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A statistics $T(X)$ is sufficient statistics for $\theta$ if the conditional distribution of the sample $X$ given the value of $T(X)$ does not depend on $ \theta$.

( this is the definition of sufficient statistics from casella and berger statistical inference)

My question is why should the conditional distribution of sample given statistic $T(X)$ not depend on $\theta$ in order for $T(X)$ to be sufficient?

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This is a great question. Here is my opinion. Let's review the definition of a statistic, it is "A statistic (singular) or sample statistic is a single measure of some attribute of a sample (e.g. its arithmetic mean value)". So, it means a statistic only depends on the data, not parameters.

Sufficient statistics help statisticians to remove all the information of $\theta$ from the data. Personally, I always think this removal process as replacing process. Since we have $T(X)$, it is sufficient enough that we no longer need $\theta$ anymore.

Hope this helps.

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  • $\begingroup$ so sufficient statistics is the information that you can get about the data without involving the parameter and that is why the condition is to not depend on the parameter. Am I right? $\endgroup$ – Dravid Hemanth Apr 10 at 11:20
  • $\begingroup$ Spot on!! Later you might encounter complete and ancillary statistics, knowing all three might help you to understand why we need sufficient statistics more. $\endgroup$ – Newbornalive Apr 10 at 23:19
  • $\begingroup$ I forgot about the minimal sufficient statistics. So, 4 definitions along with the Basu Theorem might give you a general idea of how and why statisticians use sufficient statistics and more. $\endgroup$ – Newbornalive Apr 11 at 0:29
  • $\begingroup$ Thank you so much for answering will look into Basu Theorem. :D $\endgroup$ – Dravid Hemanth Apr 11 at 4:23

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