Estimation of sample quantiles: challenges and issues in the context of income and wealth distributions

Abstract

Means, quantiles and extreme values are common statistics for the description of distributions. However, estimating sample quantiles with the default definition in different software programs leads to unequal results. This is due to the fact that software programs use different quantile definitions. Since most practitioners are not aware of this fact and use different quantile definitions interchangeably, this work compares the default definitions in the software programs SPSS, R, SAS^™ software, and Stata and additional quantile definitions that are suggested by the literature. The work especially focuses on how the quantile estimators perform in the context of describing the distribution of income and wealth. Furthermore, the possibilities of considering sampling weights in the quantile estimation and methods for producing variance estimates using the above-mentioned software are discussed.

Zusammenfassung

Mittelwerte, Quantile und Extremwerte sind übliche Statistiken, die zur Beschreibung von Verteilungen genutzt werden. Allerdings sind die Ergebnisse für Quantile, die mit verschiedener Software berechnet werden, nicht zwingend gleich. Dies ist darauf zurückzuführen, dass Quantilsdefinitionen verschiedener Software-Programme teils nicht einheitlich sind. Da diese unterschiedlichen Definitionen vielen Anwendern nicht bewusst sind und die Funktionen in der Software austauschbar genutzt werden, vergleicht diese Arbeit unterschiedliche Quantilsdefinitionen in den Software-Programmen SPSS, R, SAS^™ Software und Stata. Außerdem werden Quantilsdefinitionen betrachtet, die in vorherigen Vergleichen in der Literatur empfohlen werden. Diese Arbeit betrachtet besonders die Güte der unterschiedlichen Quantilsdefinitionen für die Beschreibung von Einkommens- und Vermögensverteilungen. Außerdem werden Möglichkeiten zur Berücksichtigung von Survey-Gewichten bei der Quantilsschätzung, sowie zur Varianzsschätzung in den genannten Software-Programmen diskutiert.

Caption Electronic Supplementary Material

Code examples and outputs based on synthetic data in the different software programs

Synthetic data and the code used to obtain the results in the aforementioned PDF-file

Tables with results from the simulation studies

Appendix A

For the sake of completeness, the expressions of quantile estimators that are introduced in Table 2 but not mentioned in the text are shown in this Appendix. Furthermore, the six properties that are used by Hyndman and Fan (1996) are summarized.

1.1 A1  Inverse of the empirical cumulative distribution function

Dielmann et al. (1994) states that this quantile estimator is neither mean nor median unbiased. For a further discussion of its properties we refer to Juritz et al. (1983).

$$\begin{aligned}\displaystyle Q_{p}=\begin{cases}X_{1}\quad&\text{if}\quad p=0;\\ X_{(i)}\quad&\text{if}\quad 0<p\leq 1\quad\text{and}\quad g=0;\\ X_{(i+1)}\quad&\text{if}\quad 0<p\leq 1\quad\text{and}\quad g\neq 0,\end{cases}\end{aligned}$$

where $i=\lfloor np\rfloor$ and $g=np-i$.

1.2 A2  Observation closest to $np$

This definition crucially depends on the rounding. While in R and SAS the rounding takes place to the next even integer, the definition in SPSS differs from the one below since it uses simple rounding.

$$\begin{aligned}\displaystyle Q_{p}=\begin{cases}X_{(1)}\quad&\text{if}\quad p\leq\frac{0.5}{n};\\ X_{(i)}\quad&\text{if}\quad\frac{0.5}{n}<p\leq 1,\quad i\text{ is even and}\quad g=0;\\ X_{(i+1)}\quad&\text{if}\quad\frac{0.5}{n}<p\leq 1,\quad i\text{ is odd and}\quad g\neq 0,\end{cases}\end{aligned}$$

where $i=\lfloor np\rfloor$ and $g=np-0.5-i$.

1.3 A3  Linear interpolation of the empirical distribution function

This definition is proposed by Parzen (1979).

$$\begin{aligned}\displaystyle Q_{p}=\begin{cases}X_{(1)}\quad&\text{if}\quad p<\frac{1}{n};\\ (1-\gamma)X_{(i)}+\gamma X_{(i+1)}\quad&\text{if}\quad\frac{1}{n}\leq p<1;\\ X_{(n)}\quad&\text{if}\quad p=1,\end{cases}\end{aligned}$$

where $i=\lfloor np_{k}\rfloor$, $p_{k}=\frac{np}{n}$, $\gamma=np_{k}-i$.

1.4 A4  Approximation to $F(E(X_{k}))$ for the normal distribution

This definition is especially preferable when the underlying distribution is normal (Blom 1958). Thus, it is often used for normal quantile-quantile plots.

$$\begin{aligned}\displaystyle Q_{p}=\begin{cases}X_{(1)}\quad&\text{if}\quad p<\frac{5/8}{n+1/4};\\ (1-\gamma)X_{(i)}+\gamma X_{(i+1)}\quad&\text{if}\quad\frac{5/8}{n+1/4}\leq p<\frac{n-3/8}{n+1/4};\\ X_{(n)}\quad&\text{if}\quad p\geq\frac{n-3/8}{n+1/4},\end{cases}\end{aligned}$$

where $i=\lfloor np_{k}+\frac{p}{4}+\frac{3}{8}\rfloor$, $p_{k}=\frac{\left(np+\frac{p}{4}+\frac{3}{8}\right)-\frac{3}{8}}{n+\frac{1}{4}}$, $\gamma=np_{k}+\frac{p}{4}+\frac{3}{8}-i$.

1.5 A5  Six desirable properties for sample quantile

 

Table 6 Replication of Table 1 in Hyndman and Fan (1996) that shows their definition of six desirable properties for a sample quantile. For more information about the properties it is referred to Hyndman and Fan (1996)