Question
Consider a r.s. {X_1, X_2, ... , X_n} with size n > 1 from a uniform distribution over [0, ], where 0 <
Consider a r.s. {X_1, X_2, ... , X_n} with size n > 1 from a uniform distribution over [0, ], where 0 < <. Find an order statistic(s) with the smallest variance. Please justify your answer.
More explanation of this question: Following the usual way, we use data {X_1, X-2, ..., X_n} to make our estimation for the parameter theta. To find an order statistic(s), we reorder the known data to {X_(1), X_(2), ..., X_(n)}. Using this kind of data to find statistic(s) is goal of this question. Also, don't forget to prove the variance of the found statistic(s) is the smallest. Variance(T({X_(1), X_(2), ..., X_(n)}) <= Variance(T{X_1, X-2, ..., X_n}) , where Variance(T{X_1, X-2, ..., X_n}) means random statistic(s) for theta. (Here, T( ) means statistics)
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