Question

Suppose that $x$ is an observed sample of size 1 from a population with probability density function $f(.)$. Based on $x$, consider testing \[ H_0: f(y) = \frac{1}{\sqrt{2\pi}} e^{-y^2/2}, y \in \mathbb{R} \text{against} H_1: f(y) = \frac{1}{2} e^{-|y|}, y \in \mathbb{R}. \] Then which one of the following statements is true?

• A. The most powerful test rejects $H_0$ if $|x|>c$ for some $c>0$
• B. The most powerful test rejects $H_0$ if $|x|<c$ for some $c>0$
• C. The most powerful test rejects $H_0$ if $| |x| - 1 |>c$ for some $c>0$
• D. The most powerful test rejects $H_0$ if $| |x| - 1 |<c$ for some $c>0$

Hint:

- The most powerful test maximizes the likelihood ratio for rejecting $H_0$ based on the observed data.
- The rejection region is based on the discrepancy between the observed sample and the expected values under the null hypothesis.

Answer 1

The Correct Option is C

1) Analyzing the distribution under $H_0$ and $H_1$:
- Under $H_0$, the distribution of $x$ is standard normal, i.e., $f_0(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$.
- Under $H_1$, the distribution is a Laplace distribution with $f_1(x) = \frac{1}{2} e^{-|x|}$.
2) Most powerful test:
The most powerful test is based on the likelihood ratio test, and in this case, it compares the likelihoods of the observed data under $H_0$ and $H_1$.
- The test rejects $H_0$ when the observed value deviates significantly from the mean (0) under the normal distribution.
- The rejection region is thus centered around $1$ (from the Laplace distribution), and it is determined by $| |x| - 1 |>c$ for some constant $c$.
Thus, the correct answer is (C).