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I know that $$\log(\hat{\psi}) \sim N\left(\log(\psi), \frac{1}{n_{11}} + \frac{1}{n_{12}} + \frac{1}{n_{21}} + \frac{1}{n_{22}}\right),$$ so what's the distribution of $\hat{\psi}$?

Note that $\displaystyle\hat{\psi} = \frac{n_{11}n_{22}}{n_{12}n_{21}}$

I tried using the $\delta$-method, and found that $$\hat{\psi} \sim N\left(\psi, \frac{1}{n_{11}} + \frac{1}{n_{12}} + \frac{1}{n_{21}} + \frac{1}{n_{22}}\right).$$ Is this correct?

Robert Z
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Adrian
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  • So $\hat {\psi} $ is both log-normal and also non random (since it is a function of some constants)? – Math-fun Dec 07 '16 at 07:06

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