1
$\begingroup$

I wish to find the distribution of $\exp(-(\exp(X))$, when $X$ is normally distributed. The first steps are straightforward: $$X \sim \operatorname{Normal}(\mu,\sigma^2),$$ $$ Y = \exp(X) \sim \log \mathcal{N}(\mu,\sigma^2) $$ However, I now want to find the distribution of a negative log-normal distribution: $Z = -Y$, and in particular, the distribution of the exponential of this distribution, but I am completely clueless how to tackle this: $$Z = \exp(-Y) \sim ???$$ I have found that $-Y$ might be done using some distributions in the Johnsons family, but I could not find how to do it. Any help would be appreciated!

  • 0
    Is this 'how to' question different to finding any other transformation? In other words, you are asking about how to transform a Lognormal, but in what way would the answer be different than for any other transformation? And at what part of that process did you get stuck? – 2017-01-12
  • 0
    Thanks for your reply. I want to know whether this specific transformation of the lognormal distribution is an already existing distribution. The problem is that I cannot find this anywhere. If this is not possible I would probably have to try to perform the transformation myself – 2017-01-12
  • 1
    But the question asks 'how to' find the transformation, and my point is that the process (steps) is not different to any other fairly simple univariate transformation. Your reply suggests that you are less interested in 'how to', and more interested in the final answer ... which is not really a 'how to' question. The answer to your "how to' question is, for example, use the "Method of Transformations / Jacobian method" – 2017-01-12

1 Answers 1

0

I think it should be easy enough to find the cumulative distribution function

$\Pr(Z \le z) = \Pr(X \ge \log_e(-\log_e(z))) = 1 - \Phi\left(\dfrac{\log_e(-\log_e(z))-\mu}{\sigma}\right)$ for $0 \lt z \lt 1$

and then the density

$p(z) = \dfrac{1}{-\sigma z \log_e(z)}\phi\left(\dfrac{\log_e(-\log_e(z))-\mu}{\sigma}\right)$ for $0 \lt z \lt 1$

based on the cdf $\Phi$ and pdf $\phi$ of a standard normal distribution.

Sketching the density for varying $\mu$ and $\sigma$ suggests it produces some slight odd results, especially varying $\sigma$ between $0.1$ and $1$ so I might doubt it has a commonly used name