I think this might be a basic scenario but I still struggle to find an appropriate methodology for my problem. Consider the following scenario:

There is a small moving object on an $xy$-millimeter-grid paper. I have a measurement system $A$ trying to estimate the $xy$ position for example by using IMU measurements. My "ground truth" measurement system $B$ could be a downwards facing laser mounted on the object in combination with me reading the laser point positions from the grid paper.

Now I collect measurements at $100$ timestamps with both system $A$ and system $B$. To estimate the uncertainty for system $A$ calculate the error between the $xy$ position estimated by $A$ and the "ground truth" reading from $B$. Over the 100 samples I can calculate the mean error $\mu$ and the std $\sigma$, for example $\mu=8.4\,mm$ and $\sigma=5\,mm$. Then I can use for example a $t$-distribution to calculate a $k=2$ confidence interval with which I can characterize the estimated true mean of system $A$'s measurement error, e.g. $\mu_{est}=8.4\,\pm\,3.5 \, mm$.

What if I want to take into account inherent uncertainties from my "ground truth" system $B$ into that confidence interval? For example, what if I know that the laser has a specific uncertainty or what if I know that my readings from the grid paper have a certain deviation as the laser point is at least $1\,mm$ thick? How would I include such knowledge into the confidence interval of measurement system $A$ in order to give a more precise uncertainty estimation?


1 Answer 1


Below I will use $X$ and $Y$ to denote random variables related to systems $A$ and $B$. You could frame the problem in terms of a matched pairs test, a two-sample test, or a prediction. That is, you could think in terms of


and perform inference on $\Delta$, the mean difference, where $\hat{\sigma}_{X-Y}$ is the estimated standard deviation of $X-Y$. You could also think in terms of


and perform inference on $\theta$, the difference in means, where $\hat{\text{se}}_{\bar{X}-\bar{Y}}$ is the standard error of $\bar{X}-\bar{Y}$. Lastly, you could assume the two random variables have the same mean and think in terms of




These quantities allow you to perform predictive inference (predictive p-values, prediction intervals) for hypotheses around a future $\bar{Y}=\bar{y}$ based on an observed $\bar{X}=\bar{x}$ or predictive inference on a future $Y_i=y_i$ based on an observed $X_i=x_i$. These aren't the only pivotal quantities for inference and prediction, but they should look familiar and account for the uncertainty in having measured both $X$ and $Y$.

I hope this helps! See my papers on confidence distributions and prediction intervals.


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