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For a normal distribution, the first two moments (mean and variance) are sufficient statistics for the entire distribution.

Suppose I have a power law distribution, and I have data on the first, second, third, and fourth order moments (mean, variance, skewness, kurtosis). How well can I approximate this distribution using the available information?

Perhaps a more precise way to state this question is: Are the first four moments sufficient statistics for a power law distribution? If not, has anybody proven that I can approximate this distribution to a given degree of uncertainty? ("Approximate" refers to the absolute difference in CDF, and I'm allowing for a loose definition of power law as given by Wikipedia:

for large values of x, $P(X>x) \sim L(x) x^{-(\alpha+1)}$ where $\alpha > 0$, and $L(x)$ is a slowly varying function, i.e. any function that satisfies $\lim_{x\rightarrow\infty} L(r\,x) / L(x) = 1$ for any positive factor $r$.)

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    $\begingroup$ Can you define your power law distribution completely, please? Some sources treat it as 1-parameter (defining the lower bound explicitly), others as 2-parameter, others more (e.g. some have a power law with an upper cut-off), or even infinite-parameter (with a general "slowly varying" L(x) term that simply has to obey some asymptotic properties) -- and are you talking about a discrete or continuous power law? (whichever you choose the answer is likely to be "no, those moments are not sufficient, & they won't even pin it down all that well" but explicit treatment will make for a clearer answer) $\endgroup$
    – Glen_b
    Commented Jul 29, 2018 at 1:37
  • $\begingroup$ Where you ask about "how well can I approximate", are you looking for an answer in terms of absolute difference in cdf or something else (like how well will it work for doing some particular task) $\endgroup$
    – Glen_b
    Commented Jul 29, 2018 at 1:48
  • $\begingroup$ Ideally it would be absolute difference in CDF, but if that is not available, I would be interested in other results that are a measure of absolute difference in CDF. $\endgroup$
    – wwl
    Commented Jul 29, 2018 at 2:32
  • $\begingroup$ Not quite sure what you intend by the second half of that comment. $\endgroup$
    – Glen_b
    Commented Jul 29, 2018 at 3:20
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    $\begingroup$ Your edit to include the distribution has specified an infinite-dimensional distributional form ($L(x)$ is nonparametric). Clearly 4 numbers cannot be sufficient for something so general. $\endgroup$
    – Glen_b
    Commented Jul 29, 2018 at 3:22

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I would recommend not utilizing the first two moments at all if you are working with a power law distribution. The mean and variance are essentially meaningless outside normal or near normal distributions. You can obtain valuable information on the asymmetry of the distribution and tail thickness using skew and kurtosis, respectively.

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    $\begingroup$ The statement about "essentially meaningless" is unnecessarily extreme, given the huge number of distributions for which these are meaningful properties. Moreover, your answer does not address the question about whether these are sufficient statistics. That doesn't mean "enough:" it has a technical meaning. $\endgroup$
    – whuber ♦
    Commented Sep 3, 2022 at 19:05

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