# Two approaches for Fitting Beta-distribution to Histogram of Survey Response. Which one is better?

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.) 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


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).

• 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) Aug 26, 2022 at 5:29
• 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. Aug 26, 2022 at 5:46

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