Question

Let \(Y_{1}

Short answer

The likelihood ratio test (LRT) for testing \(H_{0}:\theta=\theta_{0}\) against \(H_{1}:\theta \neq \theta_{0}\) is \(\Lambda= e^{-(|Y_{3}-\theta_{0}|-\Sigma_{i=1}^{5}|Y_{i}-Y_{3}|)}\).

Step-by-step solution

Understanding the Distribution and Hypotheses

Identify the given distribution and understand the hypotheses to be tested. The distribution specified is a Laplace or double-exponential distribution with pdf \(f(x; \theta)=\frac{1}{2}e^{-|x-\theta|}\) for all real \(\theta\). The hypotheses to be tested are \(H_{0}: \theta=\theta_{0}\) and \(H_{1}: \theta \neq \theta_{0}\).

Calculating Likelihood under Null Hypothesis

Under \(H_{0}\), calculate the likelihood function which is given by the product of the pdfs given the observation and the theory. The likelihood function will be \(L(\theta_{0})=\prod_{i=1}^{5}\frac{1}{2}e^{-|Y_{i}-\theta_{0}|}\).

Calculating Likelihood under Alternative Hypothesis

Under \(H_{1}\), the likelihood function is maximized when \(\theta\) equals to the median of the observations. Since the sample size is 5, the median will be the value of the third order statistic \(Y_{3}\). This gives the maximum likelihood \(L(\hat{\theta}_{ML})=\prod_{i=1}^{5}\frac{1}{2}e^{-|Y_{i}-Y_{3}|}\)

Calculating the Likelihood Ratio Test

The likelihood ratio test (LRT) statistic, denoted by \(\Lambda\), is defined as the ratio of maximized likelihood under null hypothesis to maximized likelihood under alternative hypothesis. Therefore, \(\Lambda =\frac{L(\theta_{0})}{L(\hat{\theta}_{ML})}\)

Simplify the LRT

Simplify the ratio by plugging in the likelihood functions from Steps 2 and 3. After a lot of cancellations and simplifications, we get \(\Lambda= e^{-(|Y_{3}-\theta_{0}|-\Sigma_{i=1}^{5}|Y_{i}-Y_{3}|)}\)