I have a list of experimental points
{{0.47229, 155.93, -1}, {0.802892, 121.94, 1}, {1.74747,
117.61, -1}, {2.07807, 159.14, 1}, {3.35326, 165.63, 1}, {4.29784,
165.31, -1}, {4.62844, 182.7, 1}, {5.57302, 177.46, -1}, {5.90362,
169.84, 1}, {6.8482, 187.55, -1}, {7.1788, 180.15, 1}, {8.12338,
176.97, -1}, {8.45398, 168.44, 1}, {9.39856, 199.58, -1}, {9.72917,
132.37, 1}, {11.0043, 203.1, 1}}
where each point $\{r,V,\delta V\}$ represent the radial coordinate $r$, the velocity $V$ and the error in the velocity $\delta V,$ obtained from some experiment.
I want to fit these data points to a function of the form
$$V(r)=V_{0}\dfrac{r/r_{d}+d}{r/r_{d}+1}.$$
Minimizing the $\chi^{2}$ distribution I find the best fitting parameters $(V_{0}, r_{d},d)$ by solving numerically a non-linear system of equations.
How can I determine the errors with which those best fitting parameters are determined? I mean, one should obtain, let's say for example $V_{0}=(100.00\pm x),$ $r_{d}=(1.00\pm y),$ and $d=(0.10\pm z).$ How to obtain $x,y,z$?
Given a set of data $\{r_{i}, V_{i},\delta V_{i}\}_{i\in I}$ and a fitting function $f(r,a_{j})$ with independent variable $r$ and free parameters $a_{j},$ the $\chi^{2}$ function is defined by: $$\chi^{2}(a_{j})=\sum_{i\in I}\left(\dfrac{V_{i}-f(r_{i},a_{j})}{\delta V_{i}}\right)^{2}.$$
Minimizing $\chi^{2}(a_{j})$ with respect to the parameters $a_{j}$ one obtains the best fitting values of the model. To estimate the errors in the fitted parameters $\sigma_{j}$, one has to analyze the curvature of the function $\chi^{2}(a_{j})$ around the minimum. This is, the errors $\sigma_{j}$ can be computed through $$\sigma_{j}^{2}=2\left(\dfrac{\partial^{2}\chi^{2}}{\partial^{2}a_{j}}\right)^{-1}.$$