I have a sequence of square wave pulses with an uptime following an exponential distribution. I can only observe this pulse train for a set window of time but I can do it repeatedly. Let's assume that the observation windows are spaced far enough in time that there is no correlation between window observations. What is the best way to estimate the mean of the exponential distribution?
During a window of observation, there are 4 possible types of observations:
- One or more pulses are within the window and at the start and end of the window the pulse is in the low state.
- There is a pulse in the high state at the beginning of the window
- There is a pulse in the high state at the end of the window
- The pulse is in the high state for the entire time of the observation window
Both #2 and #3 could be observed in the same window.
I have tried a maximum likelihood method where I included the probability that the edge pulse is the length measured $(x_k)$ or longer, $P(X \ge x_k)=e^{-x_k/\beta}$ which gives me the following likelihood function:
$L(\beta)=\prod_{i=1}^{n}\frac{1}{\beta}e^{-x_i/\beta}\prod_{k=1}^{m}e^{-x_k/\beta}$
where there are $n$ measures of pulses that were completely within the observation windows with a time length of $x_i$ and there are $m$ measures of pulses that extended outside of the observation windows with only $x_k$ of the pulse inside a given window. This gives me the following estimate for $\beta$
$\hat{\beta} = \frac{\sum_{i=1}^{n}x_i+\sum_{k=1}^{m}x_k}{n}$
I have run a Monti Carlo simulation and I have found that this method works better than a naive method of only counting the pulses that were completely within the observation windows. But the method derived here is biased large. ($\hat{\beta}> \beta$). We can assume that all the measurement windows are of the same time length, however, in this derivation, it did not matter.
Is there a better way of doing this? Is there a way to estimate the bias?