Question

If $X_{1}, X_{2}, \dots, X_{n}$ be independent random variable each from Gamma($\alpha, \beta$) then the jointly sufficient statistics for vector ($\alpha, \beta$) is

• A. $(\sum_{i=1}^{n} X_{i}, \prod_{i=1}^{n} X_{i})$
• B. $(\prod_{i=1}^{n} X_{i}, \sum_{i=1}^{n} X_{i})$
• C. $(\prod_{i=1}^{n} X_{i}, \prod_{i=1}^{n} X_{i})$
• D. $(\sum_{i=1}^{n} X_{i}, \sum_{i=1}^{n} X_{i})$

Hint:

For members of the exponential family like the Gamma distribution, the sufficient statistics are simply the sums of the terms that appear in the exponent of the canonical form.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:

For a random sample from Gamma(\( \alpha, \beta \)), we need the jointly sufficient statistic for the vector \( (\alpha, \beta) \).

Step 2: Detailed Explanation:

• The joint density is proportional to \( \left(\prod x_i\right)^{\alpha - 1} e^{-\sum x_i / \beta} \) times a function of the parameters only. By the factorisation theorem the data enter only through \( \prod x_i \) and \( \sum x_i \).

• Hence the pair \( \left(\prod X_i, \sum X_i\right) \) is jointly sufficient. As per the NTA final answer key the correct option is (B), which lists the pair in the order (product, sum).

Step 3: Final Answer:
The correct option is (B) \( \left(\prod X_i, \sum X_i\right) \).