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To help clarify my understanding of this statistic, I'd appreciate feedback on the rationale presented here.

Assume we have a distribution that seems potentially lognormal. Checking the median against geometric mean can be an indication of lognormality (though I don't know if this is a reliable statistical test).

x = ggplot2::diamonds$price
median(x) / exp(mean(log(x)))
#> [1] 0.996877

But I'm wondering about use of Kurtosis approach. The following function uses Pearson's measure.

require(moments)
#> Loading required package: moments
kurtosis(x)
#> [1] 5.177383

As I understand this tells us the tails are not a normal distribution. So checking the kurtosis of the log gives us:

kurtosis(log(x))
#> [1] 1.903206

Does less than 3 indicates less tail than we would expect with a lognormal distribution?

In the general case (exploring lognormality) is this a sensible approach? Would we also be wanting to apply skewness methods to robustly pin this down?

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    $\begingroup$ 1. Note that statistics that are reasonably consistent with data having been drawn from a lognormal doesn't imply that you have a lognormal; e.g. other distributions can also have median close to geometric mean. $\:$ 2. Choice of statistic will change depending on what distribution(s) you want good ability to distinguish a lognormal from -- and perhaps what you're doing it for (why you're trying to identify lognormality) $\:$ 3. "As I understand this tells us the tails are not a normal distribution" - does this imply you're primarily interesting in distinguishing a lognormal from a normal? $\endgroup$ – Glen_b Jul 12 at 2:58
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    $\begingroup$ Kurtosis can't be a good indicator of whether a distribution is lognormal for the reasons @Glen_b gives, and others. High kurtosis is consistent with zero or negative skewness, for one. For another, sample kurtosis is limited as a function of sample size and so will often deny the parentage of a lognormal even when that is a fact. The specific example of a lognormal and sample skewness and kurtosis is discussed in detail in stata-journal.com/article.html?article=st0204 (to the references there add jstor.org/stable/2236642) $\endgroup$ – Nick Cox Jul 12 at 9:26
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    $\begingroup$ The best way to check for lognormal distributions is to take logarithms and check for normal distributions! $\endgroup$ – Nick Cox Jul 12 at 9:27
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    $\begingroup$ The use of sample kurtosis--and any other high moment--is an exceptionally unreliable way to check for most distributional properties. Even its use to check for normality in the Jarque-Bera test (where the sample kurtosis should be well behaved) has long been deprecated in favor of better methods. $\endgroup$ – whuber Jul 12 at 17:19
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    $\begingroup$ A different measure of kurtosis can't solve the basic problem. Kurtosis is not a measure of how far distributions are lognormal. $\endgroup$ – Nick Cox Jul 13 at 11:40

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