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Or in other words, is there anyway prove that the t distribution doesn't belong to the exponential family without going through all that calculation? Since the density has the gamma function in it which is an integral, I figured that the calculation would be too complicated

Also, another question, if we prove that a density doesn't have a sufficient statistic ( other than the trivial one), would that be enough to prove that it doesn't belong to an exponential family?

Hijaw
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    The Gamma function appears only as a constant multiplier, so you don't need to analyze it. BTW, many functions are defined as integrals and most functions can be so defined, so don't let a definition get in the way of understanding a function. – whuber Mar 24 '21 at 11:53
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    The second question is linked with the [Pitman-Koopman-Darmois lemma](https://xianblog.wordpress.com/2017/11/15/darmois-koopman-and-pitman/) – Xi'an Mar 24 '21 at 13:46
  • Duplicate of frst Q: https://stats.stackexchange.com/questions/444468/is-the-t-distribution-a-member-of-the-exponential-family – kjetil b halvorsen Mar 24 '21 at 15:43

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