I have a binary variable, $X_t$, which takes values $1$ or $0$. A value of $1$ indicates that an event happend at time $t$.
I would like to statistically test whether we can say that the event happens $\alpha \cdot 100 \%$ of the time. This is somewhat similar to the Kupiec test from the theory in Value-at-Risk backtesting, but my setting is a lot less specified and relies on basically no assumptions.
As an example, consider the following setup: We have an observed vector of $X_t$: $X=[X_1, X_2, \dots, X_T]$. We want to know whether the event that $X_t = 1$ happens 5% of the time. If $T=1000$ and $sum(X)=62$, we could be compelled to decline the 5% hypothisis, since $62 \neq 50$. However, I want a test that takes statistical insecurity and potential finite sample properties into account.
I have considered some t-test-like approach, but I am unsure about the disitrbutional properties, and how to estimate the standard deviation in the denominator.
Additonally, I would like to test whether the occurence of the events is independent. Essentially, I would like to know whether an event happening at time $t$ affects the probability of an event happning at time $t+1$. For this, I have considered a simple autoregressive model, but I am unsure about this, as I have not seen anyone do this before.
Anwsers to one of both questiosn are more than welcome!