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I have a simple Tweedie model in GLM, or $Y_i=\beta_0+\beta_1x_1$. As usual, $Y$ is the output and $x_i$ is its dependent variable. Suppose that $p>1$, my link function would be $\frac{\mu^{1-p}}{1-p}$.

I thought this would be a simple exercise of $\frac{\mu^{1-p}}{1-p}=\beta_0+\beta_1x_1$. However, I can't solve for $\mu$ since I can't take natural log of a negative value. I'm wondering if anyone ran into this situation before.

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    $\begingroup$ At which point are you taking the logarithm of a negative value? $\endgroup$ – Sextus Empiricus Jul 8 at 20:29
  • $\begingroup$ In my model, all $\beta$s are positives. My $p$ is 1.5 since $p>1$. After I moved $1-p$ to the right side, I realized I cannot take $ln$ of both side. $\endgroup$ – tkhu Jul 8 at 21:00
  • $\begingroup$ Why are all $\beta$s positive? If the left side is negative due to that $1-p$ term then just make the right side negative as well. If you don't like that, then get rid of the $1-p$ term. $\endgroup$ – Sextus Empiricus Jul 9 at 1:19
  • $\begingroup$ My $\beta$s were positive because that's what the rxGLM summary results gave me. Turns out the package also got rid of $1-p$ on the left too, so I think $1-p$ is already baked into the estimated values. $\endgroup$ – tkhu Jul 9 at 2:56
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I was able to answer my own question. I was using rxGLM in R with Tweedie distribution. The code indicates that "Family-link: Tweedie-mu^-0.5 ". Thus, the link function is actually $\mu^{-0.5}$.

It is then simply $\mu^{-0.5}=β_0+β_1x_1$. Flipping side gives $\mu^{0.5}=\frac{1}{β_0+β_1x_1}$. Finally the answer is $\mu=\frac{1}{(β_0+β_1x_1)^2}$. I checked this by hand and it works!

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