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Let X1, . . . , Xn iid∼ U[θ − 

class="katex-mathml">1/2 , θ + 1/2 ], with θ ∈ R unknown.

(a) Find a two-dimensional minimal sufficient statistic and show it is minimal.

(b) Show that the minimal sufficient statistic is not complete.

(c) Suppose we want to estimate θ under the squared error loss L(θ, d) = (θ − d) ^2 . The sample mean X (the average of X)seems to be a reasonable estimator of θ. However, we can improve upon it by Rao-Blackwellizing it. Find this new estimator δ(X1, . . . , Xn). 

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