One day I gave a $95\%$-confidence interval to a requester who know nothing about statistics. He asked me what does it mean. Roughly, I answered "The population parameter is inside the interval $95\%$ of times". Then he asked "And when it is outside, can it be far from the interval ?".
I said no but I was a little confused. What do we know about this ? It would be interesting for instance to know the distribution of the distance from $\theta_0$ to the confidence interval conditionally to the event on which $\theta_0$ is not inside the interval. Moreover could we identify some characteristics of the data sample under which the true parameter is more likely to be to the left or to the right of the confidence interval when it is outside ? (for instance the skewness of the data sample in the case of the confidence interval about a Gaussian mean).