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A lemma concerning the distribution of sufficient statistic from exponential family

The lemma states that that you can express the probability in that way with respect to a specific measure. You do not have $$dP(t)=\exp\{\eta_1^\top(\theta)t-\xi(\theta)\} \, d t$$ but instead you ...
Sextus Empiricus's user avatar
1 vote

Prove covariance between sufficient statistic and logarithm of base measure in exponential family is equal to zero

The claim is false. As a counterexample, let $f_X(x) = xe^{-x}, x > 0$ (i.e., Gamma(2, 1) distribution) so that $T(X) = X$ and $h(X) = X$. Simulation shows that the covariance of $X$ and $\log(X)$...
Zhanxiong's user avatar
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