# Nonlinear regression standard error for two parameters?

The equation is

$$\frac 1 V = \frac{1}{g\cdot A} + \frac 1k.$$

My experiment has values for $$V$$ and $$A$$, and I used nonlinear regression in Excel to estimate $$g$$ and $$k$$. Now I need the standard error for $$g$$ and $$k$$, but I have no idea how to do that. I need to know $$g \pm \textrm{something}$$. I have $$g$$, but not the error for $$g$$. My lab uses a black-box software that spits out values, but the software does not use an equation in the form above. I would appreciate how to do it in Excel, if possible.

• Note that you can do a linear regression if you use $y = \frac{1}{V}$ and $x = \frac{1}{A}$. Regarding excel, I'm sorry but I don't know how to get the standard error using it. – Ertxiem Apr 30 at 15:01
• Just take care about an important point : what is measured is $V$ and not $\frac 1V$. Then, when you have obtained estimates of $g$ and $k$ by linearization, you must fit the model as $V=\frac{A g k}{A g+k}$. – Claude Leibovici May 1 at 8:14
This is a linear regression $$y=mx+c$$ with $$y:=1/V,\,m:=1/g,\,x:=1/A,\,c:=1/k$$. You won't find it hard to find, with either formulae or a software solution, the standard errors $$\sigma_m,\,\sigma_c$$ of $$m,\,c$$. The last step you need is $$\sigma_{f(z)}=|\partial_z f|\sigma_z$$ for differentiable functions $$f$$ of a random variable $$z$$, so $$\sigma_g=\frac{\sigma_m}{m^2}=g^2\sigma_m$$. Similarly, $$\sigma_k=\frac{\sigma_c}{c^2}=k^2\sigma_c$$.