Question

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a uniform distribution on \( [-\theta, \theta] \), where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{ X_1, X_2, \dots, X_n \} \) and \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \). Then which of the following statements is/are true?

• A. P only
• B. Q only
• C. Both P and Q
• D. Neither P nor Q

Hint:

When analyzing statistics, recall that a complete statistic provides all the information about the parameter, while an ancillary statistic has a distribution independent of the parameter.

Answer 1

The Correct Option is D

We are given a random sample \( X_1, X_2, \dots, X_n \) from a uniform distribution. We need to analyze the two statements \( P \) and \( Q \). 
Step 1: Analyzing statement P. 
Statement \( P \) asserts that \( (X_{(1)}, X_{(n)}) \) is a complete statistic. A statistic is complete if the only function of the statistic that has an expected value of zero for all values of the parameter is the zero function. In this case, \( X_{(1)} \) and \( X_{(n)} \) together form a complete statistic for \( \theta \), since they exhaust all the information about \( \theta \) in the sample. 
Step 2: Analyzing statement Q. 
Statement \( Q \) asserts that \( X_{(n)} - X_{(1)} \) is an ancillary statistic. An ancillary statistic is a statistic whose distribution does not depend on the parameter being estimated. Since \( X_{(n)} - X_{(1)} \) depends only on the sample and not on \( \theta \), it is indeed an ancillary statistic. 
Step 3: Conclusion. 
Both statements \( P \) and \( Q \) are correct, so the correct answer is (C) Both P and Q.