# Are Skewness and Kurtosis Sufficient Statistics?

I would like to prove that Skewness and Kurtosis are sufficient statistics for gaussian distribution.

Later on I will try to prove on loglogistic distribution.

Do you have any idea how to do that?

• You are going to fail --- you cannot prove what is not true – kjetil b halvorsen May 26 '17 at 12:00
• Can you give some detail? What is the reason? – Cerenimo May 26 '17 at 12:08
• You obtain absolutely no information about the location of a Gaussian from the skewness or kurtosis. Proof: they are the same for a Normal$(\mu,\sigma)$ and Normal$(\mu^\prime,\sigma)$ distribution, for any $\mu$ or $\mu^\prime$. Indeed, skewness is worthless, because it's always zero. – whuber May 26 '17 at 12:25
• Thanks a lot. That's right. How can I prove these two are sufficient statistic on Loglogistic distribution? Any idea? – Cerenimo May 26 '17 at 12:45
• Why would you think they are? What theorems exist that relate to sufficiency? – Glen_b May 26 '17 at 12:50

More details: You obtain absolutely no information about the location of a Gaussian from the skewness or kurtosis. Proof: they are the same for a $\text{Normal}(\mu,\sigma)$ and $\text{Normal}(\mu′,\sigma)$ distribution, for any μ or μ′. Indeed, skewness is worthless, because it's always zero. (stated in comments by whuber).
• This answer is presently flawed. Sample skewness is not always zero. In fact, it is non-zero with probability one (though it has a distribution that does not depend on $\mu$). – Ben Sep 1 at 0:21