Question

Suppose that \( X_1, X_2, \dots, X_n \) are independent and identically distributed random variables each having probability density function \( f(.) \) and median \( \theta \). We want to test \[ H_0: \theta = \theta_0 \quad \text{against} \quad H_1: \theta > \theta_0. \] Consider a test that rejects \( H_0 \) if \( S > c \) for some \( c \) depending on the size of the test, where \( S \) is the cardinality of the set \( \{i: X_i > \theta_0, 1 \leq i \leq n\} \). Then which one of the following statements is true?

• A. Under $H_0$, the distribution of $S$ depends on $f(.)$
• B. Under $H_1$, the distribution of $S$ does not depend on $f(.)$
• C. The power function depends on $\theta$
• D. The power function does not depend on $\theta$

Hint:

- The power function of a hypothesis test depends on how the test behaves for different values of the parameter $\theta$ under consideration.
- A test's power increases as the distance between the null hypothesis and alternative hypothesis increases.

Answer 1

The Correct Option is C

1) Understanding the power function:
The power function of a test is defined as the probability of rejecting $H_0$ when the true parameter is $\theta$, i.e., $\beta(\theta) = P(\text{Rejecting } H_0 \mid \theta)$. The power function depends on how the test behaves under different values of $\theta$.
2) Analysis:
- The test uses the statistic $S$, the number of $X_i$ greater than $\theta_0$. Under $H_0$, this statistic follows a binomial distribution with parameters $n$ and the probability $P(X_i>\theta_0)$.
- The distribution of $S$ is affected by the value of $\theta$, and so is the power of the test. Hence, the power function depends on $\theta$.
Thus, the correct answer is (C).