Discussion
Suppose \(X_{1}, \ldots, X_{n}\) are independent and identically distributed random variables each having \(U(-\theta, \theta)\) distribution. Then the minimal sufficient statistic for \(\theta\) is given by :
Answer: C
The minimal sufficient statistic is \(\max\{-X_{(1)}, X_{(n)}\}\).
The joint density is \((2\theta)^{-n}\) when every \(|x_{i}| \le \theta\), i.e. when \(\max_{i}|x_{i}| \le \theta\), and 0 otherwise. By the factorisation theorem, \(\max|X_{i}| = \max\{-X_{(1)}, X_{(n)}\}\) is sufficient, and the likelihood ratio of two samples is free of \(\theta\) exactly when this maximum is the same, so it is minimal.
\(-X_{(1)}\) or \(X_{(n)}\) alone captures only one end of the sample, and the likelihood depends on whichever of them is larger. The minimum of the two is not sufficient.
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