Consider the linear model
\[Y_{i} = \beta x_{i} + \epsilon _{i}, 1 \le i \le n,\]
where the \(\epsilon _{i}\) are independent normal random variables with \(\mathbb {E}\epsilon _{i} = 0\) and \(\mathbb {E}\epsilon _{i}^{2} = \sigma ^{2}\). Assume \(\beta \) and \(\sigma ^{2} > 0\) are deterministic but unknown, while the \(x_{i}\) are deterministic and known. Assume also that \(\sum _i x_i \ne 0\). Find the likelihood ratio test for \(\beta = 0\) versus
\[\beta > 0.\]