Robust, distribution-free inference for income share ratios under complex sampling

Abstract

The quintile share ratio of disposable income is the primary inequality indicator of the European Union. As an inequality indicator, it must be sensitive to extreme large observations. Therefore, outliers have a strong impact on the bias and the variance of the classical quintile share ratio estimator. This may mislead the interpretation of income inequality. A class of estimators which are robust against outliers is introduced. They have a bounded influence function, they may reduce the bias incurred by the robustification and they reduce variability. Based on an asymptotic framework which respects the design-based, non-parametric approach, inference for these robust estimators is developed. A large simulation study with close to reality universes derived from the Statistics of Living Conditions Surveys of the EU allows to study the performance of the proposed estimators.

Appendix: Proofs

Proof

(Lemma 1) Let $F \in \mathcal F$, where $\mathcal F$ is the set of cumulative distribution functions, and denote by $F_{\varepsilon }(y)$ the mixture distribution $F_{\varepsilon }(y)=(1-\varepsilon )F(y) + \varepsilon G(y)$, where $G(y)=1\!\!1\{ y \ge z\}$ is an elementary (degenerate) cdf. For $\beta \in \mathbb Q _1$ such that $\beta$ is not a limit point of $F^{-1}$, the influence function writes $IF(z, QSM(\cdot ;\beta ), F) = \partial / \partial \varepsilon \left[ \beta ^{-1}\int ^{\xi _{\beta }(F_{\varepsilon })} y \mathrm d F_{\varepsilon }(y)\right]$ for $\varepsilon = 0$. Thus, differentiating w.r.t. $\varepsilon$ (by means of the Leibniz integration rule) and taking $\varepsilon \downarrow 0$, yields $IF(z, QSM(\cdot ;\beta ), F) = \beta ^{-1} \int ^{\xi _{\beta }(F)} y \mathrm d G(y) - \beta ^{-1} \int ^{\xi _{\beta }(F)} y \mathrm d F(y) + \beta ^{-1}\xi _{\beta }(F)f(\xi _{\beta }(F)) \cdot \left[ \mathrm d \xi _{\beta }(F_{\varepsilon })/\mathrm d \varepsilon \right] _{\varepsilon = 0}$. Note that $[\mathrm d \xi _{\beta }(F_{\varepsilon })/ \mathrm d \varepsilon ]_{\varepsilon = 0}$ is the influence function of the $\beta$th quantile functional (see e.g., Huber 1981, 56–57) and defined as $IF(z, \xi _{\beta }(\cdot ), F) = [\beta - 1\!\!1\{\xi _{\beta }(F) \ge z \}][f(\xi _{\beta }(F))]^{-1}$. Assembling all terms, $f$ cancels out and we get the influence function which completes the proof. $\square$

Proof

(Lemma 3) Suppose $\underline{\mathcal{Y }}=0$, and $\beta _t \in \mathbb Q _1$ be associated with $Q(F;\beta _t)$, where $\forall \beta _t,t=1,\ldots ,p: 0<\beta _t<1;~ \beta _t$ is not a limit point of $F^{-1}$. The $Q(F;\beta _t)$ functional admits a first-order von Mises expansion at $F$ around $G$, which is given by $Q(G;\beta _t)=Q(F;\beta _t) + \int IF(y, Q(\cdot ;\beta _t),F) \mathrm d (G-F)(y) + R(G,F)$, with $IF$ according to Lemma 1. For ease of notation, write $z_{hijk}=IF(y_{hijk},Q(\cdot ;\beta _t),F)$ and $Z_{hijk}=IF(Y_{hijk},Q(\cdot ;\beta _t),F)$ (adopting the convention that capital letters denote random variables). Under the assumption $0<\beta _t<1$ and for $n$ sufficiently large, there exist constants $c_t$ such that $\inf _L F(c_t)>\beta _t, \forall t$ (where $L \rightarrow \infty$ according to the asymptotic framework; and the fact that $\hat{F}_L(c_t) - F_L(c_t) \rightarrow _p 0$), then $\{z_{hijk}\}$ is bounded. Moreover, and under the regularity conditions on the sampling design, i.e., Assumptions (A1) and (A2), Liapounov’s condition hold and we obtain for the weighted average

$$\begin{aligned} \int IF(y, Q(\cdot ;\beta ), F) \mathrm d \hat{F}(y) = 1/N \sum _{h=1}^{L} \sum _{i=1}^{n_h} \sum _{j=1}^{n_{hi}} \sum _{k=1}^{N_{hij}} w_{hijk} z_{hijk}=\overline{z}, \end{aligned}$$

(28)

and $\mathbb{E }\overline{z} =1/N \sum _{h=1}^{L} \sum _{i=1}^{n_h} \sum _{j=1}^{n_{hi}} \sum _{k=1}^{N_{hij}} Z_{hijk}=0$ (cf. Shao 1994, Theorem 1). Thus, by (Krewski and Rao 1981, Theorem 3.1) $\overline{z}/\sigma (Q(\cdot ;\beta _t),F) \rightarrow _d N(0,1)$ (since $\mathbb E \overline{z}=0$). For $n$ sufficiently large, we may write $\hat{Q}(\hat{F};\beta _t)=Q(F;\beta _t) + \overline{z} + R(\hat{F},F)$. Finally, by (Shao (1994), Theorem 1) $\sqrt{n}R(\hat{F},F)\rightarrow _p0$, and thus $[\hat{Q}(\hat{F};\beta _t)-Q(F;\beta _t)]/\sigma (Q(\cdot ;\beta _t),F) \rightarrow _d N(0,1)$.

In particular, the asymptotic covariance of $\sqrt{n}Q(F;\beta _i)$ and $\sqrt{n}Q(F;\beta _j)$ using the result of Lemma 1 is given by

$$\begin{aligned} \omega _{\beta _i,\beta _j}=\int IF(z, Q(\cdot ;\beta _i),F)IF(z, Q(\cdot ;\beta _j),F) \mathrm d F(z) \end{aligned}$$

(29)

Given $\beta _i \le \beta _j$ and that $1\!\!1\{x \le \xi (F;\beta _j)\}=1$ whenever $1\!\!1\{x \le \xi (F;\beta _i)\}=1$ the right-hand side of (29) becomes

$$\begin{aligned}&\left[ \xi _{\beta _i} - Q(F;\beta _i) \right] \left[ \xi _{\beta _j} - Q(F;\beta _j) \right] + \int \limits ^{\xi _{\beta _j}} \left[ \xi _{\beta _i} - Q(F;\beta _i)\right] \frac{1}{\beta _j} \left[ x - \xi _{\beta _j} \right] \mathrm d F(x) \nonumber \\&\quad + \int \limits ^{\xi _{\beta _i}} \left[ \frac{1}{\beta _j} \left( x - \xi _{\beta _j} \right) + \xi _{\beta _j} - Q(F;\beta _j)\right] \frac{1}{\beta _i} \left[ x - \xi _{\beta _i} \right] \mathrm d F(x) \end{aligned}$$

(30)

On simplifying (30), we obtain (in close relation to the “cumulative income functional” in Cowell and Victoria-Feser (2003, Appendix A.1))

$$\begin{aligned} \omega _{\beta _i, \beta _j}&= \frac{1}{\beta _i\beta _j}S(\beta _{i},F) + \xi _{\beta _{i}} Q(\beta _{j},F) + \xi _{\beta _{j}} Q(\beta _{i},F) - \frac{1}{\beta _{j}} Q(\beta _{i},F)\left( \xi _{\beta _{j}} + \xi _{\beta _{i}}\right) \nonumber \\&- Q(\beta _{i},F) Q(\beta _{j},F) + \xi _{\beta _{i}} \xi _{\beta _{j}} \left( \frac{1}{\beta _{j}} - 1\right) ,\qquad \text{ for } i \le j, \end{aligned}$$

(31)

where $\omega _{\beta _i, \beta _j}$ is a short-hand notation for $\omega _{Q(F;\beta _i), Q(F;\beta _j)}$, and $S(\beta _i,F):=\int ^{\xi (F;\beta _i)}y^2\mathrm d F(y)$.

Thus, for each $Q(F;\beta _i)$ with $0 < \beta _i < 1$ (and if $\beta _i$ is not a limit point of $F^{-1}$), $i=1,\ldots ,p$, we have $\sqrt{n} (\hat{Q}(\hat{F};\beta _i) - Q(F;\beta _i)) \rightarrow _d N(0, \omega _{\beta _i, \beta _i})$. The vector $(\hat{Q}(\hat{F};\beta _1), \ldots , \hat{Q}(\hat{F};\beta _p) )^T$ can be shown (by a Cramer-Wold device; see e.g., Serfling (1980, p.18)) to have a $p$-variate limiting normal distribution with covariance matrix $\varvec{\Omega }$ whose $i,j$th element is equal to $\omega _{\beta _i,\beta _j}$ for $i \le j$. This completes the proof. $\square$