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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
    Commented Jul 12, 2020 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
    Commented Jul 12, 2020 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
    Commented Jul 12, 2020 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 ♦
    Commented Jul 12, 2020 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
    Commented Jul 13, 2020 at 11:40

1 Answer 1

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Kurtosis is not a good way to check for lognormality. For more information on kurtosis and what it is/is not, see what is the meaning of 'tail' of kurtosis?.

If your goal is to check for lognormality, the best way is to take logs and check for normality, preferably graphically, using a qq-plot.

See also the comment thread for more information and links.

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