unlike many other indices, Zenga’s new index does not fall into the classes of L-, U-, and V -statistics. In this paper we derive desired statistical inferential results, explore their performance in a simulation study, and then employ the results to analyze data from the Bank of Italy’s Survey on Household Income and Wealth.


Introduction
Measuring and analyzing incomes, losses, risks and other (non-negative) random outcomes, which we denote by X, has been an active and fruitful research area, particularly in the fields of econometrics and actuarial science. The Gini index has been arguably the most popular measure of inequality, with a number of extensions and generalizations available in the literature. Recently, keeping in mind that the notions of 'poor' and 'rich' are relative to each other, M. Zenga constructed an new index that reflects this relativity. We shall next introduce the Gini and Zenga indices in such a way that they would be easy to compare and interpret.
To proceed, we need additional notation. Let F (x) denote the cumulative distribution function (cdf) of X, and let F −1 (s) denote the corresponding quantile function. Furthermore, let μ F denote the mean of X. In terms of the Lorenz curve L F (p) = μ −1 F p 0 F −1 (s)ds (see Pietra, 1915), the Gini index can be written as follows: where ψ(p) = 2p, which is a density function on [0,1]. Given the usual econometric interpretation of the Lorenz curve L F (p), the function is a relative measure of inequality (see Gini, 1914), called the Gini curve. Indeed, L F (p)/p is the ratio between 1) the mean income of the poorest p × 100 % of the population and 2) the mean income of the entire population; the closer to each other these two means are, the lower is the inequality. Zenga's (2007) index of inequality is defined by the formula called the Zenga curve, measures the inequality between 1) the poorest p×100 % of the population and 2) the richer remaining (i.e., (1−p)×100 %) part of it by comparing the mean incomes of these two disjoint and exhaustive sub-populations. We shall elaborate on this interpretation later, in Section 5 below.
Both the Gini and Zenga indices are averages of point inequality measures, that is, of the Gini and Zenga curves, respectively, but while in the case of the Gini index the weight (i.e., density) function ψ(p) = 2p is employed, in the case of the Zenga index the uniform weight (i.e., density) function is used. As a consequence, the Gini index underestimates comparisons between the very poor and the whole population and emphasizes comparisons which involve almost identical population subgroups. From this point of view, the Zenga index is more impartial: it is based on all comparisons between complementary disjoint population subgroups and gives the same weight to each comparison. Hence, with the same sensibility, the index detects all deviations from equality in any part of the distribution.
To illustrate the Gini curve G F (p) and its weighted version g F (p) = G F (p)ψ(p), and to also facilitate their comparisons with the Zenga curve Z F (p), we choose the Pareto distribution where x 0 > 0 and θ > 0 are parameters. (We shall use this distribution in our simulation study later in this paper as well, setting x 0 = 1 and θ = 2.06.) Corresponding to this distribution, the Lorenz curve is equal to L F (p) = 1 − (1 − p) 1−1/θ (see Gastwirth, 1971), and so the Gini curve is equal to G F (p) = ((1−p) 1−1/θ −(1−p))/p. In The rest of this paper is organized as follows. In Section 2 we define two estimators of the Zenga index and develop statistical inferential results. In Section 3 we present results of a simulation study, which explores the empirical performance of the two empirical Zenga estimators, including their coverage accuracy and length of several types of confidence intervals. In Section 4 we present an analysis of data from the Bank of Italy's Survey on Household Income and Wealth. In Section 5 we further contribute to understanding of the Zenga index by relating it to lower and upper conditional expectations. In Section 6 we provide a theoretical justification of the aforementioned two empirical Zenga estimators. In Section 7 we justify the definitions of several variance estimators as well as their uses in constructing confidence intervals.
In Section 8 we prove Theorem 2.1 of Section 2, which is the main technical result of the present paper. Some technical lemmas and their proofs are relegated to Section 9.

Estimators and statistical inference
Let X 1 , . . . , X n be independent copies of a random variable X ≥ 0, which may, for example, represent incomes in the context of economic inequality, or risks and losses in the insurance context. We use two non-parametric estimators of the Zenga index.
The first one (see Greselin and Pasquazzi, 2009) is given by the formula where X 1:n ≤ · · · ≤ X n:n are the order statistics of X 1 , . . . , X n . With X denoting the sample mean of X 1 , . . . , X n , the second estimator of the Zenga index is given by the formula The two estimators Z n and Z n are asymptotically equivalent. However, despite the fact that the estimator Z n is obviously more complex, it is more convenient to work with when establishing asymptotic results, as we shall see later in this paper.
Unless explicitly stated otherwise, our following statistical inferential results are derived under the assumption that data are outcomes of independent and identically distributed (i.i.d.) random variables. It should be noted, however, that -as is the case in many surveys concerning income analysis -households are selected using complex sampling designs. In such cases statistical inferential tools and results are quite complex. To alleviate the difficulties, in the present paper we follow the commonly accepted practice and treat income data as if they were i.i.d. Certainly, extensions of our results to complex sampling designs would be an interesting and worthwhile contribution, though it would certainly be considerably more involved than the current one, which is already quite complex as we shall see later in the paper.
Unless explicitly stated otherwise, throughout we assume that the cdf F of X is a continuous function. We note that continuous cdf's are natural choices when modeling income distributions, insurance risks and losses (see, e.g., Kleiber and Kotz, 2003). dp.
In view of Theorem 2.1, the asymptotic distribution of , which is finite (see Theorem 7.1) and can be rewritten as follows: or, alternatively, The latter expression is particularly convenient when working with distributions for which the first derivative (when it exists) of F −1 (t) is a simple function, as is the case for a large class of distributions (see, e.g., Karian and Dudewicz, 2000).
Irrespectively of what expression for the variance σ 2 F we use, it is unknown since the cdf F (x) is unknown. Replacing the cdf F (x) on the right-hand side of equation (2.4) by the empirical cdf F n (x) = n −1 n i=1 1{X i ≤ x} where 1 denotes the indicator function, we obtain the following variance estimator (see Theorem 7.2 for details): with the following expressions for the summands I X,n (i) and J X,n (i). First, we have I X,n (1) = − n k=2 X k:n − (n − 1)X 1:n ( n k=1 X k:n ) ( n k=2 X k:n ) + 1 X 1,n log 1 + X 1:n n k=2 X k:n . (2.7) Furthermore, for every i = 2, . . . , n − 1, we have and J X,n (i) = n n k=i+1 X k:n + iX i:n (2.9) Finally, J X,n (n) = 1 X n,n log n n − 1 . (2.10) With the just defined estimator S 2 X,n of the variance σ 2 F , we have the asymptotic result √ n (Z n − Z F ) S X,n → d N (0, 1). (2.11) We shall next discuss variants of statement (2.11) in the case of two populations, when samples are independent and also when paired.
We start with the independent case. Namely, let the random variables X 1 , . . . , X n ∼ F and Y 1 , . . . , Y m ∼ H be independent within and between the two samples. Just like in the case of F (x), we assume that the cdf H(x) is continuous and E[Y 2+α ] < ∞ for some α > 0. Furthermore, we assume that the sample sizes n and m are comparable in the sense that there exists η ∈ (0, 1) such that m n + m → η ∈ (0, 1) when n and m tend to infinity. Then from statement (2.3) and its counterpart for is asymptotically normal with mean zero and the variance ησ 2 F + (1 − η)σ 2 H . To estimate the variances σ 2 F and σ 2 H , we use S 2 X,n and S 2 Y,n , respectively, and obtain the following result: Consider now the case when the two samples X 1 , . . . , X n ∼ F and Y 1 , . . . , Y m ∼ H are paired. Thus, we have m = n and know that the pairs (X 1 , Y 1 ), . . . , (X n , Y n ) are independent and identically distributed, but nothing is assumed about the joint distribution of (X, Y ). As before, the cdf's F (x) and H(y) are continuous and have finite moments of the order 2 + α for some α > 0. From statement (2.3) and its analog Replacing the cdf's F (x) and H(y) everywhere on the right-hand side of the above equation by their respective estimators F n (x) and H n (y), we have (see Theorem 7.3 for details) (2.14) We conclude this section with a note that the above established asymptotic results (2.11), (2.12), and (2.14) are what we typically need when dealing with two populations, or two time periods, but extensions to more populations and/or time periods would be a worthwhile contribution.

A simulation study
We investigate the numerical performance of the estimators Z n and Z n by simulating data from Pareto distribution (1.3) with the parameters x 0 = 1 and θ = 2.06, which give the value Z F = 0.6000 that we approximately see in real income distributions. Following Davison and Hinkley (1997, Chapter 5), we compute four types of confidence intervals: normal, percentile, BCa, and t-bootstrap. For normal and studentized bootstrap confidence intervals we estimate the variance using empirical influence values.
For the estimator Z n , the influence values h(X i ) have been obtained from Theorem 2.1, and those for the estimator Z n using numerical differentiation.
In Table 3.1 we report coverage percentages of 10, 000 confidence intervals, for   Efron (1987), we have approximated the acceleration constant for the BCa confidence intervals by one-sixth times the standardized third moment of the influence values. In Table 3.2 we report summary statistics concerning the size of the 10, 000 confidence intervals. As expected, the confidence intervals based on Z n and Z n exhibit similar characteristics. We observe from Table 3.1 that all the confidence intervals suffer from some undercoverage. For example, about 97.5% of the studentized bootstrap confidence intervals with 0.99 nominal confidence level contain the true value of the Zenga index. It should be noted that the higher coverage accuracy of the studentized bootstrap confidence intervals (when compared to other ones) comes at the cost of their larger sizes, as seen in Table 3.2. Some of the studentized bootstrap confidence Table 3.2. Size of the 95% asymptotic confidence intervals from the Pareto parent distribution with x 0 = 1 and θ = 2.06 (Z F = 0.6). 4. An analysis of Italian income data Here we use the Zenga index to analyze data from the Bank of Italy's Survey on In Table 4.1 we report the values of Z n and Z n according to the geographic area of households, and we also report confidence intervals for Z F based on the two estimators.
We note that two households in the sample had negative incomes in 2006 and so we have not included them in our computations. Consequently, the point estimates of Z F are based on 7, 766 equivalent incomes with values Z n = 0.6470 and Z n = 0.6464. As pointed out by Maasoumi (1994)    Proof. We rewrite the ratio R F (p) as follows: and w 2 (x) are non-decreasing, and so by Lemma 3 on p. 1140 of Lehmann (1966)  When X is a constant, which can be interpreted as 'egalitarian' case, then R F (p) is equal to 1. The ratio R F (p) is equal to 0 for all p ∈ (0, 1) when the lower conditional expectation is equal to 0 for all p ∈ (0, 1) which means extreme inequality in the sense that, loosely speaking, there is only one individual who possesses the entire wealth.
Our wish to associate the egalitarian case with 0 and the extreme inequality with 1 leads to curve 1 − R F (p), which coincides with the Zenga curve (see equation (1.2)) when the cdf F is continuous. The area beneath the curve 1 − R F (p) is always in the interval [0, 1] as follows from Lemma 5.1.

Quantity (5.2) is a measure of inequality and coincides with the earlier defined Zenga
index when the cdf F is continuous, which we assume throughout the paper. Note that under this assumption, the lower and upper conditional expectations are equal to the absolute Bonferroni curve p −1 AL F (p) and the dual absolute Bonferroni curve is the absolute Lorenz curve. This leads us to the expression of the Zenga index Z F given by equation (1.1), which we rewrite in terms of the just introduced absolute Lorenz curve as follows: We shall extensively use expression (5.3) in the proofs below.

A closer look at the two Zenga estimators
Since samples are 'discrete populations', equations (5.2) and (5.3) lead to slightly different empirical estimators of Z F . If we choose equation (1.1), then we arrive at the estimator Z n , as seen from the proof of the following theorem.
Replacing every F on the right-hand side of equation (6.1) by F n , we obtain n k=1 X k:n 1{X k:n ≤ X i:n } n k=1 X k:n 1{X k:n > X i:n } , which simplifies to This is the estimator Z n .
If we choose equation (5.3) as the starting point for constructing an estimator for Z F , then we replace the quantile F −1 (p) by its empirical counterpart = X i:n when p ∈ (i − 1)/n, i/n in the definition of AL F (p), which gives us the empirical absolute Lorenz curve AL n (p), and then we replace each AL F (p) on the right-hand side of equation (5.3) by the just constructed AL n (p). (Note that μ F = AL F (1) ≈ AL n (1) =X.) This gives us the empirical Zenga index Z n as seen from the proof of the following theorem.
Theorem 6.2. The empirical Zenga index Z n is an estimator of Z F .
Proof. By construction, the estimator Z n is given by the equation: Hence, the proof of the lemma reduces to verifying that the right-hand sides of equations (2.2) and (6.2) coincide. For this, we split the integral in equation (6.2) into the sum of integrals over the intervals ((i − 1), i/n) for i = 1, . . . , n. For every p ∈ ((i − 1)/n, i/n), we have AL n (p) = C i,n + pX i:n , where Hence, equation (6.2) can be rewritten as Z n = n i=1 ζ i,n , where Consider the case i = 1. We have C 1,n = 0 and thus Λ 1,n = 0, which implies ζ 1,n = X X 1:n − 1 log 1 + X 1:n n k=2 X k:n .
Next, consider the case i = n. We have C n,n = X − X n:n and thus Ψ n,n = 1, which implies ζ n,n = 1 − X X n:n log n n − 1 .
When 2 ≤ i ≤ n − 1, then the integrand in the definition of ζ i,n does not have any singularity, since Ψ i,n > i/n due to n k=i+1 X k:n > 0 almost surely. Hence, after simple integration we have that, for i = 2, . . . , n − 1, With the above formulas for ζ i,n we easily check that the sum n i=1 ζ i,n is equal to the right-hand side of equation (2.2). This completes the proof of Theorem 6.2.

A closer look at variances
Following the formulation of Theorem 2.1 we claimed that the asymptotic distribu- The following theorem provides a proof of this claim. Proof. Note that n −1/2 n i=1 h(X i ) can be written as ∞ 0 e n (F (x))w F (F (x))dx, where e n (p) = √ n(E n (p) − p) is the empirical process based on the uniform on [0, 1] random The proof is based on the well known fact that, for every ε > 0, Hence, in order to prove statement (7.1), we only need to check that the integral is finite. For this, by considering the two cases p ≤ 1/2 and p > 1/2 separately, we easily show that |w F (p)| ≤ c + c log(1/p) + c log(1/ (1 − p)). Hence, for every ε > 0, there exists a constant c < ∞ such that, for all p ∈ (0, 1), (7.3) Bound (7.3) implies that integral (7.2) is finite provided that ∞ 0 (1 − F (x)) 1/2−2ε dx is finite, which is true since the moment E[X 2+α ] is finite for some α > 0 and the parameter ε > 0 can be chosen as small as desired. Hence, n −1/2 n i=1 h(X i ) → d Γ with Γ denoting the integral on the right-hand side of statement (7.1). The random variable Γ is normal because the Brownian bridge is a Gaussian process. Furthermore, Γ has mean zero because B(p) has mean zero for every p ∈ [0, 1]. The variance of Γ is We are left to show that E[Γ 2 ] < ∞. For this, we write the bound: (7.4) Since E[B 2 (F (x))] = F (x)(1 − F (x)), the finiteness of the integral on the right-hand side of bound (7.4) follows from the earlier proved statement that integral (7.2) is finite.
Theorem 7.2. The empirical variance S 2 X,n is an estimator of σ 2 F .
Proof. We construct an empirical estimator for σ 2 F by replacing every F (x) on the right-hand side of equation (2.4) by the empirical F n (x). In particular, we replace the function w F (t) by its empirical version We denote the just defined estimator of σ 2 F by S 2 X,n , and the rest of the proof consists of showing that the estimator S 2 X,n coincides with the one defined by equation (2.6). Note that min{F n (x), F n (y)} − F n (x)F n (y) = 0 when x ∈ [0, X 1:n ) ∪ [X n:n , ∞) and/or y ∈ [0, X 1:n ) ∪ [X n:n , ∞). Hence, the just defined S 2 X,n is equal to Xn:n X 1:n Xn:n X 1:n min{F n (x), F n (y)} − F n (x)F n (y) w X,n (F n (x))w X,n (F n (y))dxdy.
Proof. We proceed similarly to the proof of Theorem 7.2. We estimate the integrand After some rearrangement of terms, estimator (7.6) becomes When x ∈ [X k:n , X k+1:n ) and y ∈ [Y l:n , Y l+1:n ), then estimator (7.7) is n −1 k i=1 1{Y (i,n) ≤ Y l:n } − (k/n)(l/n), which leads us to the estimator S X,Y,n and thus completes the proof of Theorem 7.3.

Proof of Theorem 2.1
Throughout the proof we conveniently use the notation AL * F (p) for the dual absolute Lorenz curve 1 p F −1 (t)dt, which is equal to μ F −AL F (p). Likewise, we use the notation AL * n (p) for the empirical dual absolute Lorenz curve. Hence, Simple algebra gives the representation where the two remainder terms are: We shall later show (Lemmas 9.1 and 9.2 below) that the remainder terms r n,1 and r n,2 are of the order o P (1). Hence, we proceed with an analysis of the first two terms on the right-hand side of equation (8.1), for which we use the (general) Vervaat process and its dual version For mathematical and historical details on the Vervaat process, see Zitikis (1998), , and references therein. Since , adding the right-hand sides of equations (8.2) and (8.3) gives the equation V * n (p) = −V n (p). Hence, whatever upper bound we have for |V n (p)|, the same bound also holds for |V * n (p)|. In fact, the absolute value can be dropped from |V n (p)| since V n (p) is non-negative. Among other facts that we know , which is the uniform on [0, 1] empirical process, we have Bound (8.4) implies the following asymptotic representation for the first term on the right-hand side of equation (8.1): We shall later show (Lemma 9.3 below) that r n,3 = o P (1). Furthermore, we have the following asymptotic representation for the second term on the right-hand side of equation (8.1): We shall later show (Lemma 9.4 below) that r n,4 = o P (1). Hence, equations (8.1), (8.5) and (8.6) together with the statements r n,1 , . . . , r n,4 = o P (1) imply that which completes the proof of Theorem 2.1.

Negligibility of remainder terms
The following four lemmas establish the earlier noted statements that the remainder terms r n,1 , . . . , r n,4 are of the order o P (1). In the proofs of the lemmas we shall use a parameter δ ∈ (0, 1/2], possibly different from line to line but never depending on n.
Furthermore, we shall frequently use the fact that Another technical result that we shall frequently use is the fact that, for any ε > 0 as small as desired, when n → ∞.
Lemma 9.1. Under the conditions of Theorem 2.1, we have that r n,1 = o P (1).
This completes the proof of part (1).
Lemma 9.2. Under the conditions of Theorem 2.1, we have that r n,2 = o P (1).
This statement can be established following the proof of statement (9.18), with minor modifications.
This statement can be established following the proof of statement (9.18). The proof of Lemma 9.4 is finished.