Consider the two-component mixture with \(\psi_{0}\) standard normal, and \(\psi_{1}\) standard Cauchy. The null hypothesis is all Gaussian, i.e., \(\theta=(1,0)\). Show that \(\hat{\theta}_{1}>0\) if and only if \(\bar{X}_{n}>1\). By simulation or otherwise, show that \(P_{0}\left(\bar{X}_{n}>1ight) ightarrow 0\) as \(n ightarrow \infty\). What is the effect of changing the Cauchy scale parameter?