Question

Let \(X_1, X_2, X_3, X_4\) be a sample of size 4 from a U(0,\(\theta\)) distribution. Suppose that, in order to test the hypothesis \(H_0: \theta = 1\) against the alternate \(H_1: \theta \ne 1\), an UMPCR is given by, \(W_0 = \{x_{(4)} : x_{(4)}<\frac{1}{2} \text{ or } x_{(4)}>1\}\), then the size \(\alpha\) of \(W_0\) is

• A. \( \frac{1}{12} \)
• B. \( \frac{1}{16} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{3} \)

Hint:

When calculating the size of a test, always operate under the assumption that \(H_0\) is true. This can often simplify the critical region by making some parts of it impossible, as seen here where \(P(X_{(4)}>1 | \theta=1) = 0\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The size of a critical region (or test), denoted by \(\alpha\), is the probability of rejecting the null hypothesis \(H_0\) when \(H_0\) is actually true. This is also known as the Type I error rate. The test is based on the maximum order statistic, \(X_{(4)}\).

Step 2: Key Formula or Approach:
1. The size of the test is \( \alpha = P(\text{Reject } H_0 | H_0 \text{ is true}) \). 2. In this case, \( \alpha = P(W_0 | \theta=1) = P(X_{(4)}<1/2 \text{ or } X_{(4)}>1 | \theta=1) \). 3. We need the distribution of the maximum order statistic, \(X_{(n)}\), from a U(0,\(\theta\)) sample. The cumulative distribution function (CDF) of \(X_{(n)}\) is \( F_{X_{(n)}}(y) = [F_X(y)]^n \), where \(F_X(y) = y/\theta\) is the CDF of a single U(0,\(\theta\)) variable.

Step 3: Detailed Explanation:
First, let's analyze the critical region \(W_0\) under the null hypothesis \(H_0: \theta=1\). If \(\theta=1\), then all observations \(x_i\) must be in the interval (0, 1). Consequently, the maximum observation, \(x_{(4)}\), must also be less than 1. This means the event \(x_{(4)}>1\) is impossible under \(H_0\). Therefore, \( P(X_{(4)}>1 | \theta=1) = 0 \). The size of the test simplifies to: \[ \alpha = P(X_{(4)}<1/2 | \theta=1) + P(X_{(4)}>1 | \theta=1) = P(X_{(4)}<1/2 | \theta=1) + 0 \] Now, we find the distribution of \(X_{(4)}\) under \(H_0\). For a single observation \(X_i\) from U(0,1), the CDF is \( F_X(y) = y \) for \(0 \le y \le 1\). For a sample of size \(n=4\), the CDF of the maximum order statistic \(X_{(4)}\) is: \[ F_{X_{(4)}}(y) = [F_X(y)]^4 = y^4, \text{ for } 0 \le y \le 1 \] The size \(\alpha\) is the probability that \(X_{(4)}\) falls in the critical region, so we need to compute \( P(X_{(4)}<1/2) \). This is given directly by the CDF of \(X_{(4)}\) evaluated at \(y=1/2\). \[ \alpha = F_{X_{(4)}}(1/2) = (1/2)^4 \] \[ \alpha = \frac{1}{16} \]
Step 4: Final Answer:
The size \(\alpha\) of the critical region \(W_0\) is \( \frac{1}{16} \).