Confidence interval of inverse model parameter Let assume that we have a model from which we estimate a parameter with physical meaning, such as a thermal resistance, $R = 0.1$. The unit of the $R$ is kelvin per watt $ \left[ \frac{K}{W}\right]$, and has a standard error of SE $= 0.07$, which gives us a confidence interval of $[-0.037, 0.237]$. So, we say that $R$ is not significantly different from $0$.
If I now want the used the inverse value of $R$, namely the heat transfer coefficient, $Q$, with the unit watt per kelvin $ \left[ \frac{W}{K}\right]$, what is an appropriate way of calculation the confidence interval of $Q$ now?
It is clearly not as simple as calculating the inverse of the confidence interval above as I then get a confidence interval of $[-27.0, 4.2]$ and an estimate of $Q$ which is outside the confidence interval $ \left(10 \frac{W}{K} \right)$.
As negative values of $R$ does not make sense anyway, one could say that the upper bound of the confidence interval for $Q$ cannot be calculated but the lower bound of $Q$ will be $4.2 \frac{W}{K}$. Is that correct?
Any suggesting to how to deal with this is appreciated.
 A: Here's an interesting thing that is usually true. For some function $f(x)$, 
$$f\left[\text{Var}\left(x\right)\right] \ne \text{Var}\left[f\left(x\right)\right]$$ 
unless $f(x) = x$.
This means that frequentist inference on a transformed variable using confidence intervals of, for example, the form $\theta \pm \theta^{*}_{\text{CL}}s_{\theta}$ or hypothesis tests of, for example, the form $\theta \div s_{\theta}$ does not provide inference of the untransformed variable (and vice versa).
For example, it is possible for any of the following to be true:


*

*CI of $R$ spans 0, and CI of $Q$ spans 0.

*CI of $R$ spans 0, and CI of $Q$ does not span 0.

*CI of $R$ does not span 0, and CI of $Q$ spans 0.

*CI of $R$ does not span 0, and CI of $Q$ does not span 0.
(Where $Q = R^{-1}$.)
The thing to do is to decide whether you care about inference on $R$ or inference on $Q$ (or both), and to calculate their CIs (and/or test statistics) separately.
A: It is hard to work with nonlinear function of the parameters. Generally, we need to use linear or polynomial function to approximate the original function using such as Taylor series. Then you can try to get the approximation of variance from this linear or polynomial function.  
A: A quick and dirty way to handle this would be to use a crude form of the delta method: 


*

*translate the estimate and the standard errors on to the log scale (using $\log(\hat R)$ for the estimate, $\hat \sigma(f(.)) \approx \hat \sigma \times \left. \frac{df}{dx} \right|_{R=\hat R} = \hat \sigma \times 1/\hat R$ for the standard error)

*compute symmetric confidence intervals on the log scale ($\log(\hat R) \pm 1.96 \hat\sigma_{\small log}$)

*back-transform ($\exp$) these CI to the original scale, which will make them asymmetric (but sensible)


In this picture, the black dots and ranges show the estimate $\pm$ 1 SE on the original scale, transformed to the log scale; the red dots and ranges show the estimate $\pm$ 1 SE on the log scale, back-transformed to the linear scale (trying to show $\pm$ 1.96 SE on the original scale would lead to nonsensical ranges, i.e. including negative values ...)

