# Solving an equation with logarithms of negative numbers

I'm modelling a bunch of non-homogeneous poisson processes. I'm impossing on them a linear-log functional form. you end up with two parameters, theta1 and theta0 for each estimated process.

The standard way to do it is to first estimate theta1, as it's needed to solve theta0. However, for some of my processes theta1 is negative, and the equation for theta0 includes a logarithm of theta1.

$$\hat{\theta}_{0}=\log n+\log \hat{\theta}_{1}-\log \left[\mathrm{e}^{\hat{\theta}_{1} T}-1\right]$$

What can one do to solve for theta 0 for negative values of theta 1?

• If $\theta_1 \leq 0$, your functional form is clearly incorrect. If, on the other hand, you know that $\theta_1 > 0$ but your estimate $\hat{\theta}_1 \leq 0$, then you should constrain your estimate so that it is $> 0$. How much $>0$ becomes the open question! – jbowman May 11 at 17:56