Bulletin Editor Anirban DasGupta sets this problem. Student members of the IMS are invited to submit solutions (to bulletin@imstat.org with subject “Student Puzzle Corner”). The deadline is October 23, 2016.

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It is the turn of a problem on statistics this time. This is the sixteenth problem in this problem series, and we must have had a good time, because we have already come to the end of three years since the series started. Here is the exact problem, the final one in 2016, and this one is going to be a pretty good teaser:

Let $X_1, X_2, \cdots , X_n \stackrel {iid} {\sim} F$, where $F(.)$ is a CDF on the real line; $F$
is assumed to be unknown. Assume that $F$ has a density $f(x)$, that $E_F(X^2) < \infty $, denote $E_F(X)$ by $\mu $, and assume that $f(\mu ) > 0$.
Derive an asymptotically correct $95\%$ confidence interval for $\theta = \theta (F) = F(\mu )$.

Note: You have to spend some time thinking if some additional control on the density $f$ is needed to make this go through.

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The solution to the previous puzzle (number 15) is here.