For a $\chi^2$ goodness-of-fit test where a parameter is estimated (for example, $p$ is estimated if you're testing to see if the Binomial distribution is a good fit to the data), is it true that if $H_0$ is rejected with this "best guess" estimate of the parameter, then the hypothesis test will be rejected for every other possible value of the parameter (I imagine such a value is likely to be near the best guess)? I suspect not, although I can't find a specific numerical example.
So my first question is: can someone show a numerical example where the best estimate of the parameter causes us to reject $H_0$ but a different value of the parameter causes us to accept $H_0?$ Assuming the answer is yes, my second question is: Is this significant? Because what I have been taught is that we choose the "best guess" parameter in our significance test. But if a (slightly) different value of the parameter causes us to accept $H_0$ rather than reject it, then this kind of a significant fact, right?
Edit: the other distributions other than Binomial I am mainly interested in asking this question about are: Poisson, Negative Binomial and Geometric.