How sensitive is parameter estimation to uncertainty in time?

Suppose I have the following deterministic system that is a function of time:

$$y = k*t + b$$

Now let's say I have the ability to measure this system but there is a zero-mean noise component in the measurement. This is represented by the following:

$$y_m = k*t + b + v_{noise}$$

My goal is to estimate $$k$$ and $$b$$ which is represented by $$\hat{k}$$ and $$\hat{b}$$. I have access to the noisy measurement $$y_m$$ and time $$t$$. Using $$\hat{k}$$, $$\hat{b}$$, and $$t$$ I can predict the output:

$$\hat{y} = \hat{k} * t + \hat{b}$$

My estimate $$\hat{k}$$ is considered optimal if $$(y_m-\hat{y})^{2}$$ is minimized. I have essentially defined a cost function $$J = (y_m-\hat{y})^{2}$$.

--

Up until this point, the uncertainty I've been dealing with has been in the measurement. However, what would happen if there was also uncertainty in my time $$t$$?

How would I quantify how sensitive my estimate is to uncertainty in $$t$$? Obviously my estimate would be terrible if I had an inaccurate clock, but I'm having a difficult time determining how sensitive it would be. I'm also curious on the topic of stability for this. Assuming I could sample forever, would my parameter estimate converge?

• yes. I mean, the obvious estimator for this is anyway simply the time offset-divided-by-observed-value average. The rest is trivial to just write down as formulas! – Marcus Müller Sep 16 '19 at 21:32
• @MarcusMüller I tried to make the example less trivial by adding an offset parameter 'b'. In this case, it seems like observed offset will be dependent on accuracy of the time variable. Let's say for example our reference clock lags a second behind the actual time $t$. It seems like this error would leak into our parameter 'b' estimate. – Izzo Sep 16 '19 at 21:57
• I don't know if it applies here but look at "errors in variables" models. They deal with measurement error in the regressor and $t$ is a regressor in your case. I don't deal with that so I can't give good references. – mark leeds Sep 16 '19 at 23:03
• @markleeds "Errors in variables" is a good term to know, my searches weren't yielding much results before this. – Izzo Sep 17 '19 at 0:25
• Yeah, it's the name of an actual statistical-econometric method from the 50's or 60's, so maybe add "statistics" or "econometrics" to whatever you're using to search for it. This way it won't think you're truly looking for errors in variables. – mark leeds Sep 17 '19 at 2:11