Uncertainty on independent variable of a non linear model I have a question about how to estimate the uncertainty of a value calculated from a non linear model. My non-linear model is the following:
$$
y=A+\frac{B}{1+e^{k(x-x_0)}+e^{c(x-x_1)}}
$$
where $ A, B, k, c, x_0, x_1$ are fitting parameters that I find using a Levenberg–Marquardt algorithm (non linear least square fitting). What I do after is to calculate $x$ at a particular $y$, but I don't know how to estimate the uncertainty on the found $x$. What I have done so far is to calculate the standard deviation of the fitting parameters using the covariance matrix and the root mean square error (RMSE) and I tried to propagate the error, but the problem is that I cannot get an analytical solution for $x$. 
How can I proceed to estimate the uncertainty on $x$ ? Thanks
 A: Use Monte Carlo uncertainty propagation (GUM-Supplement 1):
1/ evaluate the covariance matrix of your parameters (not only the uncertainties; they are probably strongly correlated and this has a HUGE impact on the final uncertainty)
2/ draw random samples from a normal multivariate distribution using the best value of your parameters and their covariance matrix
3/ calculate x for each point of the sample
4/ estimate the mean and standard deviation of x
A: You can use bootstrap to estimate the distribution of the any x. Below is my effort to point out the things to do:


*

*Make a bootstrap sample. That is, take n samples from the original data with replacement, where n is the size of original data. Estimate all the parameters you want.

*Repeat step 1 for a number of times, say b times (The value of b may be 1000, 10000, 100000 etc). That is you make b bootstrap sample and for all the samples you estimate all the parameters. Save values of all the parameters for all b runs.

*Now you have b values of x. These values form a distribution of x. You can compute the mean and variance and translate into confidence interval, if you assume certain theoretical distribution for the parameter. The alternate way would be using empirical percentiles to construct the confidence interval. That is use 2.5th and 97.5th percentile of the distribution of x as a confidence interval of x.
If the variance co-variance matrix is necessary, then that also can be constructed from the the data you have. 
