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Positivity from the comparison between the log expectation and the logarithm of the mean of the gamma-distributed random variable, making use of Jensen's inequality.
Define complete monotonicity.
Asymptotic expansion of the log gamma function.
Complete monotonicity of the residual of the asymptotic expansion.
Complete monotonicity of the log minus digamma function.
Log minus digamma function is convex ($a_{\text{ML}}$ indeed maximizes the likelihood).
Log minus digamma function is bijective and thus invertible.
Numerical solution for the inverse of the log minus digamma function.
Add figure for log minus digamma function.
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The text was updated successfully, but these errors were encountered: