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I want to plot income data for my country among years and see how the distribution changes. The only available data I could find is frequency tables with unequal intervals.
For example,

+---------------+---------+
|Incomes        | # of ppl|
+---------------+---------+   
|16000 - 17999  | 12,0000 |
|18000 - 20999  | 14,0000 |
+---------------+---------+

I would like to create a distribution/density plot (continuous curve) from this type of data.

One method I could think about is using uniform distribution to generate $n_{people}$ of samples in each income interval. I would like to see if there are other methods to do that.

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  • $\begingroup$ One issue with using a uniform distribution is that the actual distribution in each interval won't be uniform; indeed as soon as you fit some continuous distribtion, you'll be asserting that it isn't' uniform in each bin. If the true density is slowly changing or bins are quite narrow it won't matter much but with wide bins and a density that's not close to uniform it may make quite a difference. If there's a density you want to fit, you might use an EM algorithm perhaps (perhaps even starting with uniformly distributed values), or do direct estimation with the interval-censored data...ctd $\endgroup$
    – Glen_b
    Jul 29 '17 at 9:10
  • $\begingroup$ ctd... if you're just after some smooth density and you have sufficiently narrow bins, a kernel density estimate may suffice. $\endgroup$
    – Glen_b
    Jul 29 '17 at 9:14
  • $\begingroup$ @Glen_b Thanks for the comment. Do you mean if the bins are narrow enough, I could use uniform generated data to fit kernel density estimation? $\endgroup$ Jul 29 '17 at 9:51
  • $\begingroup$ If they're narrow enough relative to the bandwidth, it's even possible you could just put them all at the bin-center. $\endgroup$
    – Glen_b
    Jul 29 '17 at 10:05
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For the most accurate results, it would probably be ideal to use a likelihood-based method, such as the one studied in this paper: Logspline density estimation for binned data.

On the other hand, as Glen_b pointed out, if the bins are narrow enough, kernel density estimation may give adequate results, by treating each observation as being located at the center of its bin. Some justification for this idea can be found in this paper: Kernel density estimation with binned data

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  • $\begingroup$ Never though about this type of question can turn into deep statistics concepts. The paper you provided are very informative. Thanks a lot. $\endgroup$ Jul 29 '17 at 20:50

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