I have a statistical problem I've been grappling with, and I'd appreciate some guidance on how to approach it. Here's the scenario:
I have a binary variable, let's call it $x$, which can be in one of two states: 0 or 1. The duration of state 1 can vary over time, but state 0 has a constant, unknown duration, which we'll call $T$, and I want to estimate it. To make this estimation, I've been employing the following process:
At random time points, denoted as $t_i$, I sample the variable $x$. If $x$ is in state 0 at $t_i$, I resample the signal after a time interval $\delta$, and I save the value recorded during the state transition at $t_i + \delta$. So, my dataset consists of a sequence of 0s and 1s corresponding to the values sampled within the intervals $t_i + \delta$.
My question is: How can I infer the value of T using this approach? Can I provide a maximum likelihood estimate (MLE) for T?
Here's a brief outline of my thoughts so far:
Define the model parameters: $T$ (the duration of state 0), $\delta$ (the time interval between samplings), and $p$ (the probability that $x$ is in state 0 at $t_i$).
Calculate the likelihood of the observed data, which involves computing the probabilities of observing specific sequences of 0s and 1s given $T$, $\delta$, and $p$.
Maximize this likelihood with respect to $T$ to obtain the MLE for $T$.
I suspect that the likelihood have some kind of exponential form; however, this process seems to become quite different than estimating the time elapsed between two Poisson events with rate $\lambda$.
Additionally, I'm aware that the MLE could be sensitive to the distribution of the time spent in state 1, which I've mentioned is variable.
I'd appreciate any insights, suggestions, or alternative approaches you might have to help me tackle this problem effectively. Thank you in advance for your assistance!