# An unbiased estimator of the ratio of two regression coefficients?

Suppose you fit a linear/logistic regression $g(y) = a_0 + a_1\cdot x_1 + a_2\cdot x_2$, with the aim of an unbiased estimate of $\frac{a_1}{a_2}$. You are very confident that both $a_1$ and $a_2$ are very positive relative to the noise in their estimations.

If you have the joint covariance of $a_1, a_2$, you could calculate, or at least simulate the answer. Are there any better ways, and in real-life problems with a lot of data, how much trouble do you get in for taking the ratio of estimates, or for taking a half-step and assuming the coefficients are independent?

• In logistic regression as described, how do you find an unbiased estimator of $a_0$ or $a_1$? The problem is unrelated with the correlation between the coefficients. Sep 18 '16 at 13:54
• Something to ponder: What if one or both of the coefficients were zero? Sep 18 '16 at 19:59
• Yeah, good point. I'm implicitly assuming that both coefficients are sufficiently positive that there's no danger of noise leading to crossed signs (re: andrewgelman.com/2011/06/21/inference_for_a). I'll edit. Sep 18 '16 at 22:33
• How precisely do you estimate $a_1$ and $a_2$ in your regression? Is a consistent estimator with small standard errors sufficient? Is it important that your estimator is unbiased? Would it work for your application to just take $\frac{\hat{a}_1}{\hat{a}_2}$ and calculate the standard-error for that using the delta method and the estimated covariance matrix for $(a_1, a_2)$ from your regression. Sep 18 '16 at 23:07
• Have you considered Fieller's theorem? Look here: stats.stackexchange.com/questions/16349/… Sep 19 '16 at 21:36

I would suggest doing error propagation on the variable type and minimize either the error or relative error of $\frac{a_1}{a_2}$. For example, from Strategies for Variance Estimation or Wikipedia
$f = \frac{A}{B}\,$
$\sigma_f^2 \approx f^2 \left[\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} \right]$
$\sigma_f \approx \left| f \right| \sqrt{ \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} }$
As a guess, you probably want to minimize $(\frac{\sigma_f}{f})^2$. It is important to understand that when one does regression to find a best parameter target, one has forsaken goodness of fit. The fit process will find a best $\frac{A}{B}$, and this is definitively not related to minimizing residuals. This has been done before by taking logarithms of a non-linear fit equation, for which multiple linear applied with a different parameter target and Tikhonov regularization.