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Suppose that $N\sim(0,1)$ and I have two functions $f$ and $g$.

Are there any theorems or techniques for comparing how close are distributions of $f(N)$ and $g(N)$?

For example, it is known that sum of lognormal random variables $ae^{bN}+ce^{dN}$ can be approximated by one lognormal variable $ue^{vN}$. Is it possible to find how good approximation is?

Thank you.

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    $\begingroup$ You may want to google ""statistical distance" or "Lévy–Prokhorov metric". $\endgroup$
    – Liding Yao
    Mar 14 at 18:02
  • $\begingroup$ @LidingYao , any simple example would be great. $\endgroup$
    – eMathHelp
    Mar 14 at 18:18
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    $\begingroup$ Check the definitions from wikipedia. $\endgroup$
    – Liding Yao
    Mar 14 at 19:49
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    $\begingroup$ Try the Kullback-Leibler divergence $\endgroup$ Apr 11 at 9:28
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    $\begingroup$ Let me know where do you need this? It depends on the application $\endgroup$
    – Amir
    Apr 12 at 17:02

2 Answers 2

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KL divergence, although neither a metric nor symmetric, is the most used measure used to measure distance between two probability distributions.

Check out more, here: https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence

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Because you appear to be looking for authoritative references discussing distances between probability distributions, here is a non-comprehensive list of references. (Making this community wiki so others can add references for other distances.)

Wasserstein distance

General dissimilarities

  • Chapter 2, Similarity Measures and Generalized Divergences, of the book Cichocki et al., Nonnegative Matrix and Tensor Factorizations, gives 40 pages of examples of different kinds of "distance measures".
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