Estimating the time between samples

Question

Let's say that an event happens every $\Delta x$ seconds and we sample the time $x$ with some error. The events are not sampled reliably so some are missing in the sample set. However, we do know the ordinal number $k_i$ of every sampled time $x_i$ so we always know exactly how many samples were skipped in between.

I'd like to estimate $\Delta x$ as accurately as possible and I'd like to calculate error bounds on the resulting value.

My intuitive solution is to calculate the difference between every sample and the first one. Then I'd sum up all of these differences and divide them by the total number of $\Delta x$ included in the sum. In other words:

$$ \Delta x \approx \frac {\sum_{i=1}^n x_i-x_0}{\sum_{i=1}^n k_i-k_0} $$

But I don't know if my intuition is correct and don't have enough knowledge to confirm or disprove it. Also, I have no idea how to calculate the error bounds.

EDIT:

Assume normal distribution for sample time errors. Should be good enough for my use case.

Answer 1

A question about model specification: is the error really iid normal? If so, then the $x_i$ are not necessarily monotic (i.e., it's possible that $x_2 < x_1$), but for many measurement systems that type of error is impossible. But let's ignore that for now.

For practical purposes, probably $$ \Delta x \approx \frac{x_n - x_0}{k_n-k_0}$$ would be fine, since the variance of the error in that estimate is dropping like $k_n-k_0$.

If you want to do more and you really think the error is normal, then this problem maps exactly to linear regression (you're trying to predict $x_i$ from $k_i$); the slope of the fit will give you your $\Delta$.