# Re-express the lognormal density in terms of its mean value

I wonder if it is possible to rewrite the lognormal density function, $$f(x)=\frac 1 {x\sigma\sqrt{2\pi}}\exp\bigg(-\frac{(\log x-\mu)^2}{2\sigma^2}\bigg)$$ so that the mean value, $\exp(\mu+\frac{ \sigma^2}2)$ is part of the expression? I want this since I'm testing the hypothesis $$H_0: \exp\bigg(\mu+\frac{ \sigma^2}2\bigg)=1$$ and I want to formulate the model that corresponds to this $H_0$.

• Your hypothesis is identical to $\mu+\sigma^2/2=0$. Call its left side $\tau$. What is to prevent you from, say, replacing all occurrences of $\mu$ in $f$ by $\tau-\sigma^2/2$? This is one of infinitely many possible reparameterizations. The difficult (and far more interesting) aspect of this question concerns finding a second parameter $\nu$ in addition to $\tau$ so that $(\tau,\nu)$ completely determines $(\mu,\sigma)$ and estimates of $\tau$ and $\nu$ are approximately independent. This depends on the nature of your data and your estimation procedure. Could you describe those?
– whuber
May 18, 2017 at 13:20
• That's true, thanks! Actually the thing I want to, is to derive the score test statistic and the wald test statistic and compare them. I know they are asymptotically equivalent but I would like to see how they differ in their expressions. May 22, 2017 at 12:10