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I'm interested, purely out of curiosity, in what methods can be used to calculate confidence intervals for discrete integer model parameters.

As an example, consider the model (which I can flesh out with code if needs be)

$$ y \in [0,1]; \\ P(y_i = 1) = \begin{cases} .25 \text{ if } i \lt \theta \\ .75 \text{ otherwise }; \end{cases} \\ i \in [1, 2, \dots, n];\\ \theta \in [1, 2, \dots, n];\\ $$

where the change point $\theta$ is an integer parameter.

I can think of a few approaches one could take here:

  • Bootstrapping
  • Using Bayesian methods, with a Uniform prior, to sample from the posterior $P(\theta | Y) \propto P(Y | \theta)$, and then treat the percentiles as bounds of the corresponding confidence interval. I think this is appropriate, given that for many models credible intervals with uniform priors are equivalent to frequentist confidence intervals.
  • Treat $\theta$ as a continuous variable...
    • ...and use Fisher information to calculate standard errors. This won't work here because $P(Y | \theta)$ is a step function.
    • For other models, it may be possible to calculate continuous (decimal) standard errors (e.g. here). I guess the resulting confidence interval, $\hat \theta \pm 1.96 \times SE$ could be rounded off to integer values.

So, what methods are available for calculating confidence intervals for integer parameters, either for models of this form or in general?

This question might also be relevant.

kjetil b halvorsen
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Eoin
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    Related: https://stats.stackexchange.com/questions/126148/generalization-of-fisher-information-for-a-discrete-parameter – kjetil b halvorsen Jun 18 '22 at 17:32
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    You can calculate a likelihood interval : ${\theta: \log\mathcal L(\theta) \ge \log\mathcal L_{max}-c}$ which is similar to the Bayesian method. Of course this does not guarantee exact coverage, but neither do the other methods. – J. Delaney Jun 18 '22 at 19:07

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