Question

Let \( X_1 \) and \( X_2 \) be a random sample from a continuous distribution with the probability density function \[ f(x) = \frac{1}{\theta} e^{-\frac{x - \theta}{\theta}}, \quad x>\theta \] If \( X_{(1) = \min \{ X_1, X_2 \} \) and \( \overline{X} = \frac{X_1 + X_2}{2} \), then which one of the following statements is TRUE?}

• A. \( ( \overline{X}, X_{(1)} ) \) is sufficient and complete
• B. \( ( \overline{X}, X_{(1)} ) \) is sufficient but not complete
• C. \( ( \overline{X}, X_{(1)} ) \) is complete but not sufficient
• D. \( ( \overline{X}, X_{(1)} ) \) is neither sufficient nor complete

Hint:

The factorization theorem is useful for checking sufficiency, and completeness requires verifying that no non-zero function can have zero expectation for all parameter values.

Answer 1

The Correct Option is B

Step 1: Check the conditions for sufficiency and completeness.
The sufficiency of \( ( \overline{X}, X_{(1)} ) \) can be determined using the factorization theorem, which states that a statistic is sufficient if the likelihood can be factored into two parts, one depending only on the data and the other on the parameter. For this problem, the statistic \( ( \overline{X}, X_{(1)} ) \) satisfies the conditions for sufficiency. Step 2: Check completeness.
Completeness requires that if the expected value of any function of the statistic equals zero for all parameter values, then that function must be zero almost surely. \( ( \overline{X}, X_{(1)} ) \) is complete because it captures all the information about \( \theta \). Step 3: Conclusion.
Thus, the correct answer is (A) \( ( \overline{X}, X_{(1) ) \) is sufficient and complete}.