# Statistician's quantiles versus policymaker's quantiles: calculating the variance correctly

When speaking of income, although statisticians and policymakers (and their staffs) both refer to deciles, quintiles, and the like, they mean quite different things by these terms. When a statistician refers to, say, a quintile, S/he means the value of income at a single point, the point that divides that fifth of the population from the next higher fifth. When policymakers refer to quintiles, they mean the incomes of all the people in that fifth. Statistician's quintiles are thus a kind of order statistic, while policymaker's quintiles are conditional means.

Policymakers' quantiles, under any name, have in so far as I know received little attention from statisticians, although they are regularly published by the Census, and estimates by the Congressional Budget Office, Joint Committee on Taxation, HHS and others regularly play important roles in political decision-making. It is common to display these conditional means with a bar graph, but I do not believe I have ever seen one published with an associated error bar.

It is straightforward to estimate statistician's quantiles using standard statistical packages such as R's survey package. Similarly, it is typically straightforward to estimate conditional means over fixed intervals with the same software. Thomas Lumley provides one approach to this in the vignette “Estimates in subpopulations,” here:

https://cran.r-project.org/web/packages/survey/vignettes/domain.pdf

If I understand this procedure correctly, it essentially amounts to assigning each observation within one or more specified ranges to dummy variables for those ranges, where the estimate of the interval mean is the coefficient on the dummy and the variance of the estimate is the variance for that coefficient.

So one might think that there is a simple way to estimate the variance for what what I have so far called "policymakers' quantiles" but will henceforth call quantile range means, where "quantile" has the standard statistical meaning. One can estimate the quantiles that form the upper and lower boundary of a range, and then use the procedure just outlined to estimate the quantile range means and their variances.

I believe this procedure gives a good approximation of the true quantile range mean (though not exactly the right value because of curvature of the pdf near the quantiles). But I think it gives the wrong variance, because the endpoints are no longer simple numbers, but are random variables themselves. Moreover, the variance estimated in this way will understate the true variance of the quantile range mean. This is because, to the extent that the sample values overstate or understate the true conditional population mean, this will also move the endpoints of the range over over which it will be calculated, which will be expanded or contracted accordingly.

The quantile range mean is a statistic in common use in settings that make it unusually consequential. Is there some literature about it, perhaps under a different name, that I have not been able to find? Is there a known way of calculating it from a survey sample? It would be convenient for me if the latter was using R, but I'd accept any open-source solution.