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We have data from a survey in which participants are asked about the probability that an event takes place. The question is probabilistic and asks the respondents to assign probabilities to pre-specified ranges. An example of such a question is:

What would you say is the percent chance that, over the next 12 months,
    -  ...
    -  you will change jobs with a probability of between 40 and 50 percent.  ___ percent chance
    -  you will change jobs with a probability of between 50 and 60 percent.  ___ percent chance
    -  ...

The response from each participant is a histogram. We plan to fit a Beta-distribution to these histograms following the approach in Engelberg, Manski, Williams (JBES 2009) who suggest to fit the CDF of the Beta_distribution to the upper right corners of the bars in the cumulative histogram. See left hand panel in the figure below. A second approach would be to fit the PDF of the Beta-distribution to the histogram directly. (To the top center of the bars. See right hand panel in the figure below.) Two approaches to fit a continuous distribution to a histogram Our impression is that both approaches give similar results. As an example, consider a respondent who assigns the following probabilities to three intervals: 20, 50, 30. The estimated shape parameters (a,b) of the Beta-distributions are:

Fitting CDF: 
a = 2.3649366335876776
b = 2.0642275819299365
Fitting PDF: 
a = 2.2934099704558495
b = 2.055981090639008

Results

Question: What are the advantages and disadvantages of these two approaches? Is there a reason why fitting a CDF is more common in the literature (at least this was our impression).

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  • $\begingroup$ To treat your survey responses as if they represent binned observations from some distribution $G$, they would be interval censored (you only know that the underlying $G$-distributed values were in some interval). If you believe your beta model (that $G$ is $B(\alpha,\beta)$), the obvious thing to try would be to use (say) maximum likelihood estimation of the underlying beta parameters from that interval-censored data. (There are other possibilities, like minimum chi-square, but MLE has some nice properties, so is common) $\endgroup$
    – Glen_b
    Aug 26, 2022 at 5:29
  • $\begingroup$ This should, for example, be doable (using MLE or a number of other fitting methods) in R via the function fitdistcens in the fitdistrplus package, which is on CRAN. $\endgroup$
    – Glen_b
    Aug 26, 2022 at 5:46

1 Answer 1

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The questions the participants were asked exactly define points on a cumulative distribution function, which by definition is $P(X<x)$ (which you get by summing up the probabilities for the intervals below $x$ that you've asked about).

In contrast, the histogram is only an approximation (through a piecewise constant step function) to the probability density function. I.e. your participants did not actually tell you that the density in the middle of the interval is the point on the histogram you marked. I.e. you are adding extra unnecessary approximations here, which with a sufficiently large number of intervals (which obviously all need the same width) in the right areas of the pdf might not make too much of a difference in practice. However, while there's a potential downside (that may or may not make a difference), there's no real advantage to it, as far as I can see.

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  • $\begingroup$ Thanks, +1. The argument that the response only defines points on a cumulative distribution function is important. We had the impression that this argument is less relevant in practice because Engelberg et al. (2009) assume that all responses are single-peaked. This rules out any "weird" behavior within the intervals. Can you agree on that? (Sorry if this questions is silly, we are no experts in this field.) $\endgroup$ Aug 25, 2022 at 15:12
  • $\begingroup$ If you fit a beta-distribution, then you are indeed assuming that you either have no peak (for Beta(1,1)), one or two peaks at the ends of the interval (if either or both parameters are <1) or a single peak in (0,1). However, that still means that within each interval the curve could have a lot of shapes and putting values at the middle of intervals may end up missing the actual density by some amount. $\endgroup$
    – Björn
    Aug 25, 2022 at 17:32
  • $\begingroup$ I understand, thanks. $\endgroup$ Aug 26, 2022 at 14:38
  • $\begingroup$ One advantage of the PDF approach we thought about is that it requires less observations. In the 3-interval example in the question, there are 3 midpoints but only two upper right corners (the upper right corner of the right-most interval coincides with the limit of the distribution). The CDF approach requires that respondents use 3 or more intervals. The PDF approach only 2 or more. $\endgroup$ Aug 26, 2022 at 14:39
  • $\begingroup$ I don't think you've gained any information though. I.e. it's not really an advantage. But you could easily try some simulations to compare the methods and see which behaves better. As stated, my bet would be on the CDF based approach. $\endgroup$
    – Björn
    Aug 26, 2022 at 15:00

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