Confidence interval for the ratio of two Gaussian means

Question

I have data set of $\sim\! 125$ points of a certain variable $A$ that goes approx. proportional with variable $B$. The quotient $A/B$ is approx. independent of $B$; however the distribution is not very Gaussian as it has some tail at low values of $A/B$, but still not very "pathological". I calculate the mean: $<A/B_{data}>=X$, which should be approx. Gaussian as it is the average of more than 100 data points. I also know the standard deviation of the mean: $\sigma[<A/B_{data}>]=\sigma[X]$.

On the other hand, I have a theory that predicts a certain behavior (continuous function) of A with B, and I calculate $<A/B_{theory}>=Y$ by integration in the same range of $B$ values of my data sample. This random variable is approx. Gaussian too, and I know $\sigma[<A/B_{theory}>] = \sigma[Y]$.

Because there is an important mismatch between measurements and theory, I want to quantify it calculating the ratio: $R=X/Y$. I know that, even when $X$ and $Y$ are Gaussian, $R$ follows the "ratio" distribution.

I want to calculate a likelihood $68\%$ confidence interval for the ratio $R$. This is what I tried:

1 - Fieller's theorem, as explained in this question, seems close to giving me an answer, but because $Y$ is not the average of a certain number of measurements $N$, but comes from an integral, I don't know how to apply it to this case. Can I "assume" a certain $N$?

2 - I tried expressing the variance of the ratio distribution as a function of the variances of $X$ and $Y$ but I failed.

3 - I tried a "maximum likelihood" method, but the problem is that I have only "one measurement" of $R$, so I'm not confident that what I did is appropriate, and I found in no book any similar procedure.

Furthermore: I know that if $0<\sigma[Y]<<Y$ then it is reasonable to approximate: $(\sigma(R)/R)² \approx (\sigma(X)/X)² + (\sigma(Y)/Y)²$. However, for some of the theories I test, the assumption $0<\sigma[Y]<<Y$ is not valid, so I want to avoid it. Finally, I am looking for an analytic solution, not a numerical solution like bootstrapping for example.

Answer 1

As I re-read this question, I am not sure of some things...

• Is it true that the dataset has values of variables $A$ and $B$, occurring in pairs $(A_i, B_i), i=1,2,\ldots,175$?

• If so, is it true that $A/B_{data}$ denotes the set of values of $A_1/B_1, A_2/B_2, \ldots, A_{175}/B_{175}$?

• Is it true that the notation $< Z >$ is meant to denote the mean of a variable $Z$?

The above three assumptions are the only way I can make sense out of the question. If I am right, then my first suggestion is definitely to construct a scatterplot of the data -- points plotted with the $B_i$ values on the vertical axis and the $A_i$ values on the horizontal axis. Look at the pattern of points. If they scatter around a straight line, then your idea that $A$ and $B$ are approximately proportional, if that line goes through the origin.

If the point pattern is not fairly linear, that would explain why the theory and the observations don't match. If it is fairly linear, you can fit a linear regression model:

$$ A_i \approx c + d\cdot B_i $$

Look at the usual statistical tests for the regression coefficients. If the intercept $c$ is statistically close to $0$, then the slope $d$ and its confidence interval give you kind of an estimate of the $A/B$ ratio.

The above regression approach is really conditional on the $B_i$. Treating the $B_i$ as random is what is called an errors-in-variables problem. But before digging into all that, the scatterplot I suggest is nearly essential for understanding what is going on in the relationship between these variables.