# How to obtain simultaneous confidence intervals for correlated frequencies?

I observed 400 episodes of nursing care in a hospital. I tracked the movement of the nurses between 5 rooms $A,B,C,D$ and $E$. The maximum likelihood estimates of the probabilities of moving from room $i\rightarrow j$ are given by:

$$P_{ij}=\displaystyle \dfrac{\text{# of times from room i\rightarrow j}}{\displaystyle \text{Total # of transitions to any room}}$$

• Is there a way of defining a confidence interval on this maximum likelihood estimate $P_{ij}$?
• And for all maximum likelihood estimates of all possible room combinations ($n=25$ combinations)?

This is confusing me because the $P_{ij}$ where $i=A..E$ and $j=A..E$ are dependent on each other. How can I account for this?

Reference:

I have come across this reference: http://arxiv.org/pdf/0905.4131v1.pdf This suggests that for n observations $X_i$, the empirical maximum likelihood estimate $\hat P_{ij}$ minus the actual probability $P_{ij}$ would tend to a normal distribution with mean 0 and standard deviation $\epsilon$.

$$\sqrt{n}|\hat{P_{ij}}-P_{ij}|\sim N(0,\epsilon)\quad \text{as}\quad n\rightarrow \infty$$

How do I calculate $\epsilon$?

• I think you have may have misinterpreted this paper. Its Theorem 2.1 makes it clear that this is a multivariate Normal distribution and explicitly gives all variances and covariances. From those you can immediately obtain asymptotic joint confidence regions.
– whuber
Commented Jun 30, 2013 at 17:28
• @whuber Thank you for looking through that paper. Could you please clarify your last statement? I'm unsure of how I can deduce the joint confidence regions from that for my sample data. Finding the MLE is easy enough but the CI is puzzling me? I'd really appreciate your input on this
– HCAI
Commented Jun 30, 2013 at 17:58
• What do you want the confidence interval for? It is difficult to present/plot/appreciate a five-dimensional confidence region! Commented Mar 21, 2021 at 21:30