Assuming that $X$ has a PDF of the normal distribution form with sd = 1. But we don't know the mean. We want to test the hypothesis $H_0: \theta_0 = 0$ versus $H_1: \theta_0 = \theta_2^*$, and assess the power ofthe test that rejects $H_0$, as $\theta_2^*$ increases away from 0.
The above is the last part of the assignment. The previous parts of the same question establish rejection rule. In the first part, we show that the expectation of the likelihood ratio statistic is equal to 1, under the null hypothesis.
In the part b of the whole question, inequality in statistics $P (ω : P (ω) ≤ \alpha) ≤ \alpha.$
we show that a random variable $P = min(1, 1/E)$ is a p-value. In part c of the whole question, I need to construct a rejection rule with size $\alpha$ for the test. Then based on part a and part b, we can construct a rejection rule that is:
if $\min (1, \frac{1}{L(w)}) \leq \alpha$, we then reject the null hypothesis.
How to use this rejection rule to set up R codes to see the power?
