Question:
Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function
\[
f(x|\theta) = \begin{cases}
\frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\
0, & \text{otherwise}
\end{cases}
\]
Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?
Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?
\[ f(x|\theta) = \begin{cases} \frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?
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A statistic is sufficient if it encapsulates all the information about the parameter. A statistic is complete if no other statistic can provide more information.
Updated On: Nov 20, 2025
- \( \bar{X} \)
- \( \bar{X}^2 + 3 \)
- \( (X_1, \sum_{i=2}^n X_i) \)
- \( (X_1, \bar{X}) \)
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The Correct Option is C, D
Solution and Explanation
Step 1: Understanding the problem.
The likelihood function based on the random sample from the given distribution is used to determine sufficient and complete statistics. The sufficiency of a statistic is generally determined by the factorization theorem, and completeness depends on whether the statistic captures all the information about \( \theta \).
Step 2: Analyzing the options.
- (A) \( \bar{X} \): This is a sufficient statistic for \( \theta \) since it captures the necessary information for estimating \( \theta \).
- (B) \( \bar{X}^2 + 3 \): This is still sufficient because \( \bar{X} \) is sufficient, and any transformation of a sufficient statistic will also be sufficient. However, it is not complete as it is a transformation of a statistic.
- (C) \( (X_1, \sum_{i=2}^n X_i) \): This statistic is sufficient but not complete. While it is sufficient due to the factorization theorem, it does not fully capture the information about \( \theta \) since \( \sum_{i=2}^n X_i \) does not provide any new information beyond \( X_1 \) and \( \bar{X} \).
- (D) \( (X_1, \bar{X}) \): This is a complete and sufficient statistic for \( \theta \).
Step 3: Conclusion.
The correct answers are (C) and (D) because \( (X_1, \bar{X}) \) is a complete and sufficient statistic, while \( (X_1, \sum_{i=2}^n X_i) \) is sufficient but not complete.
The likelihood function based on the random sample from the given distribution is used to determine sufficient and complete statistics. The sufficiency of a statistic is generally determined by the factorization theorem, and completeness depends on whether the statistic captures all the information about \( \theta \).
Step 2: Analyzing the options.
- (A) \( \bar{X} \): This is a sufficient statistic for \( \theta \) since it captures the necessary information for estimating \( \theta \).
- (B) \( \bar{X}^2 + 3 \): This is still sufficient because \( \bar{X} \) is sufficient, and any transformation of a sufficient statistic will also be sufficient. However, it is not complete as it is a transformation of a statistic.
- (C) \( (X_1, \sum_{i=2}^n X_i) \): This statistic is sufficient but not complete. While it is sufficient due to the factorization theorem, it does not fully capture the information about \( \theta \) since \( \sum_{i=2}^n X_i \) does not provide any new information beyond \( X_1 \) and \( \bar{X} \).
- (D) \( (X_1, \bar{X}) \): This is a complete and sufficient statistic for \( \theta \).
Step 3: Conclusion.
The correct answers are (C) and (D) because \( (X_1, \bar{X}) \) is a complete and sufficient statistic, while \( (X_1, \sum_{i=2}^n X_i) \) is sufficient but not complete.
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