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I'm looking for a measure that measure the similarity of two distributions in the following forms: $$ S = \frac{a \mu_1 + b \sigma_1}{a \mu_2 + b \sigma_2}. $$

The above formula I proposed is not rigorous. But it can show that if two distributions are similar, then $S$ would turn to 1. Does there exist a measure like this?

Wynn
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  • Hi Wynn! Welcome to CV. Have you considered a more general test of similarity such as the Kolmogorov-Smirnov (KS) test? – Alex J Mar 20 '23 at 02:59
  • make an assumption of the underlying distribution (maybe consider to test that hypothesis) and then use any distance for distributions, like KL divergence – Alberto Mar 20 '23 at 11:03
  • Although your question is unclear (because you don't explain what this "similarity" is intended to mean or how it is to be applied), it is reminiscent of the symmetrized divergence between two Normal distributions. Could you edit your post to help us understand the context of your question? – whuber Mar 22 '23 at 13:02

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