Question

Let \( X_1, \dots, X_n \) be a random sample from a uniform distribution over the interval \( \left( -\frac{\theta}{2}, \frac{\theta}{2} \right) \), where \( \theta>0 \) is an unknown parameter. Then which of the following options is/are correct?

• A. \( 2 \max\{ X_1, \dots, X_n \} \) is the maximum likelihood estimator of \( \theta \)
• B. \( \left( \min\{ X_1, \dots, X_n \}, \max\{ X_1, \dots, X_n \} \right) \) is a sufficient statistic
• C. \( \left( \min\{ X_1, \dots, X_n \}, \max\{ X_1, \dots, X_n \} \right) \) is a complete statistic
• D. \( 2 \frac{n+1}{n} \max\{ |X_1|, \dots, |X_n| \} \) is a uniformly minimum variance unbiased estimator of \( \theta \)

Hint:

For uniform distributions, the maximum of the sample provides the maximum likelihood estimator for the parameter, and combinations of the sample minimum and maximum provide sufficient and complete statistics.

Answer 1

The Correct Option is B, D

Step 1: Maximum likelihood estimator of \( \theta \).
The likelihood function for a uniform distribution over \( \left( -\frac{\theta}{2}, \frac{\theta}{2} \right) \) is maximized by the largest observed value. Therefore, the maximum likelihood estimator for \( \theta \) is: \[ \hat{\theta} = 2 \max\{ X_1, \dots, X_n \}. \] Hence, option (A) is correct. Step 2: Sufficient statistic.
By the factorization theorem, the pair \( \left( \min\{ X_1, \dots, X_n \}, \max\{ X_1, \dots, X_n \} \right) \) is a sufficient statistic for \( \theta \), so option (B) is correct. Step 3: Completeness of the statistic.
The statistic \( \left( \min\{ X_1, \dots, X_n \}, \max\{ X_1, \dots, X_n \} \right) \) is not complete for the uniform distribution, so option (C) is incorrect. Step 4: Unbiased estimator.
The statistic \( 2 \frac{n+1}{n} \max\{ |X_1|, \dots, |X_n| \} \) is known to be the uniformly minimum variance unbiased estimator (UMVUE) for \( \theta \), so option (D) is correct. Thus, the correct answer is \( \boxed{(B), (D)} \).