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In my plot below I am going to compare the pdf of my sample in log scale to the normpdf in log scale . From the plot I can see that the sample pdf roughly follows a standardized normal distribution. my question what is the advantage of comparing pdf in log scales? wouldn't it be the same thing to compare via a histogram?

sample pdf

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    $\begingroup$ The log-scale allows you to get a decent visualisation in the tails. Without taking logs, you would get the normal bell curve, with the peak density at $0$ about $90$ times the value at $\pm 3$ and $3000$ times the value at $\pm 4$ so differences between your sample and the theoretical distribution in the tails would almost be invisible whether they were substantial or not. An alternative approach is a Q–Q plot $\endgroup$
    – Henry
    Feb 13 at 0:24

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The big advantage of using a log-scale here is that it allows you to see the relative comparison properly in the tails where the relevant probability density is extremely low. If you looked at this on the standard scale (e.g., with a histogram) the values would be so low that it would be impossible to visually discern the relative sizes of the small values.

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You can use empirical distribution fitting (ECDF) on the arithmetic scale to "find" a best fitting distribution. For example, take a look at the shape of log-normal distributions here with varying parameters called location and scale -- which may be similar to your fitted pdf before taking the log transform. Theoretical probability distributions are fit to empirical data all the time.

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