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 ?
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2 Answers
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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 $A$.
A little more of context could be useful to give a more helpful answer.
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$\begingroup$ Thanks for the reply. How do I obtain the probability that \hat{\theta}\not\in(\hat{\eta}_{\min},\hat{\eta}_{\max}) assuming statement A holds ? In confidence interval estimation, we usually test whether some population parameter belongs to a confidence interval formed by sample statistics with a certain probability. There is some associated test statistic (involving the parameter and its estimate) which follows a certain distribution that is used to contstruct the confidence interval for the parameter. Here, the context is different. $\endgroup$ Commented Dec 26, 2016 at 12:56
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$\begingroup$ @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. $\endgroup$– PereCommented Dec 26, 2016 at 15:16
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$\begingroup$ The question could also have something to do with properties of MLEs but that is not evident from the question as currently posed. $\endgroup$ Commented Dec 26, 2016 at 15:32
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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.
