*Bounty: 50*

I’m estimating parameters for a complex, “implicit” nonlinear model $f(mathbf{x}, boldsymbol{theta})$. It’s “implicit” in the sense that I don’t have an explicit formula for $f$: its value is the output of a complex fluid dynamics code (CFD). After NLS regression, I had a look at residuals, and they don’t look very normal at all. Also, I’m having a lot of issues with estimating their variance-covariance matrix: methods available in `nlstools`

fail with an error.

I’m suspecting the assumption of normally distributed parameter estimators is not valid: thus I would like to use some nonparametric method to estimate confidence intervals, $p$-values and confidence regions for the three parameters of my model. I thought of bootstrap, but other approaches are welcome, so long as they don’t rely on normality of parameter estimators. Would this work:

- given data set $D={P_i=(mathbf{x}_i,f_i)}_{i=1}^N$, generate datasets $D_1,dots,D_m$ by sampling with replacement from $D$
- For each $D_i$, use NLS (Nonlinear Least Squares) to estimate model parameters $boldsymbol{theta}^*_i=(theta^*_{1i},theta^*_{2i},theta^*_{3i})$
- I now have empirical distributions for the NLS parameters estimator. The sample mean of this distribution would be the bootstrap estimate for my parameters; 2.5% and 97.5% quantiles would give me confidence intervals. I could also make scatterplots matrices of each parameter against each other, and get an idea of the correlation among them. This is the part I like the most, because I believe that one parameter is weakly correlated with the others, while the remaining are extremely strongly correlated among themselves.

Is this correct? Then how do I compute the $p-$values – what is the null for nonlinear regression models? For example, for parameter $theta_{3}$, is it that $theta_{3}=0$, and the other two are not? How would I compute the $p-$value for such an hypothesis from my bootstrap sample $boldsymbol{theta}^*_1,dots,boldsymbol{theta}^*_m$? I don’t see the connection with the null…

Also, each NLS fit takes me quite some time (let’s say a few hours) because I need to run my fluid dynamics code $ptimes N$ times, where $N$ is the size of $D$ and $p$ is about 40 in my case. The total CPU time for bootstrap is then $40times N times m$ the time of a single CFD run, which is a lot. I would need a faster way. What can I do? I thought of building a metamodel for my CFD code (for example, a Gaussian Process model) and use that for bootstrapping, instead than CFD. What do you think? Would that work?

Get this bounty!!!