This question is a duplicate of a question I previously asked, here: Dimensional analysis for a qdf/quantile function corresponing to the pdf/CDF (size distribution of income example). I believe I have found the answer to my original question, and that this answer establishes that my previous question was not a duplicate of that earlier one, but it appears to me that once a question is marked as a duplicate it is no longer possible to post the answer; hence this redundancy.

I am going to frame this question in terms of a pdf and quantile functions for the size distribution of income, because that is where I have currently encountered it, but similar questions in quite different contexts have puzzled me on several occasions in the past.

Suppose I have a pdf f(x) or a CDF F(x) that represents the size distribution of income. Income is a flow of money (dollars or some unit of currency), per some unit of time, to particular individuals. I therefor think the proper unit of measurement is dollars per person-year.

In order for the probability to come out in dimensionless units, f(x) must be measured in person-years per dollar. Integrating this over x also yields a dimensionless share for the CDF because the units of an integral are the product of the units of the integrand f(x), here measured in person-years/dollar, and of the differential dx, here measured in dollars/person-year.

Integrating xf(x) therefore puts the units on the mean income in $/person-year; this seems reasonable.

From this I have drawn conclusions on the units and interpretation of the quantile density function q(y), quantile function Q(y), and the integral of yq(y)dy (i.e. the the inverse function analogue of the partial moment function of order 1) which if correct would be very helpful to me, but in which I have limited confidence. I’d be grateful if someone could confirm or deny these conclusions, perhaps in the context of some discussion/explanation of interpretation of dimensional analysis in the the context of distribution functions.

I believe the units of the y axis and of q(y) are necessarily the same as those in the previous analysis, i.e. the y values are still measured in person-years per dollar and x = q(y) is still measured in $/person-year. This implies that integrals of q(y) over some interval in general, and Q(y) (which is equal to the integral of q(y) from 0 to y) in particular, are both measured in unitless shares. I am somewhat dubious of this and my earlier conclusion, however, because I believe that F(x) is in unitless shares of total income, while Q(y) is in unitless shares of the population for a year. This seems odd to me, however, as I wonder if the fact that these two shares are shares of different things should not be reflected in the units.

Finally, this would have the integral of yq(y)dy in units of person-years per dollar. But I am having trouble interpreting this. The corresponding x quantity is the mean (or for a bounded x, the partial mean) of income per person-year. So this would be the mean number of person-years--for what? for a single dollar? for the mean income?

Consider the truncated version of this function, where the integral of yq(y)dyfrom zero to q* is divided by F(q*). The corresponding truncated function of x yields the mean for those with incomes of x* or less, and $/person-year seems like reasonable units for a mean income. But I want the the y version to come out in number of people, or number of person-years, corresponding to the share of person-years over the same interval, and the "per dollar" in the denominator does not have an interpretation which is obvious to me.

  • 2
    $\begingroup$ What is the question? $\endgroup$
    – Firebug
    Apr 17, 2018 at 11:33
  • $\begingroup$ @Firebug The actual questions are in the last three paragraphs, and boil down to 1. Given the units on the pdf, do I have the right units on the quantile and quantile moment functions; and 2. If correct, I am finding those units hard to interpret. That is partly because some of them were wrong. See below. $\endgroup$
    – andrewH
    Apr 20, 2018 at 3:12

1 Answer 1


Only paragraphs (6) through (9) below actually constitute an answer to my original questions (contained in the last three paragraphs above). Skip to them if you like. The rest just details the path I took to get to that answer, as I found it easier to work through the correct interpretation on the pdf/CDF side and then construct the qdf/Q answers by analogy. I am going to supply names to several functions, to keep track of them. If these functions have established names I would very much appreciate it if people would inform me of them.

  1. Consider a pdf. The area under a pdf is a unitless share, so the units on the y axis must be the inverse of the units on the x axis.

  2. Cumulating the area under the pdf gives the CDF, therefor also unitless.

  3. Integrating the x times pdf(x) from zero to y* gives the partial mean function (pmf), measured in the units of y = f(x), a number between 0 and the mean. Dividing the full range into disjoint sub-ranges and integrating over the ranges yields partial means over the sub-ranges that sum to the mean.

  4. Dividing the partial mean function by F(x) gives the (full) mean of the distribution truncated at x*. Call this function the Conditional Mean Function, (CMF(x)). Again dividing into sub-ranges and integrating between x* and x** yields the conditional mean of each range, given that the observation is in the range, and is equal to CMF(x**) - CMF(x*). The sum of the conditional means is always greater than the mean and is essentially meaningless. Instead, the overall mean is the mean of the conditional means, weighted by the share of person-years that fall in that income range.

  5. The CMF(y) is measured in the same units as the partial mean function, $/person-year, as the denominator F(x) in the definition above is unitless. Multiplying the CMF times the number of person-years that earned in that income range yields the Cumulative Aggregate Value Function, (CAVF(x)). As applied income distribution, the CAVF could be called the Cumulative Aggregate Income Function. It is denominated in dollars and represents the cumulative total income of all persons earning less than x*. Note that this multiplication and the resulting function was missing in the analysis within my question.

  6. Now, turning to the quantile side, we proceed strictly in parallel. y has units of person-years per dollar, the same units as pdf(x). The qdf q(y) has the same units as x, $/person-year, required to make quantile function Q(y) unitless.

  7. Integrating yq(y) over subranges yields the Partial First Quantile-Moment Function (PFQ-M(y)), a quantity measured in person-years per dollar. These numbers represent the portion (n.b. portion, not proportion) of the overall mean person-years per dollar that falls within the given quantile sub-range. The overall mean person-years per dollar, on a whole-population basis for the U.S. in 2017, is a very small number, approximately 0.00002, and the partial Q-Ms are of course smaller yet. (I am not sure that quantile-moment is an appropriate nomenclature here. There is some literature that uses this term for what I would call quantile range moments – ordinary non-quantile moments over ranges defined by quantiles. The hyphen is intended to distinguish this usage from that one, but I welcome better terminology throughout.)

  8. Taking the PFQ-Ms for quantiles up to y = q* and multiplying by the total number of dollars earned by persons that fall within those ranges yields the Cumulative Aggregate Base Function, CABF(y). As applied in the income context, I would call this the Cumulative Total Population Function. It is denominated in person-years, and represents the total number of person-years of earners. To get population share for those ranges, just divide through by total population.

And that, I believe, is the answer to my question. If this answer is contained in the supposed duplicate referenced above, and the worthy Dr. Huber has a hat that he does not value, I hereby volunteer to eat it on live internet video.

  1. Oh, and as to whether I should be troubled by unitless income shares being different from unitless population shares, I believe the answer is no: All shares, where the numerator and denominator are in the same units, are unitless like elasticities and for the same reason.

It might be of interest to some that the partial mean function defined in (3) above divided by the mean gives the Lorenz curve; and that the pmf(x) divided by the CDF(x*) gives the Lorenz curve for the truncated distribution, for x < x*. On the quantile side, the PFQ-FM(y) from (7) above divided by the first quantile-moment is also equal to the Lorenz curve.

There ought to be a parallel result for a truncated quantile distribution defined by PFQ-M(y)/CABF(y*) for y < y*, but I have not quite worked through the interpretation of a truncated quantile-moment.


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