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

Abstract

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 fascinating 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, Zenga (2007) 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 use the results to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW).

1. Introduction

Measuring and analyzing incomes, losses, risks, and other 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 is arguably the most popular measure of inequality, with a number of extensions and generalizations available in the literature. Keeping in mind that the notions of poor and rich are relative to each other, Zenga [1] constructed an index that reflects this relativity. We will next recall the definitions of the Gini and Zenga indices.

Let 𝐹(𝑥)=𝐏[𝑋≤𝑥] denote the cumulative distribution function (cdf) of the random variable 𝑋, which we assume to be nonnegative throughout the paper. Let 𝐹⁻¹(𝑝)=inf{𝑥∶𝐹(𝑥)≥𝑝} denote the corresponding quantile function. The Lorenz curve 𝐿_𝐹(𝑝) is given by the formula (see [2])

𝐿_𝐹1(𝑝)=𝜇_𝐹^𝑝₀𝐹⁻¹(𝑠)𝑑𝑠, (1.1) where 𝜇_𝐹=𝐄[𝑋] is the unknown true mean of 𝑋. Certainly, from the rigorous mathematical point of view we should call 𝐿_𝐹(𝑝) the Lorenz function, but this would deviate from the widely accepted usage of the term “Lorenz curve”. Hence, curves and functions are viewed as synonyms throughout this paper.

The classical Gini index 𝐺_𝐹 can now be written as follows:

𝐺_𝐹=¹₀𝐿(1−^𝐹(𝑝)𝑝)𝜓(𝑝)𝑑𝑝, (1.2) where 𝜓(𝑝)=2𝑝. Note that 𝜓(𝑝) is a density function on [0,1]. Given the usual econometric interpretation of the Lorenz curve [3], the function

𝐺_𝐹𝐿(𝑝)=1−^𝐹(𝑝)𝑝, (1.3) which we call the Gini curve, is a relative measure of inequality (see [4]). Indeed, 𝐿_𝐹(𝑝)/𝑝 is the ratio between (i) the mean income of the poorest 𝑝×100% of the population and (ii) the mean income of the entire population: the closer to each other these two means are, the lower is the inequality.

Zenga's [1] index 𝑍_𝐹 of inequality is defined by the formula

𝑍_𝐹=¹₀𝑍_𝐹(𝑝)𝑑𝑝, (1.4) where the Zenga curve 𝑍_𝐹(𝑝) is given by

𝑍_𝐹𝐿(𝑝)=1−^𝐹(𝑝)𝑝⋅1−𝑝1−𝐿_𝐹.(𝑝) (1.5) The Zenga curve measures the inequality between (i) the poorest 𝑝×100% of the population and (ii) the richer remaining (1−𝑝)×100% part of the population by comparing the mean incomes of these two disjoint and exhaustive subpopulations. We will elaborate on this interpretation later, in Section 5.

The Gini and Zenga indices 𝐺_𝐹 and 𝑍_𝐹 are (weighted) averages of the Gini and Zenga curves 𝐺_𝐹(𝑝) and 𝑍_𝐹(𝑝), respectively. However, while in the case of the Gini index the weight function (i.e., the density) 𝜓(𝑝)=2𝑝 is employed, in the case of the Zenga index the uniform weight function 𝜓(𝑝)=1 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, the Zenga index 𝑍_𝐹 detects, with the same sensibility, 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

𝑥𝐹(𝑥)=1−(⁰𝑥)^𝜃,𝑥≥𝑥₀, (1.6) where 𝑥₀>0 and 𝜃>0 are parameters. Later in this paper, we will use this distribution in a simulation study, setting 𝑥₀=1 and 𝜃=2.06. Note that when 𝜃>2, then the second moment of the distribution is finite. The “heavy-tailed” case 1<𝜃<2 is also of interest, especially when modeling incomes of countries with very high economic inequality. We will provide additional details on the case in Section 5.

Note. Pareto distribution (1.6) is perhaps the oldest model for income distributions. It dates back to Pareto [5], and Pareto [6]. Pareto's original empirical research suggested him that the number of tax payers with income 𝑥 is roughly proportional to 𝑥^−(𝜃+1), where 𝜃 is a parameter that measures inequality. For historical details on the interpretation of this parameter in the context of measuring economic inequality, we refer to Zenga [7]. We can view the parameter 𝑥₀>0 as the lowest taxable income. In addition, besides being the greatest lower bound of the distribution support, 𝑥₀ is also the scale parameter of the distribution and thus does not affect our inequality indices and curves, as we will see in formulas below.

Note. The Pareto distribution is positively supported, 𝑥≥𝑥₀>0. In real surveys, however, in addition to many positive incomes we may also observe some zero and negative incomes. This happens when evaluating net household incomes, which are the sums of payroll incomes (net wages, salaries, fringe benefits), pensions and net transfers (pensions, arrears, financial assistance, scholarships, alimony, gifts). Paid alimony and gifts are subtracted in forming the incomes. However, negative incomes usually happen in the case of very few statistical units. For example, in the 2006 Bank of Italy survey we observe only four households with nonpositive incomes, out of the total of 7,766 households. Hence, it is natural to fit the Pareto model to the positive incomes and keep in mind that we are actually dealing with a conditional distribution. If, however, it is desired to deal with negative, null, and positive incomes, then instead of the Pareto distribution we may switch to different ones, such as Dagum distributions with three or four parameters [8–10].

Corresponding to Pareto distribution (1.6), the Lorenz curve is given by the formula 𝐿_𝐹(𝑝)=1−(1−𝑝)^1−1/𝜃 (see [11]), and thus the Gini curve becomes 𝐺_𝐹(𝑝)=((1−𝑝)^1−1/𝜃−(1−𝑝))/𝑝. In Figure 1(a) we have depicted the Gini and weighted Gini curves. The corresponding Zenga curve is equal to 𝑍_𝐹(𝑝)=(1−(1−𝑝)^1/𝜃)/𝑝 and is depicted in Figure 1(b), alongside the Gini curve 𝐺_𝐹(𝑝) for an easy comparison. Figure 1(a) allows us to appreciate how the Gini weight function 𝜓(𝑝)=2𝑝 disguises the high inequality between the mean income of the very poor and that of the whole population, and overemphasizes comparisons between almost identical subgroups. The outcome is that the Gini index 𝐺_𝐹 underestimates inequality. In Figure 1(b) we see the difference between the Gini and Zenga inequality curves. For example, 𝐺_𝐹(𝑝) for 𝑝=0.8 yields 0.296, which tells us that the mean income of the poorest 80% of the population is 29.6% lower than the mean income of the whole population, while the corresponding ordinate of the Zenga curve is 𝑍_𝐹(0.8)=0.678, which tells us that the mean income of the poorest 80% of the population is 67.8% lower than the mean income of the remaining (richer) part of the population.

(a)

(b)

Figure 1 

The Gini curve 𝐺𝐹(𝑝)(dashed; (a) and (b)), the weighted Gini curve 𝑔𝐹(𝑝)(solid; (a)), and the Zenga curve 𝑍𝐹(𝑝)(solid; (b)) in the Pareto case with 𝑥0=1 and 𝜃=2.06.

The rest of this paper is organized as follows. In Section 2 we define two estimators of the Zenga index 𝑍_𝐹 and develop statistical inferential results. In Section 3 we present results of a simulation study, which explores the empirical performance of two Zenga estimators, 𝑍_𝑛 and 𝑍_𝑛, including coverage accuracy and length of several types of confidence intervals. In Section 4 we present an analysis of the the Bank of Italy Survey on Household Income and Wealth (SHIW) data. In Section 5 we further contribute to the understanding of the Zenga index 𝑍_𝐹 by relating it to lower and upper conditional expectations, as well as to the conditional tail expectation (CTE), which has been widely used in insurance. In Section 6 we provide a theoretical background 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. Technical lemmas and their proofs are relegated to Section 9.

2. Estimators and Statistical Inference

Unless explicitly stated otherwise, our statistical inferential results are derived under the assumption that data are outcomes of independent and identically distributed (i.i.d.) random variables.

Hence, let 𝑋₁,…,𝑋_𝑛 be independent copies of 𝑋. We use two nonparametric estimators for the Zenga index 𝑍_𝐹. The first one [12] is given by the formula

𝑍_𝑛1=1−𝑛^𝑛−1_𝑖=1𝑖⁻¹∑^𝑖_𝑘=1𝑋_𝑘∶𝑛(𝑛−𝑖)⁻¹∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛, (2.1) where 𝑋_1∶𝑛≤⋯≤𝑋_𝑛∶𝑛 are the order statistics of 𝑋₁,…,𝑋_𝑛. With 𝑋 denoting the sample mean of 𝑋₁,…,𝑋_𝑛, the second estimator of the Zenga index 𝑍_𝐹 is given by the formula

𝑍_𝑛=−^𝑛_𝑖=2∑^𝑖−1_𝑘=1𝑋_𝑘∶𝑛−(𝑖−1)𝑋_𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛𝑖log()+𝑖−1_{𝑛−1𝑖=1}(𝑋𝑋_𝑖∶𝑛∑−1−^{𝑖−1𝑘=1}𝑋_𝑘∶𝑛−(𝑖−1)𝑋_𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛𝑋)log(1+^𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛). (2.2) The two estimators 𝑍_𝑛 and 𝑍_𝑛 are asymptotically equivalent. However, despite the fact that the estimator 𝑍_𝑛 is more complex, it will nevertheless be more convenient to work with when establishing asymptotic results later in this paper.

Unless explicitly stated otherwise, we assume throughout that the cdf 𝐹(𝑥) 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., [13]).

Theorem 2.1. If the moment 𝐄[𝑋^2+𝛼] is finite for some 𝛼>0, then one has the asymptotic representation √𝑛(𝑍_𝑛−𝑍_𝐹1)=√𝑛^𝑛_𝑖=1ℎ(𝑋_𝑖)+𝑜_𝐏(1), (2.3) where 𝑜_𝐏(1) denotes a random variable that converges to 0 in probability when 𝑛→∞, and ℎ(𝑋_𝑖)=^∞₀(𝟏{𝑋_𝑖≤𝑥}−𝐹(𝑥))𝑤_𝐹(𝐹(𝑥))𝑑𝑥 (2.4) with the weight function 𝑤_𝐹1(𝑡)=−𝜇_𝐹^𝑡₀(1𝑝𝐿−1)^𝐹(𝑝)(1−𝐿_𝐹(𝑝))²1𝑑𝑝+𝜇_𝐹¹_𝑡(1𝑝1−1)1−𝐿_𝐹(𝑝)𝑑𝑝. (2.5)

In view of Theorem 2.1, the asymptotic distribution of √𝑛(𝑍_𝑛−𝑍_𝐹) is centered normal with the variance 𝜎²_𝐹=𝐄[ℎ²(𝑋)], which is finite (see Theorem 7.1) and can be written as follows:

𝜎²_𝐹=^∞₀^∞₀(min{𝐹(𝑥),𝐹(𝑦)}−𝐹(𝑥)𝐹(𝑦))𝑤_𝐹(𝐹(𝑥))𝑤_𝐹(𝐹(𝑦))𝑑𝑥𝑑𝑦. (2.6) Alternatively,

𝜎²_𝐹=¹₀(_[0,𝑢)𝑡𝑤_𝐹(𝑡)𝑑𝐹⁻¹(𝑡)−_[𝑢,1)(1−𝑡)𝑤_𝐹(𝑡)𝑑𝐹⁻¹(𝑡))²𝑑𝑢. (2.7) The latter expression of 𝜎²_𝐹 is particularly convenient when working with distributions for which the first derivative (when it exists) of the quantile 𝐹⁻¹(𝑡) is a relatively simple function, as is the case for a large class of distributions (see, e.g., [14]). However, irrespectively of what expression for the variance 𝜎²_𝐹 we use, the variance is unknown since the cdf 𝐹(𝑥) is unknown, and thus 𝜎²_𝐹 needs to be estimated empirically.

2.1. One Sample Case

Replacing the population cdf everywhere on the right-hand side of (2.6) by the empirical cdf 𝐹_𝑛(𝑥)=𝑛⁻¹∑^𝑛_𝑖=1𝟏{𝑋_𝑖≤𝑥}, where 𝟏 denotes the indicator function, we obtain (Theorem 7.2) the following estimator of the variance 𝜎²_𝐹:

𝑆²_𝑋,𝑛=^𝑛−1_𝑘=1^𝑛−1_𝑙=1(min{𝑘,𝑙}𝑛−𝑘𝑛𝑙𝑛)×𝑤_𝑋,𝑛(𝑘𝑛)𝑤_𝑋,𝑛(𝑙𝑛)(𝑋_{𝑘+1∶𝑛}−𝑋_𝑘∶𝑛)(𝑋_{𝑙+1∶𝑛}−𝑋_𝑙∶𝑛), (2.8) where

𝑤_𝑋,𝑛(𝑘𝑛)=−^𝑘_𝑖=1𝐼_𝑋,𝑛(𝑖)+^𝑛_𝑖=𝑘+1𝐽_𝑋,𝑛(𝑖) (2.9) with the following expressions for the summands 𝐼_𝑋,𝑛(𝑖) and 𝐽_𝑋,𝑛(𝑖)∶ first,

𝐼_𝑋,𝑛∑(1)=−^𝑛𝑘=2𝑋_𝑘∶𝑛−(𝑛−1)𝑋_1∶𝑛(∑^𝑛_𝑘=1𝑋_𝑘∶𝑛)(∑^𝑛_𝑘=2𝑋_𝑘∶𝑛)+1𝑋_1,𝑛𝑋log(1+^1∶𝑛∑^𝑛_𝑘=2𝑋_𝑘∶𝑛). (2.10) Furthermore, for every 𝑖=2,…,𝑛−1,

𝐼_𝑋,𝑛∑(𝑖)=𝑛^{𝑖−1𝑘=1}𝑋_𝑘∶𝑛−(𝑖−1)𝑋_𝑖∶𝑛(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛)²𝑖log()−𝑖−1(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛−(𝑛−𝑖)𝑋_𝑖∶𝑛)(∑^𝑛_𝑘=1𝑋_𝑘∶𝑛)(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛)(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛)(∑^𝑛_𝑘=𝑖𝑋_𝑘∶𝑛)1+(𝑋_𝑖∶𝑛∑+𝑛^{𝑖−1𝑘=1}𝑋_𝑘∶𝑛−(𝑖−1)𝑋_𝑖∶𝑛(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛)²𝑋)log(1+^𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛𝐽),_𝑋,𝑛𝑛(𝑖)=∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛𝑖log()−∑𝑖−1_{𝑛𝑘=𝑖+1}𝑋_𝑘∶𝑛−(𝑛−𝑖)𝑋_𝑖∶𝑛𝑋_𝑖∶𝑛(∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛)𝑋log(1+^𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛). (2.11)(2.12) Finally,

𝐽_𝑋,𝑛1(𝑛)=𝑋_𝑛,𝑛𝑛log(𝑛−1). (2.13) With the just defined estimator 𝑆²_𝑋,𝑛 of the variance 𝜎²_𝐹, we have the asymptotic result:

√𝑛(𝑍_𝑛−𝑍_𝐹)𝑆_𝑋,𝑛→_𝑑𝒩(0,1), (2.14) where →_𝑑 denotes convergence in distribution.

2.2. Two Independent Samples

We now discuss a variant of statement (2.14) in the case of two populations when samples are independent. Namely, let the random variables 𝑋₁,…,𝑋_𝑛∼𝐹 and 𝑌₁,…,𝑌_𝑚∼𝐻 be independent within and between the two samples. Just like in the case of the cdf 𝐹(𝑥), here we also 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, which means that there exists 𝜂∈(0,1) such that

𝑚𝑛+𝑚→𝜂∈(0,1) (2.15) when both 𝑛 and 𝑚 tend to infinity. From statement (2.3) and its counterpart for 𝑌_𝑖∼𝐻 we then have that the quantity 𝑛𝑚/(𝑛+𝑚)((𝑍_𝑋,𝑛−𝑍_𝑌,𝑚)−(𝑍_𝐹−𝑍_𝐻)) is asymptotically normal with mean zero and the variance 𝜂𝜎²_𝐹+(1−𝜂)𝜎²_𝐻. To estimate the variances 𝜎²_𝐹 and 𝜎²_𝐻, we use 𝑆²_𝑋,𝑛 and 𝑆²_𝑌,𝑛, respectively, and obtain the following result:

(𝑍_𝑋,𝑛−𝑍_𝑌,𝑚)−(𝑍_𝐹−𝑍_𝐻)(1/𝑛)𝑆²_𝑋,𝑛+(1/𝑚)𝑆²_𝑌,𝑚→_𝑑𝒩(0,1). (2.16)

2.3. Paired Samples

Consider now the case when the two samples 𝑋₁,…,𝑋_𝑛∼𝐹 and 𝑌₁,…,𝑌_𝑚∼𝐻 are paired. Thus, we have that 𝑚=𝑛, and we also have that the pairs (𝑋₁,𝑌₁),…,(𝑋_𝑛,𝑌_𝑛) are independent and identically distributed. Nothing is assumed about the joint distribution of (𝑋,𝑌). As before, the cdf's 𝐹(𝑥) and 𝐻(𝑦) are continuous and both have finite moments of order 2+𝛼, for some 𝛼>0. From statement (2.3) and its analog for 𝑌 we have that √𝑛((𝑍_𝑋,𝑛−𝑍_𝑌,𝑛)−(𝑍_𝐹−𝑍_𝐻)) is asymptotically normal with mean zero and the variance 𝜎²_𝐹,𝐻=𝐄[(ℎ(𝑋)−ℎ(𝑌))²]. The latter variance can of course be written as 𝜎²_𝐹−2𝐄[ℎ(𝑋)ℎ(𝑌)]+𝜎²_𝐻. Having already constructed estimators 𝑆²_𝑋,𝑛 and 𝑆²_𝑌,𝑛, we are only left to construct an estimator for 𝐄[ℎ(𝑋)ℎ(𝑌)]. (Note that when 𝑋 and 𝑌 are independent, then 𝐏[𝑋≤𝑥,𝑌≤𝑦]=𝐹(𝑥)𝐻(𝑦) and thus the expectation 𝐄[ℎ(𝑋)ℎ(𝑌)] vanishes.) To this end, we write the equation

𝐄[ℎ(𝑋)ℎ(𝑌)]=^∞₀^∞₀(𝐏[𝑋≤𝑥,𝑌≤𝑦]−𝐹(𝑥)𝐻(𝑦))𝑤_𝐹(𝐹(𝑥))𝑤_𝐻(𝐻(𝑦))𝑑𝑥𝑑𝑦. (2.17) Replacing the cdf's 𝐹(𝑥) and 𝐻(𝑦) everywhere on the right-hand side of the above equation by their respective empirical estimators 𝐹_𝑛(𝑥) and 𝐻_𝑛(𝑦), we have (Theorem 7.3)

𝑆_{𝑋,𝑌,𝑛}=^𝑛−1_𝑘=1^𝑛−1_𝑙=1(1𝑛^𝑘_𝑖=1𝟏{𝑌_(𝑖,𝑛)≤𝑌_𝑙∶𝑛𝑘}−𝑛𝑙𝑛)×𝑤_𝑋,𝑛(𝑘𝑛)𝑤_𝑌,𝑛(𝑙𝑛)(𝑋_{𝑘+1∶𝑛}−𝑋_𝑘∶𝑛)(𝑌_{𝑙+1∶𝑛}−𝑌_𝑙∶𝑛), (2.18) where 𝑌_(1,𝑛),…,𝑌_(𝑛,𝑛) are the induced (by 𝑋₁,…,𝑋_𝑛) order statistics of 𝑌₁,…,𝑌_𝑛. (Note that when 𝑌≡𝑋, then 𝑌_(𝑖,𝑛)=𝑌_𝑖∶𝑛 and so the sum ∑^𝑘_𝑖=1𝟏{𝑌_(𝑖,𝑛)≤𝑌_𝑙∶𝑛} is equal to min{𝑘,𝑙}; hence, estimator (2.18) coincides with estimator (2.8), as expected.) Consequently, 𝑆²_𝑋,𝑛−2𝑆_{𝑋,𝑌,𝑛}+𝑆²_𝑌,𝑛 is an empirical estimator of 𝜎²_𝐹,𝐻, and so we have that

√𝑛(𝑍_𝑋,𝑛−𝑍_𝑌,𝑛)−(𝑍_𝐹−𝑍_𝐻)𝑆²_𝑋,𝑛−2𝑆_{𝑋,𝑌,𝑛}+𝑆²_𝑌,𝑛→_𝑑𝒩(0,1). (2.19)

We conclude this section with a note that the above established asymptotic results (2.14), (2.16), and (2.19) 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. For hints and references on the topic, we refer to Jones et al. [15] and Brazauskas et al. [16].

3. A Simulation Study

Here we investigate the numerical performance of the estimators 𝑍_𝑛 and 𝑍_𝑛 by simulating data from Pareto distribution (1.6) with 𝑥₀=1 and 𝜃=2.06. These choices give the value 𝑍_𝐹=0.6, which is approximately seen in real income distributions. As to the (artificial) choice 𝑥₀=1, we note that since 𝑥₀ is the scale parameter in the Pareto model, the inequality indices and curves are invariant to it. Hence, all results to be reported in this section concerning the coverage accuracy and size of confidence intervals will not be affected by the choice 𝑥₀=1.

Following Davison and Hinkley [17, 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) are obtained from Theorem 2.1, and those for the estimator 𝑍_𝑛 using numerical differentiation as in Greselin and Pasquazzi [12].

In Table 1 we report coverage percentages of 10,000 confidence intervals, for each of the four types: normal, percentile, BCa, and 𝑡-bootstrap. Bootstrap-based approximations have been obtained from 9,999 resamples of the original samples. As suggested by Efron [18], 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 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 1 that all confidence intervals suffer from some undercoverage. For example, with sample size 800, 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 the other ones) comes at the cost of their larger sizes, as seen in Table 2. Some of the studentized bootstrap confidence intervals extend beyond the range [0,1] of the Zenga index 𝑍_𝐹, but this can easily be fixed by taking the minimum between the currently recorded upper bounds and 1, which is the upper bound of the Zenga index 𝑍_𝐹 for every cdf 𝐹. We note that for the BCa confidence intervals, the number of bootstrap replications of the original sample has to be increased beyond 9,999 if the nominal confidence level is high. Indeed, for samples of size 800, it turns out that the upper bound of 1,598 (out of 10,000) of the BCa confidence intervals based on 𝑍_𝑛 and with 0.99 nominal confidence level is given by the largest order statistics of the bootstrap distribution. For the confidence intervals based on 𝑍_𝑛, the corresponding figure is 1,641.

Table 1 

Coverage proportions of confidence intervals from the Pareto parent distribution with 𝑥₀=1 and 𝜃=2.06 (𝑍_𝐹=0.6).

Table 2 

Size of the 95% asymptotic confidence intervals from the Pareto parent distribution with 𝑥₀=1 and 𝜃=2.06 (𝑍_𝐹=0.6).

4. An Analysis of Italian Income Data

In this section we use the Zenga index 𝑍_𝐹 to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW). The sample of the 2006 wave of this survey contains 7,768 households, with 3,957 of them being panel households. For detailed information on the survey, we refer to the Bank of Italy [19] publication. In order to treat data correctly in the case of different household sizes, we work with equivalent incomes, which we have obtained by dividing the total household income by an equivalence coefficient, which is the sum of weights assigned to each household member. Following the modified Organization for Economic Cooperation and Development (OECD) equivalence scale, we give weight 1 to the household head, 0.5 to the other adult members of the household, and 0.3 to the members under 14 years of age. 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 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.

In Table 3 we report the values of 𝑍_𝑛 and 𝑍_𝑛 according to the geographic area of the 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.

Table 3 

Confidence intervals for 𝑍_𝐹 in the 2006 Italian income distribution.

Note. Removing the negative incomes from our current analysis is important as otherwise we would need to develop a much more complex methodology than the one offered in this paper. To give a flavour of technical challenges, we note that the Gini index may overestimate the economic inequality when negative, zero, and positive incomes are considered. In this case the Gini index needs to be renormalized as demonstrated by, for example, Chen et al. [20]. Another way to deal with the issue would be to analyze the negative incomes and their concentration separately from the zero and positive incomes and their concentration.

Consequently, the point estimates of 𝑍_𝐹 are based on 7,766 equivalent incomes with 𝑍_𝑛=0.6470 and 𝑍_𝑛=0.6464. As pointed out by Maasoumi [21], however, good care is needed when comparing point estimates of inequality measures. Indeed, direct comparison of the point estimates corresponding to the five geographic areas of Italy would lead us to the conclusion that the inequality is higher in the central and southern areas when compared to the northern area and the islands. But as we glean from pairwise comparisons of the confidence intervals, only the differences between the estimates corresponding to the northwestern and southern areas and perhaps to the islands and the southern area may be deemed statistically significant.

Moreover, we have used the paired samples of the 2004 and 2006 incomes of the 3,957 panel households in order to check whether during this time period there was a change in inequality among households. In Table 4 we report the values of 𝑍_𝑛 based on the panel households for these two years, and the 95% confidence intervals for the difference between the values of the Zenga index for the years 2006 and 2004. These computations have been based on formula (2.19). Having removed the four households with at least one negative income in the paired sample, we were left with a total of 3,953 observations. We see that even though we deal with large sample sizes, the point estimates alone are not reliable. Indeed, for Italy as the whole and for all geographic areas except the center, the point estimates suggest that the Zenga index decreased from the year 2004 to 2006. However, the 95% confidence intervals in Table 4 suggest that this change is not significant.

Table 4 

95% confidence intervals for the difference of the Zenga indices between 2006 and 2004 in the Italian income distribution.

5. An Alternative Look at the Zenga Index

In various contexts we have notions of rich and poor, large and small, risky and secure. They divide the underlying populations into two parts, which we view as subpopulations. The quantile 𝐹⁻¹(𝑝), for some 𝑝∈(0,1), usually serves as a boundary separating the two subpopulations. For example, we may define rich if 𝑋>𝐹⁻¹(𝑝) and poor if 𝑋≤𝐹⁻¹(𝑝). Calculating the mean value of the former subpopulation gives rise to the upper conditional expectation 𝐄[𝑋∣𝑋>𝐹⁻¹(𝑝)], which is known in the actuarial risk theory as the conditional tail expectation (CTE). Calculating the mean value of the latter subpopulation gives rise to the lower conditional expectation 𝐄[𝑋∣𝑋≤𝐹⁻¹(𝑝)], which is known in the econometric literature as the absolute Bonferroni curve, as a function of 𝑝.

Clearly, the ratio

𝑅_𝐹(𝑝)=𝐄[𝑋∣𝑋≤𝐹⁻¹(𝑝)]𝐄[𝑋∣𝑋>𝐹⁻¹(𝑝)] (5.1) of the lower and upper conditional expectations takes on values in the interval [0,1]. When 𝑋 is equal to any constant, which can be interpreted as the 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). This 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 function 1−𝑅_𝐹(𝑝), which coincides with the Zenga curve (1.5) when the cdf 𝐹(𝑥) is continuous. The area

1−¹₀𝐄[𝑋∣𝑋≤𝐹⁻¹(𝑝)]𝐄[𝑋∣𝑋>𝐹⁻¹(𝑝)]𝑑𝑝 (5.2) beneath the function 1−𝑅_𝐹(𝑝) is always in the interval [0,1]. Quantity (5.2) is a measure of inequality and coincides with the earlier defined Zenga index 𝑍_𝐹 when the cdf 𝐹(𝑥) is continuous, which we assume throughout the paper.

Note that under the continuity of 𝐹(𝑥), the lower and upper conditional expectations are equal to the absolute Bonferroni curve 𝑝⁻¹AL_𝐹(𝑝) and the dual absolute Bonferroni curve (1−𝑝)⁻¹(𝜇_𝐹−AL_𝐹(𝑝)), respectively, where

AL_𝐹(𝑝)=^𝑝₀𝐹⁻¹(𝑡)𝑑𝑡 (5.3) is the absolute Lorenz curve. This leads us to the expression of the Zenga index 𝑍_𝐹 given by (1.4), which we now rewrite in terms of the absolute Lorenz curve as follows:

𝑍_𝐹=1−¹₀(1𝑝−1)AL_𝐹(𝑝)𝜇_𝐹−AL_𝐹(𝑝)𝑑𝑝. (5.4) We will extensively use expression (5.4) in the proofs below. In particular, we will see in the next section that the empirical Zenga index 𝑍_𝑛 is equal to 𝑍_𝐹 with the population cdf 𝐹(𝑥) replaced by the empirical cdf 𝐹_𝑛(𝑥).

We are now in the position to provide additional details on the earlier noted Pareto case 1<𝜃<2, when the Pareto distribution has finite 𝐄[𝑋] but infinite 𝐄[𝑋²]. The above derived asymptotic results and thus the statistical inferential theory fail in this case. The required adjustments are serious and rely on the use of the extreme value theory, instead of the classical central limit theorem (CLT). Specifically, the task can be achieved by first expressing the absolute Lorenz curve AL_𝐹(𝑝) in terms of the conditional tail expectation (CTE):

CTE_𝐹1(𝑝)=1−𝑝¹_𝑝𝐹⁻¹(𝑡)𝑑𝑡 (5.5) using the equation AL_𝐹(𝑝)=𝜇_𝐹−(1−𝑝)CTE_𝐹(𝑝). Hence, (5.4) becomes

𝑍_𝐹=1−¹₀1𝑝(CTE_𝐹(0)CTE_𝐹(𝑝)−(1−𝑝))𝑑𝑝, (5.6) where CTE_𝐹(0) is of course the mean 𝜇_𝐹. Note that replacing the population cdf 𝐹(𝑥) by its empirical counterpart 𝐹_𝑛(𝑥) on the right-hand side of (5.6) would not lead to an estimator that would work when 𝐄[𝑋²]=∞, and thus when the Pareto parameter 1<𝜃<2. A solution to this problem is provided by Necir et al. [22], who have suggested a new estimator of the conditional tail expectation CTE_𝐹(𝑝) for heavy-tailed distributions. Plugging in that estimator instead of the CTE on the right-hand side of (5.6) produces an estimator of the Zenga index when 𝐄[𝑋²]=∞. Establishing asymptotic results for the new “heavy-tailed” Zenga estimator would, however, be a complex technical task, well beyond the scope of the present paper, as can be seen from the proofs of Necir et al. [22].

6. A Closer Look at the Two Zenga Estimators

Since samples are “discrete populations”, (5.2) and (5.4) lead to slightly different empirical estimators of 𝑍_𝐹. If we choose (5.2) and replace all population-related quantities by their empirical counterparts, then we will arrive at the estimator 𝑍_𝑛, as seen from the proof of the following theorem.

Theorem 6.1. The empirical Zenga index 𝑍_𝑛 is an empirical estimator of 𝑍_𝐹.

Proof. Let 𝑈 be a uniform on [0,1] random variable independent of 𝑋. The cdf of 𝐹⁻¹(𝑈) is 𝐹. Hence, we have the following equations: 𝑍_𝐹=1−𝐄_𝑈(𝐄_𝑋[𝑋∣𝑋≤𝐹⁻¹(𝑈)]𝐄_𝑋[𝑋∣𝑋>𝐹⁻¹)(𝑈)]=1−_(0,∞)1−𝐹(𝑥)𝐹(𝑥)𝐄[𝑋𝟏{𝑋≤𝑥}]𝐄[𝑋𝟏{𝑋>𝑥}]𝑑𝐹(𝑥)=1−_(0,∞)1−𝐹(𝑥)∫𝐹(𝑥)^(0,𝑥]𝑦𝑑𝐹(𝑦)∫_(𝑥,∞)𝑦𝑑𝐹(𝑦)𝑑𝐹(𝑥). (6.1) Replacing every 𝐹 on the right-hand side of (6.1) by 𝐹_𝑛, we obtain 11−𝑛^𝑛−1_𝑖=11−𝐹_𝑛(𝑋_𝑖∶𝑛)𝐹_𝑛(𝑋_𝑖∶𝑛)∑^𝑛_𝑘=1𝑋_𝑘∶𝑛𝟏{𝑋_𝑘∶𝑛≤𝑋_𝑖∶𝑛}∑^𝑛_𝑘=1𝑋_𝑘∶𝑛𝟏{𝑋_𝑘∶𝑛>𝑋_𝑖∶𝑛}, (6.2) which simplifies to 11−𝑛^𝑛−1_𝑖=11−𝑖/𝑛∑𝑖/𝑛^𝑖𝑘=1𝑋_𝑘∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛. (6.3) This is the estimator 𝑍_𝑛 [12].

If, on the other hand, we choose (5.4) as the starting point for constructing an empirical estimator for 𝑍_𝐹, then we first replace the quantile 𝐹⁻¹(𝑝) by its empirical counterpart

𝐹_𝑛⁻¹(𝑝)=inf{𝑥∶𝐹_𝑛(𝑥)≥𝑝}=𝑋_𝑖∶𝑛when𝑝∈((𝑖−1)𝑛,𝑖𝑛] (6.4) in the definition of AL_𝐹(𝑝), which leads to the empirical absolute Lorenz curve AL_𝑛(𝑝), and then we replace each AL_𝐹(𝑝) on the right-hand side of (5.4) by the just constructed AL_𝑛(𝑝). (Note that 𝜇_𝐹=AL_𝐹(1)≈AL_𝑛(1)=𝑋.) These considerations produce 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: 𝑍_𝑛=1−¹₀(1𝑝−1)AL_𝑛(𝑝)𝑋−AL_𝑛(𝑝)𝑑𝑝. (6.5) Hence, the proof of the lemma reduces to verifying that the right-hand sides of (2.2) and (6.5) coincide. For this, we split the integral in (6.5) 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 𝐶_𝑖,𝑛=1𝑛^𝑖−1_𝑘=1𝑋_𝑘∶𝑛−𝑖−1𝑛𝑋_𝑖∶𝑛. (6.6) Hence, (6.5) can be rewritten as 𝑍_𝑛=∑^𝑛_𝑖=1𝜁_𝑖,𝑛, where 𝜁_𝑖,𝑛=1𝑛−^𝑖/𝑛_{(𝑖−1)/𝑛}(1𝑝Λ−1)^𝑖,𝑛+𝑝Ψ_𝑖,𝑛−𝑝𝑑𝑝 (6.7) with Λ_𝑖,𝑛=𝐶_𝑖,𝑛𝑋_𝑖∶𝑛,Ψ_𝑖,𝑛=𝑋−𝐶_𝑖,𝑛𝑋_𝑖∶𝑛. (6.8) Consider first the case 𝑖=1. We have 𝐶_1,𝑛=0 and thus Λ_1,𝑛=0, which implies 𝜁_1,𝑛=(𝑋𝑋_1∶𝑛𝑋−1)log(1+^1∶𝑛∑^𝑛_𝑘=2𝑋_𝑘∶𝑛). (6.9) Next, we 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−𝑋𝑋_𝑛∶𝑛𝑛)log(𝑛−1). (6.10) 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+1𝑋_𝑘∶𝑛>0 almost surely. Hence, after simple integration we have that, for 𝑖=2,…,𝑛−1, 𝜁_𝑖,𝑛=(𝑖−1)𝑋_𝑖∶𝑛−∑^𝑖−1_𝑘=1𝑋_𝑘∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛𝑖log()𝑖−1+(𝑋𝑋_𝑖∶𝑛−1+(𝑖−1)𝑋_𝑖∶𝑛−∑^𝑖−1_𝑘=1𝑋_𝑘∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛+𝑖𝑋_𝑖∶𝑛𝑋)log(1+^𝑖∶𝑛∑^𝑛_𝑘=𝑖+1𝑋_𝑘∶𝑛). (6.11) With the above formulas for 𝜁_𝑖,𝑛 we easily check that the sum ∑^𝑛_𝑖=1𝜁_𝑖,𝑛 is equal to the right-hand side of (2.2). This completes the proof of Theorem 6.2.

7. A Closer Look at the Variances

Following the formulation of Theorem 2.1 we claimed that the asymptotic distribution of √𝑛(𝑍_𝑛−𝑍_𝐹) is centered normal with the finite variance 𝜎²_𝐹=𝐄[ℎ²(𝑋)]. The following theorem proves this claim.

Theorem 7.1. When 𝐄[𝑋^2+𝛼]<∞ for some 𝛼>0, then 𝑛^−1/2∑^𝑛_𝑖=1ℎ(𝑋_𝑖) converges in distribution to the centered normal random variable Γ=^∞₀ℬ(𝐹(𝑥))𝑤_𝐹(𝐹(𝑥))𝑑𝑥, (7.1) where ℬ(𝑝) is the Brownian bridge on the interval [0,1]. The variance of Γ is finite and equal to 𝜎²_𝐹.

Proof. Note that 𝑛^−1/2∑^𝑛_𝑖=1ℎ(𝑋_𝑖) can be written as ∫^∞₀𝑒_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 variables 𝑈_𝑖=𝐹(𝑋_𝑖),𝑖=1,…,𝑛. We will next show that ^∞₀𝑒_𝑛(𝐹(𝑥))𝑤_𝐹(𝐹(𝑥))𝑑𝑥→_𝑑^∞₀ℬ(𝐹(𝑥))𝑤_𝐹(𝐹(𝑥))𝑑𝑥. (7.2) The proof is based on the well-known fact that, for every 𝜀>0, the following weak convergence of stochastic processes takes place: {𝑒_𝑛(𝑝)𝑝^1/2−𝜀(1−𝑝)^1/2−𝜀,0≤𝑝≤1}⇒{ℬ(𝑝)𝑝^1/2−𝜀(1−𝑝)^1/2−𝜀,0≤𝑝≤1}. (7.3) Hence, in order to prove statement (7.2), we only need to check that the integral ^∞₀𝐹(𝑥)^1/2−𝜀(1−𝐹(𝑥))^1/2−𝜀𝑤_𝐹(𝐹(𝑥))𝑑𝑥 (7.4) is finite. For this, by considering, for example, the two cases 𝑝≤1/2 and 𝑝>1/2 separately, we first easily verify the bound |𝑤_𝐹(𝑝)|≤𝑐+𝑐log(1/𝑝)+𝑐log(1/(1−𝑝)). Hence, for every 𝜀>0, there exists a constant 𝑐<∞ such that, for all 𝑝∈(0,1), |𝑤_𝐹𝑐(𝑝)|≤𝑝^𝜀(1−𝑝)^𝜀. (7.5) Bound (7.5) implies that integral (7.4) is finite when ∫^∞₀(1−𝐹(𝑥))^1/2−2𝜀𝑑𝑥<∞, 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=1ℎ(𝑋_𝑖)→_𝑑Γ with Γ denoting the integral on the right-hand side of statement (7.2). The random variable Γ is normal because the Brownian bridge ℬ(𝑝) 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 equal to 𝜎²_𝐹 because 𝐄[ℬ(𝑝)ℬ(𝑞)]=min{𝑝,𝑞}−𝑝𝑞 for all 𝑝,𝑞∈[0,1]. We are left to show that 𝐄[Γ²]<∞. For this, we write the bound: 𝐄[Γ²]=^∞₀^∞₀𝐄[ℬ(𝐹(𝑥))ℬ(𝐹(𝑦))]𝑤_𝐹(𝐹(𝑥))𝑤_𝐹(𝐹(𝑦))𝑑𝑥𝑑𝑦≤(^∞₀𝐄[ℬ²(𝐹(𝑥))]𝑤_𝐹(𝐹(𝑥))𝑑𝑥)². (7.6) Since 𝐄[ℬ²(𝐹(𝑥))]=𝐹(𝑥)(1−𝐹(𝑥)), the finiteness of the integral on the right-hand side of bound (7.6) follows from the earlier proved statement that integral (7.4) is finite. Hence, 𝐄[Γ²]<∞ as claimed, which concludes the proof of Theorem 7.1.

Theorem 7.2. The empirical variance 𝑆²_𝑋,𝑛 is an estimator of 𝜎²_𝐹.

Proof. We construct an empirical estimator for 𝜎²_𝐹 by replacing every 𝐹 on the right-hand side of (2.6) by the empirical 𝐹_𝑛. Consequently, we replace the function 𝑤_𝐹(𝑡) by its empirical version 𝑤_𝑋,𝑛(𝑡)=−^𝑡₀(1𝑝−1)AL_𝑛(𝑝)(𝑋−AL_𝑛(𝑝))²𝑑𝑝+¹_𝑡(1𝑝1−1)𝑋−AL_𝑛(𝑝)𝑑𝑝. (7.7) We denote the resulting estimator of 𝜎²_𝐹 by 𝑆²_𝑋,𝑛. The rest of the proof consists of verifying that this estimator coincides with the one defined by (2.8). 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 𝑆²_𝑋,𝑛 is equal to ^{𝑋𝑛∶𝑛}_𝑋1∶𝑛^{𝑋𝑛∶𝑛}_𝑋1∶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. (7.8) Since 𝐹_𝑛(𝑥)=𝑘/𝑛 when 𝑥∈[𝑋_𝑘∶𝑛,𝑋_{𝑘+1∶𝑛}), we therefore have that 𝑆²_𝑋,𝑛=^𝑛−1_𝑘=1^𝑛−1_𝑙=1(min{𝑘,𝑙}𝑛−𝑘𝑛𝑙𝑛)×𝑤_𝑋,𝑛(𝑘𝑛)𝑤_𝑋,𝑛(𝑙𝑛)(𝑋_{𝑘+1∶𝑛}−𝑋_𝑘∶𝑛)(𝑋_{𝑙+1∶𝑛}−𝑋_𝑙∶𝑛). (7.9) Furthermore, 𝑤_𝑋,𝑛(𝑘𝑛)=−₀^𝑘/𝑛(1𝑝−1)AL_𝑛(𝑝)(𝑋−AL_𝑛(𝑝))²𝑑𝑝+¹_𝑘/𝑛(1𝑝1−1)𝑋−AL_𝑛(𝑝)𝑑𝑝=−^𝑘_𝑖=1𝐼_𝑋,𝑛(𝑖)+^𝑛_𝑖=𝑘+1𝐽_𝑋,𝑛(𝑖), (7.10) where, using notations (6.6) and (6.8), the summands on the right-hand side of (7.10) are 𝐼_𝑋,𝑛1(𝑖)=𝑋_𝑖∶𝑛^𝑖/𝑛_{(𝑖−1)/𝑛}(1𝑝Λ−1)^𝑖,𝑛+𝑝(Ψ_𝑖,𝑛−𝑝)²𝑑𝑝 (7.11) for all 𝑖=1,…,𝑛−1, and 𝐽_𝑋,𝑛1(𝑖)=𝑋_𝑖∶𝑛^𝑖/𝑛_{(𝑖−1)/𝑛}(1𝑝1−1)Ψ_𝑖,𝑛−𝑝𝑑𝑝 (7.12) for all 𝑖=2,…,𝑛. When 𝑖=1, then Λ_𝑖,𝑛=0. Hence, we immediately arrive at the expression for 𝐼_𝑋,𝑛(1) given by (2.10). When 2≤𝑖≤𝑛−1, then 𝐼_𝑋,𝑛Λ(𝑖)=^𝑖,𝑛𝑋_𝑖∶𝑛Ψ²_𝑖,𝑛𝑖log(𝑖−1)−(Λ_𝑖,𝑛+Ψ_𝑖,𝑛)(Ψ_𝑖,𝑛−1)𝑛𝑋_𝑖∶𝑛Ψ_𝑖,𝑛(Ψ_𝑖,𝑛−(𝑖−1)/𝑛)(Ψ_𝑖,𝑛+1−𝑖/𝑛)𝑋_𝑖∶𝑛Λ(1+^𝑖,𝑛Ψ²_𝑖,𝑛Ψ)log(^𝑖,𝑛−(𝑖−1)/𝑛Ψ_𝑖,𝑛−𝑖/𝑛), (7.13) and, after some algebra, we arrive at the right-hand side of (2.11). When 2≤𝑖≤𝑛−1, then we have the expression 𝐽_𝑋,𝑛1(𝑖)=𝑋_𝑖∶𝑛Ψ_𝑖,𝑛𝑖log(1𝑖−1)−𝑋_𝑖∶𝑛1(1−Ψ_𝑖,𝑛Ψ)log(^𝑖,𝑛−(𝑖−1)/𝑛Ψ_𝑖,𝑛−𝑖/𝑛), (7.14) which, after some algebra, becomes the expression recorded in (2.12). When 𝑖=𝑛, then Ψ_𝑖,𝑛=1, and so we see that 𝐽_𝑋,𝑛(𝑛) is given by (2.13). This completes the proof of Theorem 7.2.

Theorem 7.3. The empirical mixed moment 𝑆_{𝑋,𝑌,𝑛} is an estimator of 𝐄[ℎ(𝑋)ℎ(𝑌)].

Proof. We proceed similarly to the proof of Theorem 7.2. We estimate the integrand 𝐏[𝑋≤𝑥,𝑌≤𝑦]−𝐹(𝑥)𝐻(𝑦) using 1𝑛^𝑛_𝑖=1𝟏{𝑋_𝑖≤𝑥,𝑌_𝑖1≤𝑦}−𝑛^𝑛_𝑖=1𝟏{𝑋_𝑖1≤𝑥}𝑛^𝑛_𝑖=1𝟏{𝑌_𝑖≤𝑦}. (7.15) After some rearrangement of terms, estimator (7.15) becomes 1𝑛^𝑛_𝑖=1𝟏{𝑋_𝑖∶𝑛≤𝑥,𝑌_(𝑖,𝑛)1≤𝑦}−𝑛^𝑛_𝑖=1𝟏{𝑋_𝑖∶𝑛1≤𝑥}𝑛^𝑛_𝑖=1𝟏{𝑌_𝑖∶𝑛≤𝑦}. (7.16) When 𝑥∈[𝑋_𝑘∶𝑛,𝑋_{𝑘+1∶𝑛}) and 𝑦∈[𝑌_𝑙∶𝑛,𝑌_{𝑙+1∶𝑛}), then estimator (7.16) is equal to 𝑛⁻¹∑^𝑘_𝑖=1𝟏{𝑌_(𝑖,𝑛)≤𝑌_𝑙∶𝑛}−(𝑘/𝑛)(𝑙/𝑛), which leads us to the estimator 𝑆_{𝑋,𝑌,𝑛}. This completes the proof of Theorem 7.3.

8. Proof of Theorem 2.1

Throughout the proof we use the notation AL^∗_𝐹(𝑝) for the dual absolute Lorenz curve ∫¹_𝑝𝐹⁻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.

Proof. Simple algebra gives the equations √𝑛(𝑍_𝑛−𝑍_𝐹)=−√𝑛¹₀(1𝑝−1)(AL_𝑛(𝑝)AL^∗_𝑛−(𝑝)AL_𝐹(𝑝)AL^∗_𝐹√(𝑝))𝑑𝑝=−𝑛¹₀(1𝑝−1)AL_𝑛(𝑝)−AL_𝐹(𝑝)AL^∗_𝐹+√(𝑝)𝑑𝑝𝑛¹₀(1𝑝−1)AL_𝐹(𝑝)AL_𝐹^∗2(𝑝)(AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝))𝑑𝑝+𝑂_𝐏(𝑟_𝑛,1)+𝑂_𝐏(𝑟_𝑛,2) (8.1) with the remainder terms 𝑟_𝑛,1=√𝑛¹₀(1𝑝−1)(AL_𝑛(𝑝)−AL_𝐹1(𝑝))(AL^∗_𝑛−1(𝑝)AL^∗_𝐹𝑟(𝑝))𝑑𝑝,_𝑛,2=√𝑛¹₀(1𝑝−1)AL_𝐹(𝑝)AL^∗_𝐹(𝑝)(AL^∗_𝑛(𝑝)−AL^∗_𝐹1(𝑝))(AL^∗_𝑛−1(𝑝)AL^∗_𝐹(𝑝))𝑑𝑝. (8.2) We will later show (Lemmas 9.1 and 9.2) that the remainder terms 𝑟_𝑛,1 and 𝑟_𝑛,2 are of the order 𝑜_𝐏(1). Hence, we now proceed with our analysis of the first two terms on the right-hand side of (8.1), for which we use the (general) Vervaat process 𝑉_𝑛(𝑝)=^𝑝₀(𝐹_𝑛⁻¹(𝑡)−𝐹⁻¹(𝑡))𝑑𝑡+^𝐹−1₀^(𝑝)(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥 (8.3) and its dual version 𝑉^∗_𝑛(𝑝)=¹_𝑝(𝐹_𝑛⁻¹(𝑡)−𝐹⁻¹(𝑡))𝑑𝑡+^∞_{𝐹−1(𝑝)}(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥. (8.4) For mathematical and historical details on the Vervaat process, see Zitikis [23], Davydov and Zitikis [24], Greselin et al. [25], and references therein. Since ∫¹₀(𝐹_𝑛⁻¹(𝑡)−𝐹⁻¹(𝑡))𝑑𝑡=𝑋−𝜇_𝐹 and ∫^∞₀(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥=−(𝑋−𝜇_𝐹), adding the right-hand sides of (8.3) and (8.4) gives the equation 𝑉^∗_𝑛(𝑝)=−𝑉_𝑛(𝑝). Hence, whatever upper bound we have for |𝑉_𝑛(𝑝)|, the same bound holds for |𝑉^∗_𝑛(𝑝)|. In fact, the absolute value can be dropped from |𝑉_𝑛(𝑝)| since 𝑉_𝑛(𝑝) is always nonnegative. Furthermore, we know that 𝑉_𝑛(𝑝) does not exceed (𝑝−𝐹_𝑛(𝐹⁻¹(𝑝)))(𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)). Hence, with the notation 𝑒_𝑛(𝑝)=√𝑛(𝐹_𝑛(𝐹⁻¹(𝑝))−𝑝), which is the uniform on [0,1] empirical process, we have that √𝑛𝑉_𝑛(𝑝)≤|𝑒_𝑛(𝑝)||𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|. (8.5) Bound (8.5) implies the following asymptotic representation for the first term on the right-hand side of (8.1): −√𝑛¹₀(1𝑝−1)AL_𝑛(𝑝)−AL_𝐹(𝑝)AL^∗_𝐹=√(𝑝)𝑑𝑝𝑛¹₀(1𝑝1−1)AL^∗_𝐹((𝑝)^𝐹−1₀^(𝑝)(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥)𝑑𝑝+𝑂_𝐏(𝑟_𝑛,3), (8.6) where 𝑟_𝑛,3=¹₀(1𝑝1−1)AL^∗_𝐹(𝑝)|𝑒_𝑛(𝑝)||𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝. (8.7) We will later show (Lemma 9.3) that 𝑟_𝑛,3=𝑜_𝐏(1). Furthermore, we have the following asymptotic representation for the second term on the right-hand side of (8.1): √𝑛¹₀(1𝑝−1)AL_𝐹(𝑝)AL_𝐹^∗2(𝑝)(AL^∗_𝑛(𝑝)−AL^∗_𝐹√(𝑝))𝑑𝑝=−𝑛¹₀(1𝑝−1)AL_𝐹(𝑝)AL_𝐹^∗2(𝑝)(^∞_{𝐹−1(𝑝)}(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥)𝑑𝑝+𝑂_𝐏(𝑟_𝑛,4), (8.8) where 𝑟_𝑛,4=¹₀(1𝑝−1)AL_𝐹(𝑝)AL_𝐹^∗2(𝑝)|𝑒_𝑛(𝑝)||𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝. (8.9) We will later show (Lemma 9.4) that 𝑟_𝑛,4=𝑜_𝐏(1). Hence, (8.1), (8.6) and (8.8) together with the aforementioned statements that 𝑟_𝑛,1,…,𝑟_𝑛,4 are of the order 𝑜_𝐏(1) imply that √𝑛(𝑍_𝑛−𝑍_𝐹)=√𝑛¹₀(1𝑝1−1)AL^∗_𝐹((𝑝)^𝐹−1₀^(𝑝)(𝐹_𝑛−√(𝑥)−𝐹(𝑥))𝑑𝑥)𝑑𝑝𝑛¹₀(1𝑝−1)AL_𝐹(𝑝)AL_𝐹^∗2((𝑝)^∞_𝐹⁻¹_(𝑝)(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥)𝑑𝑝+𝑜_𝐏=1(1)√𝑛^𝑛_𝑖=1ℎ(𝑋_𝑖)+𝑜_𝐏(1). (8.10) This completes the proof of Theorem 2.1.

9. Negligibility of Remainder Terms

The following four lemmas establish the above 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 will use a parameter 𝛿∈(0,1/2], possibly different from line to line but never depending on 𝑛. Furthermore, we will frequently use the fact that

𝐄[𝑋^𝑞]<∞implies¹₀|𝐹_𝑛⁻¹(𝑡)−𝐹⁻¹(𝑡)|^𝑞𝑑𝑡=𝑜_𝐏(1). (9.1) Another technical result that we will frequently use is the fact that, for any 𝜀>0 as small as desired,

sup_𝑥∈𝐑√𝑛|𝐹_𝑛(𝑥)−𝐹(𝑥)|𝐹(𝑥)^1/2−𝜀(1−𝐹(𝑥))^1/2−𝜀=𝑂_𝐏(1) (9.2) when 𝑛→∞.

Lemma 9.1. Under the conditions of Theorem 2.1, 𝑟_𝑛,1=𝑜_𝐏(1).

Proof. We split the remainder term 𝑟_𝑛,1=√𝑛∫¹₀⋯𝑑𝑝 into the sum of 𝑟^∗_𝑛,1(𝛿)=√𝑛∫₀^1−𝛿⋯𝑑𝑝 and 𝑟^∗∗_𝑛,1(𝛿)=√𝑛∫¹_1−𝛿⋯𝑑𝑝. The lemma follows if (1)for every 𝛿>0, the statement 𝑟^∗_𝑛,1(𝛿)=𝑜_𝐏(1) holds when 𝑛→∞, (2)𝑟^∗∗_𝑛,1(𝛿)=ℎ(𝛿)𝑂_𝐏(1) for a deterministic ℎ(𝛿)↓0 when 𝛿↓0, where 𝑂_𝐏(1) does not depend on 𝛿.
To prove part (1), we first note that when 0<𝑝<1−𝛿, then AL^∗_𝐹(𝑝)≥∫¹_1−𝛿𝐹⁻¹(𝑡)𝑑𝑡, which is positive, and AL^∗_𝑛(𝑝)≥∫¹_1−𝛿𝐹⁻¹(𝑡)𝑑𝑡+𝑜_𝐏(1) due to statement (9.1) with 𝑞=1. Hence, we are left to show that, when 𝑛→∞,
√𝑛₀^1−𝛿1𝑝|AL_𝑛(𝑝)−AL_𝐹(𝑝)||AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝)|𝑑𝑝=𝑜_𝐏(1). (9.3) Since AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝)=(𝑋−𝜇_𝐹)−(AL_𝑛(𝑝)−AL_𝐹(𝑝)), statement (9.3) follows if √𝑛|𝑋−𝜇_𝐹|₀^1−𝛿1𝑝|AL_𝑛(𝑝)−AL_𝐹(𝑝)|𝑑𝑝=𝑜_𝐏√(1),𝑛₀^1−𝛿1𝑝|AL_𝑛(𝑝)−AL_𝐹(𝑝)|²𝑑𝑝=𝑜_𝐏(1). (9.4)(9.5) We have √𝑛|𝑋−𝜇_𝐹|=𝑂_𝐏(1) and |AL_𝑛(𝑝)−AL_𝐹(𝑝)|≤√𝑝(∫¹₀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|²𝑑𝑝)^1/2. Since ∫¹₀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|²𝑑𝑝=𝑜_𝐏(1) and ∫₀^1−𝛿𝑝⁻¹√𝑝𝑑𝑝<∞, we have statement (9.4). To prove statement (9.5), we use bound (8.5) and reduce the proof to showing that 1√𝑛₀^1−𝛿1𝑝|^𝐹−1₀^(𝑝)√𝑛(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥|²𝑑𝑝=𝑜_𝐏1(1),√𝑛₀^1−𝛿1𝑝|𝑒_𝑛(𝑝)|²|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|²𝑑𝑝=𝑜_𝐏(1). (9.6)(9.7) To prove statement (9.6), we use statement (9.2) and observe that ₀^1−𝛿1𝑝(^𝐹−1₀^(𝑝)𝐹(𝑥)^1/2−𝜀𝑑𝑥)²𝑑𝑝≤𝑐(𝐹,𝛿)₀^1−𝛿1𝑝𝑝^1−2𝜀𝑑𝑝<∞. (9.8) To prove statement (9.7), we use the uniform on [0,1] version of statement (9.2) and Hölder's inequality, and in this way reduce the proof to showing that 1√𝑛(₀^1−𝛿1𝑝^2𝜀𝑎𝑑𝑝)^1/𝑎(₀^1−𝛿|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|^2𝑏𝑑𝑝)^1/𝑏=𝑜_𝐏(1) (9.9) for some 𝑎,𝑏>1 such that 𝑎⁻¹+𝑏⁻¹=1. We choose the parameters 𝑎 and 𝑏 as follows. First, since 𝐄[𝑋^2+𝛼]<∞, we set 𝑏=(2+𝛼)/2. Next, we choose 𝜀>0 on the left-hand side of statement (9.9) so that 2𝜀𝑎<1, which holds when 𝜀<𝛼/(4+2𝛼) in view of the equation 𝑎⁻¹+𝑏⁻¹=1. Hence, statement (9.9) holds and thus statement (9.7) follows. This completes the proof of part (1).
To establish part (2), we first estimate |𝑟^∗∗_𝑛,1(𝛿)| from above using the bounds AL^∗_𝐹(𝑝)≥(1−𝑝)𝐹⁻¹(1/2) and AL^∗_𝑛(𝑝)≥(1−𝑝)𝐹_𝑛⁻¹(1/2), which hold since 𝛿≤1/2. Hence, we have reduced our task to verifying the statement √𝑛∫¹_1−𝛿|AL_𝑛(𝑝)−AL_𝐹(𝑝)|𝑑𝑝=ℎ(𝛿)𝑂_𝐏(1). Using the Vervaat process 𝑉_𝑛(𝑝) and bound (8.5), we reduce the proof of the statement to showing that the integrals
¹_1−𝛿(^𝐹−1₀^(𝑝)√𝑛|𝐹_𝑛(𝑥)−𝐹(𝑥)|𝑑𝑥)𝑑𝑝,_11−𝛿|𝑒_𝑛(𝑝)||𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝 (9.10)(9.11) are of the order ℎ(𝛿)𝑂_𝐏(1) with possibly different ℎ(𝛿)↓0 in each case. In view of statement (9.2), we have the desired statement for integral (9.10) if the quantity ¹_1−𝛿(^𝐹−1₀^(𝑝)(1−𝐹(𝑥))^1/2−𝜀𝑑𝑥)𝑑𝑝 (9.12) converges to 0 when 𝛿↓0, in which case we use it as ℎ(𝛿). The inner integral of (9.12) does not exceed ∫^∞₀(1−𝐹(𝑥))^1/2−𝜀𝑑𝑥, which is finite for all sufficiently small 𝜀>0 since 𝐄[𝑋^2+𝛼]<∞ for some 𝛼>0. This completes the proof that quantity (9.10) is of the order ℎ(𝛿)𝑂_𝐏(1). To show that quantity (9.11) is of a similar order, we use the uniform on [0,1] version of statement (9.2) and reduce the task to showing that ∫¹_1−𝛿|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝 is of the order ℎ(𝛿)𝑂_𝐏(1). By the Cauchy-Bunyakowski-Schwarz inequality, we have that ¹_1−𝛿|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝≤√𝛿(¹₀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|²𝑑𝑝)^1/2. (9.13) Since 𝐄[𝑋²]<∞, we have ∫¹₀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|²𝑑𝑝=𝑜_𝐏(1), and so setting ℎ(𝛿)=√𝛿 establishes the desired asymptotic result for integral (9.11). This also completes the proof of part (2), and also of Lemma 9.1.

Lemma 9.2. Under the conditions of Theorem 2.1, 𝑟_𝑛,2=𝑜_𝐏(1).

Proof. Like in the proof of Lemma 9.1, we split the remainder term 𝑟_𝑛,2=√𝑛∫¹₀⋯𝑑𝑝 into the sum of 𝑟^∗_𝑛,2(𝛿)=√𝑛∫₀^1−𝛿⋯𝑑𝑝 and 𝑟^∗∗_𝑛,2(𝛿)=√𝑛∫¹_1−𝛿⋯𝑑𝑝. To prove the lemma, we need to show the following. (1)For every 𝛿>0, the statement 𝑟^∗_𝑛,2(𝛿)=𝑜_𝐏(1) holds when 𝑛→∞. (2)𝑟^∗∗_𝑛,2(𝛿)=ℎ(𝛿)𝑂_𝐏(1) for a deterministic ℎ(𝛿)↓0 when 𝛿↓0, where 𝑂_𝐏(1) does not depend on 𝛿.
To prove part (1), we first estimate |𝑟^∗_𝑛,2(𝛿)| from above using the bounds 𝑝⁻¹AL_𝐹(𝑝)≤𝐹⁻¹(1−𝛿)<∞, AL^∗_𝐹(𝑝)≥∫¹_1−𝛿𝐹⁻¹(𝑡)𝑑𝑡>0, and AL^∗_𝑛(𝑝)≥∫¹_1−𝛿𝐹⁻¹(𝑡)𝑑𝑡+𝑜_𝐏(1). This reduces our task to showing that, for every 𝛿>0,
√𝑛₀^1−𝛿|AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝)|²𝑑𝑝=𝑜_𝐏(1). (9.14) Since AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝)=(𝑋−𝜇_𝐹)−(AL_𝑛(𝑝)−AL_𝐹(𝑝)) and √𝑛(𝑋−𝜇_𝐹)²=𝑜_𝐏(1), statement (9.14) follows from √𝑛₀^1−𝛿|AL_𝑛(𝑝)−AL_𝐹(𝑝)|²𝑑𝑝=𝑜_𝐏(1), (9.15) which is an elementary consequence of statement (9.5). This establishes part (1).
To prove part (2), we first estimate |𝑟^∗∗_𝑛,2(𝛿)| from above using the bounds AL^∗_𝐹(𝑝)≥(1−𝑝)𝐹⁻¹(1/2) and AL^∗_𝑛(𝑝)≥(1−𝑝)𝐹_𝑛⁻¹(1/2), and in this way reduce the task to showing that
√𝑛¹_1−𝛿11−𝑝|AL^∗_𝑛(𝑝)−AL^∗_𝐹(𝑝)|𝑑𝑝=ℎ(𝛿)𝑂_𝐏(1). (9.16) Using the Vervaat process, statement (9.16) follows if ¹_1−𝛿11−𝑝(^∞_{𝐹−1(𝑝)}√𝑛|𝐹_𝑛(𝑥)−𝐹(𝑥)|𝑑𝑥)𝑑𝑝=ℎ(𝛿)𝑂_𝐏(1),_11−𝛿11−𝑝|𝑒_𝑛(𝑝)||𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝=ℎ(𝛿)𝑂_𝐏(1) (9.17)(9.18) with possibly different ℎ(𝛿)↓0 in each case. Using statement (9.2), we have that statement (9.17) holds with ℎ(𝛿) defined as the integral ¹_1−𝛿11−𝑝(^∞_{𝐹−1(𝑝)}(1−𝐹(𝑥))^1/2−𝜀𝑑𝑥)𝑑𝑝, (9.19) which converges to 0 when 𝛿↓0 as the following argument shows. First, we write the integrand as the product of (1−𝐹(𝑥))^𝜀 and (1−𝐹(𝑥))^1/2−2𝜀. Then we estimate the first factor by (1−𝑝)^𝜀. The integral ∫^∞₀(1−𝐹(𝑥))^1/2−2𝜀𝑑𝑥 is finite for all sufficiently small 𝜀>0 since 𝐄[𝑋^2+𝛼]<∞ for some 𝛼>0. Since ∫¹_1−𝛿(1−𝑝)^−1+𝜀𝑑𝑝↓0 when 𝛿↓0, integral (9.19) converges to 0 when 𝛿↓0. The proof of statement (9.17) is finished.
We are left to prove statement (9.18). Using the uniform on [0,1] version of statement (9.2), we reduce the task to showing that
¹_1−𝛿1(1−𝑝)^1/2+𝜀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝=ℎ(𝛿)𝑂_𝐏(1). (9.20) In fact, we will see below that 𝑂_𝐏(1) can be replaced by 𝑜_𝐏(1). Using Hölder's inequality, we have that the right-hand side of (9.20) does not exceed (¹_1−𝛿1(1−𝑝)^{(1/2+𝜀)𝑎}𝑑𝑝)^1/𝑎(¹_1−𝛿|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|^𝑏𝑑𝑝)^1/𝑏 (9.21) for some 𝑎,𝑏>1 such that 𝑎⁻¹+𝑏⁻¹=1. We choose the parameters 𝑎 and 𝑏 as follows. Since 𝐄[𝑋^2+𝛼]<∞, we set 𝑏=2+𝛼, and so the right-most integral of (9.21) is of the order 𝑜_𝐏(1). Furthermore, 𝑎=(2+𝛼)/(1+𝛼)<2, which can be made arbitrarily close to 2 by choosing sufficiently small 𝛼>0. Choosing 𝜀>0 so small that (1/2+𝜀)𝑎<1, we have that the left-most integral in (9.21) converges to 0 when 𝛿↓0. This establishes statement (9.18) and completes the proof of Lemma 9.2.

Lemma 9.3. Under the conditions of Theorem 2.1, 𝑟_𝑛,3=𝑜_𝐏(1).

Proof. We split the remainder term 𝑟_𝑛,3=∫¹₀⋯𝑑𝑝 into the sum of 𝑟^∗_𝑛,3=∫₀^1/2⋯𝑑𝑝 and 𝑟^∗∗_𝑛,3=∫¹_1/2⋯𝑑𝑝. The lemma follows if the two summands are of the order 𝑜_𝐏(1).
To prove 𝑟^∗_𝑛,3=𝑜_𝐏(1), we use the bound AL^∗_𝐹(𝑝)≥∫¹_1/2𝐹⁻¹(𝑝)𝑑𝑝 and the uniform on [0,1] version of statement (9.2), and in this way reduce our task to showing that
₀^1/21𝑝^1/2+𝜀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝=𝑜_𝐏(1). (9.22) This statement can be established following the proof of statement (9.20), with minor modifications.
To prove 𝑟^∗∗_𝑛,3=𝑜_𝐏(1), we use the bound AL^∗_𝐹(𝑝)≥(1−𝑝)𝐹⁻¹(1/2), the fact that sup_𝑡|𝑒_𝑛(𝑡)|=𝑂_𝐏(1), and statement (9.1) with 𝑞=1. The desired result for 𝑟^∗∗_𝑛,3 follows, which finishes the proof of Lemma 9.3.

Lemma 9.4. Under the conditions of Theorem 2.1, 𝑟_𝑛,4=𝑜_𝐏(1).

Proof. We split 𝑟_𝑛,4=∫¹₀⋯𝑑𝑝 into the sum of 𝑟^∗_𝑛,4=∫₀^1/2⋯𝑑𝑝 and 𝑟^∗∗_𝑛,4=∫¹_1/2⋯𝑑𝑝, and then show that the two summands are of the order 𝑜_𝐏(1).
To prove 𝑟^∗_𝑛,4=𝑜_𝐏(1), we use the bounds 𝑝⁻¹AL_𝐹(𝑝)≤𝐹⁻¹(1/2)<∞ and AL^∗_𝐹(𝑝)≥∫¹_1/2𝐹⁻¹(𝑝)𝑑𝑝>0 together with the uniform on [0,1] version of statement (9.2). This reduces our task to showing that ∫₀^1/2|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝=𝑜_𝐏(1), which holds due to statement (9.1) with 𝑞=1.
To prove 𝑟^∗∗_𝑛,4=𝑜_𝐏(1), we use the bound AL^∗_𝐹(𝑝)≥(1−𝑝)𝐹⁻¹(1/2) and the uniform on [0,1] version of statement (9.2), and in this way reduce the proof to showing that
¹_1/21(1−𝑝)^1/2+𝜀|𝐹_𝑛⁻¹(𝑝)−𝐹⁻¹(𝑝)|𝑑𝑝=𝑜_𝐏(1). (9.23) This statement can be established following the proof of statement (9.20). The proof of Lemma 9.4 is finished.

Acknowledgments

The authors are indebted to two anonymous referees and the editor in charge of the manuscript, Madan L. Puri, for their constructive criticism and suggestions that helped them to improve the paper. The research has been partially supported by the 2009 F.A.R. (Fondo di Ateneo per la Ricerca) at the University of Milan Bicocca, and the Natural Sciences and Engineering Research Council (NSERC) of Canada.