With Respect
I am still working with my dataset about the citations of a group of precedents, and the distributions of those cases over different measurements.
I was interested to read of Benford's law, a distribution of initial digits of the numbers that represent the result of a count or measure. I expected that this would apply to the total number of citations per case, a classic count(lognormally distributed). I did not expect it, at first thought, to apply to the ratio of the number of cases "distinguishing" a case to the number "following" it (in a gamma distribution) . That ratio, though, had a right-skewed distribution and the charted initial digit distribution of the values(33/46) that had an initial digit fit the Benford's law distribution roughly and not much worse than that of the total count. I was unsure if it would apply to my other two measures, of the share of total citations taken by specific types of citation. From empirical testing, they very clearly don't follow it. Is this because there is a fixed maximum -1, or 100%- to those distributions?
What most interests me is the question -could Benford's law apply to the difference in rank orders between the same cases on two different measures. Obviously it can't apply to the rankings themselves (since there have to be as many from 20 to 29 as from 10 to 19 and so on) but could it apply to the differences between two rankings of the same cases? Would it apply in the particular case of zero correlation where extrapolation from the vertical (on zero) distribution if there is perfect positive correlation and flat distribution if there is perfect negative correlation leads me to expect there would be a pyramid or bell shaped distribution? Again the actual initial digit distribution is right-skewed like the Benford's Law distribution. What is the general relation of Benford's law to the actual distribution of values, and in particular to the normal and lognormal distributions?
I would like an explanation of how Benford's law relates to different patterns of distributions, and just what its being followed or not implies about a distribution, particularly a distribution of rank order differences.
Yours Sincerely,
Alan E. Dunne
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