I have a time series of binomial data $y_i$ with a time dependent $p(t)$. The data is generated iteratively, hence $t \in 1,...,T$. Any suggestions on how to estimate $p(t)$? I feel that the naive approach would be to use some sort of smoothing method (e.g. smoothing splines) to the raw data, but this allows for $p(t)$ outside the [0,1] interval.
Note that $p(t)$ seem to have some periodicity so a logistic model would not be suitable. The periodicity is not regular though, so I don't think incorporating some sort of Fourier analysis would work.