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Context: I have binary data $x_{it}\in\{0,1\}$ where $i\in\{1,...,N\}$ indexes trials and $t\in\{1,...,T\}$ indexes time (independent across trials; not independent across time). It's from a simulation I ran ($N$ times, over $T$ periods for each) and I don't have an explicit description of the stochastic process $\{x_{\cdot t}\}_{t=1}^T$ follow.

My question: I visualized the frequency that $x_{\cdot,t}$ always stays at 1 after time $t$, plotted for each $t$. I would also like to plot a "confidence band," but I don't know how/if this can be done.

My work so far: for each $(i,t)$, $S_{it} = \prod_{\tau=t}^{T}x_{i\tau}$ codes the event that $x_{i\tau}=1$ for all $\tau\geq t$. I plotted $\frac{1}{N}\sum_{n=1}^N 1\{S_{\cdot t}=1\}$ over each $t$ to visualize the frequency that $x_{\cdot t}$ stays at 1 after time $t$. (Is this proper?) Regarding visualizing a confidence band, I've tried looking on guides on how to do this for "survival curves" (since what I am drawing seems related to that) but I don't know if they are applicable (and even so, I am not entirely sure how to translate the methods I have found so far to my problem). Any guidance would be very much appreciated.

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  • $\begingroup$ You might find this discussion on forecasting binary data helpful. stats.stackexchange.com/q/15068/9568 $\endgroup$
    – Jessica Collins
    Commented Oct 19 at 13:51
  • $\begingroup$ You say that you "* don't have an explicit description of the stochastic process*". But you are starting with a Bernoulli process $B(1,p)$ with p unknown. Are you then also implying that $p$ is not constant over time? (i.e. $p$ is different at each time $t$?) Or that even for a given time $t$, p varies accross the $N$ samples? Some clarification would be helpful. Also0, since ytou visualized your data, it would be nice if you could attach the graph to your question. $\endgroup$
    – jginestet
    Commented Oct 19 at 20:01

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