Consider the family of variates $(X_1, X_2, ..., X_n)$ with $X_i\sim Unif\left[\theta-\frac 1 2, \theta+\frac 1 2\right]$. If $X_{(1)} \le ... \le X_{(n)}$ denotes the order statistics, what is the distribution of $X_i|X_{(1)},X_{(n)}$? Hint: The conditional distribution is not uniform.
Why is it not uniform? In the limiting case, if the minimum is (for example) 0 and the maximum is 1, shouldn't the distribution be uniform on [0, 1]? And why, if the minimum and maximum are less than 1 apart, wouldn't the conditional distribution be the uniform distribution $\left[\frac{X_{(1)}+X_{(n)}} 2-\frac 1 2, \frac{X_{(1)}+X_{(n)}} 2+ \frac 1 2 \right]$?
This is part of a larger problem, so if it would help that I write more, I can post more information. Here is the rest of the problem:
Suppose that we wish to estimate $\theta$ under quadratic loss $L(\theta, a)=(\theta-a)^2$. The sample mean $\bar X$ is one reasonable estimate of $\theta$, but turns out to be inadmissible. Show that the estimator $\delta(X_1,...,X_n)=E(\bar X|X_{(1)}, X_{(n)})$ has strictly better MSE than the MSE of $\bar X_i$ and moreover that $\delta(\textbf X)=\frac 1 2 (X_{(1)}+X_{(n)})$.
I think I need the conditional distribution to calculate the expectation to find $\delta$.