Simple adjustments of observed distributions for missing income and missing people

Abstract

Correcting household survey distribution data for missing income or for undersampling may give an idea of the extent of possible biases in measuring inequality, especially when there are reasons to expect the missing income and people to belong to the top of the distribution. There are simple ways to do so when only an aggregate estimate of how much is missing is available. Atkinson had provided a formula to correct the Gini coefficient for the missing income, which was later generalized by Alvaredo (Econ. Lett. 110(3), 274–277 2011). This paper concentrates on the whole distribution and explores various alternative adjustment methods based on three key parameters: how much income, how many people are missing and on what range of income the correction should bear.

Appendices

Appendix A: Proof of (7)

Integrating (5) yields:

$$ L_{3}^{*}(p)=\left\{ \begin{array}{ll} \frac{1}{1+\mu}L(p)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,p\leq\overline{p}\\ \frac{1}{1+\mu}L(p)\,+\,\,\frac{\delta}{1+\mu}.[L(p)-L(\overline{p})]\,-x(\overline{p}).(p-\overline{p})/\overline{x}^{*}\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,p>\overline{p} \end{array}\right. $$

(1)

But δ can be expressed as:

$$\delta=\frac{\mu\overline{x}}{(1-\overline{p})[X(\overline{p})-x(\overline{p})]}=\frac{\mu}{1-L(\overline{p})-x(\overline{p})(1-\overline{p})} $$

and Eq. 7 follows:

$$L_{3}^{*}(p)=\left\{ \begin{array}{ll} \frac{1}{1+\mu}L(p)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,p\leq\overline{p}\\ \frac{1}{1+\mu}L(p)\,+\,\,\frac{\mu}{1+\mu}.\frac{L(p)-L(\overline{p})\,-x(\overline{p}).(p-\overline{p})/\overline{x}}{1-L(\overline{p})-x(\overline{p})(1-\overline{p})/\overline{x}}\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,p>\overline{p} \end{array}\right. $$

We now show that the second term in the bottom row of that expression is that of the Lorenz curve of the distribution of $z=x-x(\overline {p}$). Define $q=(p-\overline {p})/(1-\overline {p})$, which thus varies in [0, 1]. The quantile function of z is given by $z(q)=x[(1-\overline {p})q+\overline {p}]-x(\overline {p})$. The Lorenz curve of z is then given by:

$$L_{z}(q)=\frac{1}{X(\overline{p})-x(\overline{p})}\left[{{\int}_{0}^{q}}x[(1-\overline{p})u+\overline{p}]du-{{\int}_{0}^{q}}x(\overline{p})du\right] $$

With a change of variable in the first integral it can be shown that:

$$L_{z}(q)=\frac{1}{X(\overline{p})-x(\overline{p})}\left[\frac{1}{1-\overline{p}}{\int}_{\overline{p}}^{(1-\overline{p})q+\overline{p}}x(p)dp-qx(\overline{p})\right] $$

Or:

$$L_{z}(q)=\frac{\overline{x}}{X(\overline{p})-x(\overline{p})}\left[\frac{L[p]-L(\overline{p})}{1-\overline{p}}-\frac{(p-\overline{p})}{1-\overline{p}}\,\frac{x(\overline{p})}{\overline{x}}\right]\,\,\,\,\,\,\,\,\,\,\,\,\,with\,\,\,\,\,\,\,\,\,\,q=\frac{(p-\overline{p})}{1-\overline{p}} $$

Then noticing that $X(\overline {p})-x(\overline {p})=\frac {1-L(\overline {p})}{1-\overline {p}}\overline {x}-x(\overline {p})$ the preceding expression can be rewritten as:

$$L_{z}(q)=\frac{L[p]-L(\overline{p})-(p-\overline{p})x(\overline{p})/\overline{x}}{1-L(\overline{p})-(1-\overline{p})x(\overline{p})/\overline{x}}\,\,\,\,\,\,\,\,\,\,q=\frac{(p-\overline{p})}{1-\overline{p}} $$

which is the term that appears in the the second row of Eq. 7. Integrating this expression with respect to q in (0,1), or p in $(\overline {p},1)$ yields $(1-p)(1-\widetilde {G})/2$, where $\widetilde {G}$ is the Gini coefficient associated to z. Doing the same with L(p) in (0,1) yields (1 − G)/2. Integrating L^∗(p) thus leads to:

$$\frac{1-G^{*}}{2}=\frac{1}{1+\mu}\frac{(1-G)}{2}+\frac{\mu}{1+\mu}(1-p)\frac{(1-\widetilde{G})}{2} $$

Eq. 8 follows.

Appendix B: Proof of (12)

A useful property to be used in what follows is:

$$ \frac{1}{\overline{x}^{*}}{\intop_{a}^{b}}x(\alpha u+\beta)du=\frac{1}{\overline{x}^{*}}\frac{1}{\alpha}\intop_{\alpha a+\beta}^{\alpha b+\beta}x(v)dv=\frac{1}{1+\mu}\frac{1}{\alpha}[L(\alpha b+\beta)-L(\alpha a+\beta)] $$

(2)

Integrating the top part of the expression of the quantile function:

$$x^{*}(p)=\left\{ \begin{array}{ll} x(\frac{p}{1-\lambda})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,p\leq\overline{p}.(1-\lambda)\\ x\left[\frac{p(1-\overline{p})+\lambda\overline{p})}{1-\overline{p}(1-\lambda)}\right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,p>\overline{p}.(1-\lambda) \end{array}\right. $$

dividing by the mean of the corrected distribution and using (18) leads to:

$$ L_{4}^{*}(p)=\frac{1-\lambda}{1+\mu}L\left( \frac{p}{1-\lambda}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,\,\,p\leq\overline{p}.(1-\lambda) $$

(3)

Doing the same on the bottom part of the expression of the quantile function yields:

$$L_{4}^{*}(p)-L_{4}^{*}[\overline{p}.(1-\lambda)]=\frac{1}{1+\mu}\frac{1-\overline{p}(1-\lambda)}{1-\overline{p}}\left[L\left( \frac{p(1-\overline{p})+\lambda\overline{p})}{1-\overline{p}(1-\lambda)}\right)-L(\overline{p})\right] $$

Then, noticing that Eq. 19 implies $L^{*}[\overline {p}.(1-\lambda )]=\frac {1-\lambda }{1+\mu }L(\overline {p}$), Eq. 12 follows.