Harmonic means for reciprocal distributions

Reciprocal or inverse normal distributionAn interesting post on ExploringDataBlog on the properties of the distribution of 1/X. Hmm, maybe not the most enticing way of presenting it, since there does not seem anything special in a generic inversion! What attracted me to this post (via Rbloggers) is the fact that a picture shown there was one I had obtained about twenty years ago when looking for a particular conjugate prior in astronomy, a distribution I dubbed the inverse normal distribution (to distinguish it from the inverse Gaussian distribution).  The author, Ron Pearson [who manages to mix the first name and the second name of two arch-enemies of 20th Century statistics!] points out that well-behaved distributions usually lead to heavy tailed reciprocal distributions. Of course, the arithmetic mean of a variable X is the inverse of the harmonic mean of the inverse variable 1/X, so looking at those distributions makes sense. The post shows that, for the inverse normal distribution, depending on the value of the normal mean, the harmonic mean has tails that vary between a Cauchy and a normal distributions…

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