Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a population \( f(x; \theta) \), where \( \theta \) is a parameter. Then which one of the following statements is NOT true?

• A. \( \sum_{i=1}^{n} X_i \) is a complete and sufficient statistic for \( \theta \), if \[ f(x; \theta) = \frac{e^{-\theta} \theta^x}{x!}, \, x = 0, 1, 2, \dots, \, \theta>0 \]
• B. \( \left( \sum_{i=1}^{n} X_i, \sum_{i=1}^{n} X_i^2 \right) \) is a complete and sufficient statistic for \( \theta \), if \[ f(x; \theta) = \frac{1}{\sqrt{2 \pi \theta}} e^{-\frac{1}{2 \theta}(x-\theta)^2}, \, -\infty<x<\infty, \, \theta>0 \]
• C. \( f(x; \theta) = \theta x^{\theta - 1}, 0<x<1, \, \theta>0 \) has the monotone likelihood ratio property in \( \sum_{i=1}^{n} X_i \)
• D. \( X_{(n)} - X_{(1)} \) is ancillary statistic for \( \theta \) if \( f(x; \theta) = 1, 0<\theta<x<\theta + 1 \), where \( X_{(1)} = \min\{X_1, X_2, \dots, X_n\} \) and \( X_{(n)} = \max\{X_1, X_2, \dots, X_n\} \)

Hint:

In many distributions, especially normal and Poisson distributions, the sample sum or mean is a complete and sufficient statistic for the parameter \( \theta \). Additional moments, like the sum of squares, may not always add more information.

Answer 1

The Correct Option is B

Step 1: Analyzing Statement (A).
The given distribution in (A) is a Poisson distribution, where \( \sum_{i=1}^{n} X_i \) is a sufficient and complete statistic for \( \theta \) because it captures all the information about \( \theta \) for the Poisson distribution. Thus, statement (A) is true.

Step 2: Analyzing Statement (B).
In statement (B), \( \left( \sum_{i=1}^{n} X_i, \sum_{i=1}^{n} X_i^2 \right) \) is stated to be a complete and sufficient statistic for \( \theta \), where the distribution is normal. However, the sample mean \( \sum_{i=1}^{n} X_i \) alone is sufficient and complete for \( \theta \) in a normal distribution, while \( \sum_{i=1}^{n} X_i^2 \) does not add any additional information about \( \theta \). Hence, this statement is NOT true.

Step 3: Analyzing Statement (C).
The given distribution in statement (C) corresponds to a Beta distribution. The sum of the sample values \( \sum_{i=1}^{n} X_i \) follows the monotone likelihood ratio property for the Beta distribution, meaning that the statistic \( \sum_{i=1}^{n} X_i \) is appropriate for hypothesis testing and follows the likelihood ratio property. Therefore, statement (C) is true.

Step 4: Analyzing Statement (D).
Statement (D) refers to the distribution of the difference between the maximum and minimum values of the sample, \( X_{(n)} - X_{(1)} \), which is an ancillary statistic. This difference does not depend on \( \theta \), making it an ancillary statistic in this case. Hence, statement (D) is true.

Step 5: Conclusion.
From the analysis above, we conclude that statement (B) is the only one that is NOT true. The correct answer is (B).