On the estimation of the Lorenz curve under complex sampling designs

Abstract

This paper focuses on the estimation of the concentration curve of a finite population, when data are collected according to a complex sampling design with different inclusion probabilities. A (design-based) Hájek type estimator for the Lorenz curve is proposed, and its asymptotic properties are studied. Then, a resampling scheme able to approximate the asymptotic law of the Lorenz curve estimator is constructed. Applications are given to the construction of (i) a confidence band for the Lorenz curve, (ii) confidence intervals for the Gini concentration ratio, and (iii) a test for Lorenz dominance. The merits of the proposed resampling procedure are evaluated through a simulation study.

Appendix

Proof of Proposition 2

Suppose that $y < t$. From (25) it is not difficult to see that

$$\begin{aligned} E[ ( {\mathcal {W}}^H (t) - {\mathcal {W}}^H (y) )^2 ]= & {} C_1 (y, \, y) + C_1 (t, \, t) - 2 C_1 (t, \, y) \nonumber \\&+ f ( C_2 (y, \, y) + C_2 (t, \, t) - 2 C_2 (t, \, y) ) \nonumber \\= & {} {\mathbb {E}} [ X_1 ] \left( T_{-1} (t) - T_{-1} (y) \right) + f \left( F(t) - F(y) \right) \nonumber \\&- \frac{f^{3}}{d} \left\{ \left( \frac{T_{1} (y)}{{\mathbb {E}} [ X_{1} ]} - F(y) \right) \left( \frac{T_{1} (t)}{{\mathbb {E}} [ X_{1} ]} - F(t) \right) \right\} ^2 \nonumber \\&- 2 {\mathbb {E}} [ X_1 ] ( T_{-1} (y) - T_{-1} (y) ) ( F(y) - F(t)) \nonumber \\&+ 2 {\mathbb {E}} [ X_1 ] \left( {\mathbb {E}} [ X_1^{-1} ] +1 \right) ( F(t) - F( y) )^2 \nonumber \\&+ f \{ (F(t) - F(y) ) - (F(t) - F(y) )^2 \} . \end{aligned}$$

(71)

Assumption C1 implies that

$$\begin{aligned} \left| T_{\alpha } (t) - T_{\alpha } (y) \right| \le M_{\alpha } \left| F (t) - F(y) \right| \end{aligned}$$

so that from (71) it is not difficult to see that

$$\begin{aligned} E[ ( {\mathcal {W}}^H (t) - {\mathcal {W}}^H (y) )^2 ] \le C \left| F(t) - F(y) \right| . \end{aligned}$$

(72)

C being an appropriate constant. Inequality (72) also holds when $y>t$. Hence, in terms of the process ${\mathcal {B}}^H$ introduced above we may write

$$\begin{aligned} {\mathbb {E}} \left[ \left( {\mathcal {B}}^H (t) - {\mathcal {B}}^H (y) \right) ^2 \right] \le C \vert t-y \vert \quad \forall \, y, \, t \in [0, \, 1] . \end{aligned}$$

(73)

Inequality (73) and the Gaussianity of ${\mathcal {B}}^H (t) - {\mathcal {B}}^H (y)$, in their turn, imply that

$$\begin{aligned} {\mathbb {E}} \left[ \left( {\mathcal {B}}^H (t) - {\mathcal {B}}^H (y) \right) ^2 \right] \le \frac{C}{\left| \log ( t-y ) \right| ^{\beta }} \quad \forall \, \beta > 1 , \quad \forall \, y, \, t \in [0, \, 1] . \end{aligned}$$

(74)

Observing that $P ( {\mathcal {B}}^H (0) =0) = P ({\mathcal {B}}^H (1) =0) =1$, Proposition 2 now follows from (74) and Leadbetter and Weissner (1969). $\square$

Proof of Proposition 6

Let

$$\begin{aligned} R^{*}_{n} (z) = P_{P^{*}} \left( \left. Z^{*}_{n,m} \le z \, \right| {\varvec{Y}}_N, \, {\varvec{X}}_N, \, {\varvec{D}}_N, \, N^*_1 , \ldots N^*_M \right) \end{aligned}$$

be the (resampling) d.f. of $Z^{*}_{n,m}$ (48). By Dvoretzky–Kiefer–Wolfowitz inequality (cfr. Massart 1990), we have first

$$\begin{aligned} P \left( \left. \sup _{z} \left| {\widehat{R}}^{*}_{n,M} (z) - R^{*}_{n} (z) \right| > \epsilon \, \right| {\varvec{Y}}_N, \, {\varvec{X}}_N, \, {\varvec{D}}_N, \, N^*_1 , \ldots N^*_M \right) \le 2 \exp \left\{ -2 M \epsilon ^2 \right\} . \nonumber \\ \end{aligned}$$

(75)

Using the Borel–Cantelli first lemma, and taking into account that $R^{*}_{n} (z)$ converges uniformly to $P ( \sup _p \vert {\mathcal {L}}^H (p) \vert \le z )$, (51) immediately follows. Statement (52) follows from (51) and the absolute continuity of the distribution of $\sup _p \vert {\mathcal {L}}^H (p) \vert$ (cfr. Lifshits 1982). $\square$

Proof of Proposition 7

Proof of (57) and (58) is similar to Proposition 6. As far as (59) is concerned, it is a consequence of Th. 2.5.5. in Sen and Singer (1993) (pp. 90–91). $\square$