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I am trying to check if the noncentral chi-square distribution is log-concave in its noncentrality parameter. Specifically, given

$p(y ; \lambda, \sigma^2) = \frac{1}{2\sigma^2}\exp\left(-\frac{y+\lambda}{2\sigma^2}\right) I_0\left(\frac{1}{\sigma^2}\sqrt{\lambda y_i}\right)$

is $\lambda \rightarrow \log p(y;\lambda,\sigma^2)$ concave? In addition, is $\lambda^2 \rightarrow \log p(y;\lambda,\sigma^2)$ concave?

I think the first is affirmative, but could not find the second in the literature.

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