Estimation of conditional inequality curves and measures via estimating the conditional quantile function

TL;DR

The paper extends inequality analysis to conditional settings by defining the curves $q_Z$ and $q_D$ and their integrated measures $qZI$ and $qDI$, conditional on covariates through the quantile function $Q_x$. It develops a family of estimators—notably IOQR and IAQR—that enforce noncrossing via isotonic regression and introduces a smooth approximated QR (AQR/IAQR) to improve numerical performance, while comparing with constrained and stepwise noncrossing methods. The authors prove consistency of plug-in estimators for $Q_x$, $q_Z$, and $q_{ZI}$ (and their $q_D$ counterparts) under standard regularity conditions and demonstrate finite-sample behavior through FLD-based simulations, showing competitive or superior performance of the noncrossing approaches, particularly for small samples. A real-data application to salary inequality across age/experience demonstrates the practical utility of conditional inequality measures and their sensitivity to the chosen measure, with code available for replication on GitHub.

Abstract

The classical concept of inequality curves and measures is extended to conditional inequality curves and measures and a curve of conditional inequality measures is introduced. This extension provides a more nuanced analysis of inequality in relation to covariates. In particular, this enables comparison of inequalities between subpopulations, conditioned on certain values of covariates. To estimate the curves and measures, a novel method for estimating the conditional quantile function is proposed. The method incorporates a modified quantile regression framework that employs isotonic regression to ensure that there is no quantile crossing. The consistency of the proposed estimators is proved while their finite sample performance is evaluated through simulation studies and compared with existing quantile regression approaches. Finally, practical application is demonstrated by analysing salary inequality across different employee age groups, highlighting the potential of conditional inequality measures in empirical research. The code used to prepare the results presented in this article is available in a dedicated GitHub repository.

Figures (11)

• Figure 1: Isotonic regression on $\hat{\beta_0}(p)$ and $\hat{\beta_1}(p)$ compared to the true value of ${\beta_0}(p)$ (left) and ${\beta_1}(p)$ (right) for sample of data generated from flattened logistic distribution $FLD(2, 3, \kappa)$ with $\kappa\in[0,10]$ (see Section \ref{['sec:fld']}). Estimation based on 33 equidistant orders of quantiles
• Figure 2: Scatterplot of data generated from FLD$(0.5, 0.2, 0.3 x)$, where $x\in(0,30)$
• Figure 3: Scatterplot of the exponent of the data drawn from FLD depicted in Figure \ref{['fig:scatter_plot_fld']}
• Figure 4: True values of the indices $qZI$ and $qDI$ of $EFLD$, for nine different sets of parameters and $x$ varying between 1 and 30
• Figure 5: Ratio of MSE of the plug-in estimator of the conditional inequality measures of samples drawn from $EFLD$ for samples of size 50, 100, 500 and 1000 and for nine combinations of parameters of $EFLD$), obtained with certain method, compared to the lowest MSE in each case. The order of the sets of parameters is the same as in Figure \ref{['fig:qzi_qdi_true_values']}

Theorems & Definitions (13)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5