# Confidence interval estimation

Suppose there are two parameters $\theta$ and $\eta$. I have shown that $\theta\in(\eta_{\min},\eta_{\max})$ if statement $A$ holds. Let $\hat{\theta}$ and $\hat{\eta}$ be the MLE's of $\theta$ and $\eta$ respectively. Can I say that if $\hat{\theta}\not\in(\hat{\eta}_{\min},\hat{\eta}_{\max})$, then statement $A$ doesn't hold ?

Maximum likelihood estimators are random variables because they are functions of the sample, which is a random variable. With a test-like approach, the question would be "What is the probability of $\hat{\theta}\not\in(\hat{\eta}_{\min},\hat{\eta}_{\max})$ assuming that statement $A$ does hold?". If that probability is zero, $\hat{\theta}\not\in(\hat{\eta}_{\min},\hat{\eta}_{\max})$ implies that statement $A$ doesn't hold, but if it's different than zero it could just be used to build a test on $A$ or estimate probability of $A$.
• @user143423 That's why I suggested giving more context. Knowing what statement $A$ is, what your parameters are and how your confidence interval is computed is needed to compute probabilities. – Pere Dec 26 '16 at 15:16
Here statement $A$ refers to the specific missing mechanism (NMAR or MAR or MCAR) of a variable in a missing data model for an incomplete contingency table. The parameters $\theta$ and $\eta$ refer to the vectors of fully observed and partially observed cell probabilities respectively in the table. It can be shown theoretically that $\theta\in(\eta_{\min},\eta_{\max})$ always under the model. The intention is to say something about the type of missing mechanism of the variable if the above inclusion doesn't hold for the MLE's of the parameters under the model.