More on Likelihood Ratio Test, the following problem is originally from Casella and Berger (2001), exercise 8.12. Problem For samples of size n=1,4,16,64,100 from a normal population with mean \mu and known variance \sigma^2, plot the power function of the following LRTs (Likelihood Ratio Tests). Take \alpha = .05. H_0:\mu\leq 0 versus H_1:\mu>0 H_0:\mu=0 versus H_1:\mu\neq 0 Solution The LRT statistic is given by \lambda(\mathbf{x})=\frac{\displaystyle\sup_{\mu\leq 0}\mathcal{L}(\mu|\mathbf{x})}{\displaystyle\sup_{-\infty<\mu<\infty}\mathcal{L}(\mu|\mathbf{x})}, \;\text{since }\sigma^2\text{ is known}.
The denominator can be expanded as follows: $$ \begin{aligned} \sup_{-\infty<\mu<\infty}\mathcal{L}(\mu|\mathbf{x})&=\sup_{-\infty<\mu<\infty}\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x_i-\mu)^2}{2\sigma^2}\right]\\ &=\sup_{-\infty<\mu<\infty}\frac{1}{(2\pi\sigma^2)^{1/n}}\exp\left[-\displaystyle\sum...