Decomposing the Quantile Ratio Index with Applications to Australian Income and Wealth Data

Abstract

The quantile ratio index introduced by Prendergast and Staudte 2017 is a simple and effective measure of relative inequality for income data that is resistant to outliers. It measures the average relative distance of a randomly chosen income from its symmetric quantile. Another useful property of this index is investigated here: given a partition of the income distribution into a union of sets of symmetric quantiles, one can find the conditional inequality for each set as measured by the quantile ratio index and readily combine them in a weighted average to obtain the index for the entire population. When applied to data for various years, one can track how these contributions to inequality vary over time, as illustrated here for Australian income and wealth data.

Some confidence intervals and insights for the proportion below the relative poverty line

“…In Table 5 are the equalized disposal weekly incomes (DWI) in Australia reported by the Australian Bureau of Statistics (ABS 2016) for 5 years in grouped format. This data set has been previously looked at by Prendergast and Staudte (2019) to obtain estimates for a quantile-based measure of inequality. We consider two approaches to obtaining estimates of H p and their corresponding standard errors.…”

“…In the first approach, we reconstruct the full data set from the available grouped data by simulating data values from the uniform distribution for the bounded intervals and for the final interval we generate data from the Pareto(3) distribution. This method was used by Prendergast and Staudte (2019) to construct a complete data set for (labelled 1998-1992) illustrative purposed. As the second approach, we use the GLD percentile matching method to estimate the underlying density of the grouped data.…”

Interval Estimators for Inequality Measures Using Grouped Data

“…More details for the Atkinson index can be found in Atkinson (1970), Biewen & Jenkins (2006) and Shorrocks (1980). Prendergast & Staudte (2018, 2019 introduced the quantile ratio index (QRI) which uses the ratio of symmetric quantiles and which is simpler than similarly defined inequality measures given by Prendergast & Staudte (2016b). The QRI is denoted as…”

Insights and inference for the proportion below the relative poverty line

“…In Table 6 are the equalized disposal weekly incomes (DWI) in Australia reported by the Australian Bureau of Statistics (ABS, 2016) for five years in grouped format. This data set has been previously looked at by Prendergast & Staudte (2019) to obtain estimates for the QRI. Here we extend it for estimating H and its standard error while comparing it with the other measures.…”

“…In the first approach, we reconstruct the full data set from the available grouped data by simulating data values from the uniform distribution for the bounded intervals and for the final interval we generate data from the Pareto(3) distribution. This method was used by Prendergast & Staudte (2019) to construct a complete data set for illustrative purposed. As the second approach, we use the GLD percentile matching method to estimate the underlying density of the grouped data as detailed in Section 3.2.2.…”