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I need some help to understand how to utilize sufficient statistic from a data.

Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I understand, a sufficient statistic for $X$ is $(\log(x), x)$ or $(E[\log(X)], E[X])$ if I'm going to update my knowledge of the observed values.

However, I don't understand how to get the parameterization $(a, b)$ of the underlying gamma distribution if I know sufficient statistics. With the normal distribution and some discrete ones it’s more or less clear, but here I’m stuck.

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    $\begingroup$ $(E[\log(X)], E[X])$ is not a sufficient statistic but $(\overline{\log x}, \overline{x})$ might be. $(\sum \log{ x_i}, \sum{x_i})$ or $(\prod{ x_i}, \sum{x_i})$ might be too. They do not give you estimates of the parameters directly, but can be used to give you estimates of the parameters particularly if you want a maximum likelihood estimator. $\endgroup$
    – Henry
    Commented Mar 22 at 20:58
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    $\begingroup$ Could you please explain what you mean by "get the parameterization"? Are you perhaps asking how to estimate the parameters from values of these statistics? $\endgroup$
    – whuber ♦
    Commented Mar 22 at 22:22
  • $\begingroup$ @henry Thank you! Your answer was exactly the missing piece of the puzzle. I think I had misunderstanding or overexpectations in regards to sufficient statistics, just thought I’m missing some arcane knowledge. To double check if I'm correct in understanding that my sufficient statistics will depend on the target - if I'm going to use the MLE approximation it will be $(\overline{\ln x}, \overline{x})$ and if the closed form approach is $(\overline{\ln x}, \overline{x}\overline{\ln x}, \overline{x})$? $\endgroup$
    – tessob
    Commented Mar 23 at 7:17
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    $\begingroup$ @whuber Yes, I asked about how to estimate the parameters $(a, b)$ of the gamma distribution having sufficient statistic, but now it seems to me that I already know the answer to this question. $\endgroup$
    – tessob
    Commented Mar 23 at 7:22

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