# How are regression parameter confidences found *for each parameter*?

I strongly feel that this question has been answered or explained somewhere, however, I'm struggling to find the correct terminology and resources to learn about this, and I believe this post may have utility in that others with the same issue will find it.

I have some model that has physically meaningful parameters $$k_f$$ and $$M_f$$:

$$f(t) = M_f[1 - \frac{8}{\pi} \cdot \sum^{\infty}_0{(\frac{1}{-(2n+1)^2}\cdot e^{k_f \cdot t})} ]$$

I can fit this model to a few thousand data points by minimizing the RSS as target. At this point, though, I have $$k_f$$ and $$M_f$$ without any idea about how accurate they are to the data!

Normally, with this many data points, I would apply bootstrap resampling and get a standard error this way, and this part I (mostly) feel that I understand. However, if the model just grossly cant describe the data, the error will still be low. In the most extreme case, some made up model like $$f(t) = 19 + M_f + k_f$$ would still get a very low error because bootstrap resampling doesn't change the fittings at all (all fittings are pretty much equally bad!).

I've read a bit about methods that weight the fittings to residuals or otherwise try to use them to calculate parameter error (and aren't monte carlo methods like bootstrap), but I hardly understand them. From these, it sounds like what I'm trying to do is a regression residual analysis. Is this correct? In other words, I'd like to be pointed towards introductory reading or an explanation on how to do this kind of regression analysis geared towards someone with experience in math topics like differential equations but not statistics.

• The error of the model or of the parameter estimates? – Michael M Jun 19 at 6:31
• Of the parameter estimates! Sorry for the ambiguity. If the parameter estimate errors are known, could you not simply plug them back into the model with linear error propagation to get the error of the model? – Will Jun 19 at 19:57