Question

Let \(X_1,X_2,\ldots,X_n\ (n\geq2)\) be a random sample from the probability density function \(f(x)\). Consider the following hypotheses:
\[ H_0: f(x)=\frac{1}{\sqrt{2\pi}}\,e^{-\frac{x^2}{2}};\quad -\infty<x<\infty \]
\[ H_1: f(x)=\frac{1}{2}\,e^{-|x|};\quad -\infty<x<\infty. \]
For testing \(H_0\) against \(H_1\), let \(R\) denote the critical region based on the likelihood ratio test having level \(0.05\). Then, for some constant \(c\), the region \(R\) is

• A. \(\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n(x_i-1)^2>c\}\)
• B. \(\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n(x_i-1)^2<c\}\)
• C. \(\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n(|x_i|-1)^2>c\}\)
• D. \(\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n(|x_i|-1)^2<c\}\)

Hint:

Take the log likelihood ratio and complete the square in \(|x_i|\); remember \(x_i^2=|x_i|^2\), so the exponent naturally becomes a function of \(|x_i|-1\).

Answer 1

The Correct Option is C

Step 1: Write down the two densities.
$H_0$: standard normal $f_0(x)$. $H_1$: standard Laplace $f_1(x)$. LRT rejects for large $\Lambda=L_1/L_0$.

Step 2: Form the log ratio.
\[ \ln\Lambda = n\ln\frac{\sqrt{2\pi}}{2} + T, \qquad T=\sum_{i=1}^n\left(\frac{x_i^2}{2}-|x_i|\right). \]

Step 3: Substitute $y_i=|x_i|$.
\[ T = \frac{1}{2}\sum_{i=1}^n(y_i-1)^2 - \frac{n}{2}. \]

Step 4: Translate the rejection rule.
\[ \sum_{i=1}^n(|x_i|-1)^2 > c. \]

Step 5: Rule out the wrong options.
$x_i$ instead of $|x_i|$ ignores the natural transformation; reversed inequality rejects when data looks normal, backward.

Final Answer:
\[ \boxed{R=\left\{(x_1,\ldots,x_n):\sum_{i=1}^n(|x_i|-1)^2>c\right\}} \]