This is a question from Statistical Theory I have encountered. I have almost solved it, but have some trouble interpreting the solution. Something seems weird, and I am not sure whether I am entirely correct.
X is a random variable with distribution $\frac3{\theta^3} x^2 {1}_{(0,\theta)}$. We want to test the hypotheses: $H_0: \theta = 1$ vs. $H_1: \theta = 1.1$ with significance level $\alpha$ = $P(\text{reject }H_0| H_0 \text{ is true})$. We gather $n$ = 100 observations.
Okay, we perform the likelihood ratio test: $$ \Lambda = \frac{L(x_i,\theta_1)}{L(x_i\theta_0)} = \frac{(\frac{3}{1.1^3})^n(\Pi x_i)^2 1_{(0,1.1)}}{3^n(\Pi x_i)^2 1_{(0,1)}} = \begin{cases} c_n, & \text{if $\max x_i \leq 1$} \\ +\infty, & \text{if $\max x_i \geq 1$} \\ \end{cases} $$ Where $c_n = (\frac 1{1.1^{3n}}) = c_{100} = c$
I get trouble attempting to match the hypothesis level: the test should go like this: reject $H_0$ is $\Lambda$ > $C = C(\alpha)$.
If $C < c$ we always reject $H_0$. Therefore significance level is 1. if $C > c$ we reject the null hypothesis only if $\max x_i > 1$. But under $H_0$ this never happens, therefore $\alpha = 0$
The power of the test is $\pi = 1 - P(\text{do not reject } H_0| H_1 \text{ is true}) = \text{either } 1 \text{ or some other constant, which I have calculated to be ~~$1-0.75^n$, depending on $C >< c$}$
So, first of all, can somebody please verify whether I have made any significant mistakes? (I may have gotten some number wrong, but am I conceptually correct?)
How do I calculate $C = C(\alpha)?$ What is the precise significance and power of this test for, say $\alpha = 0.01$? The fact the the test is so "$\alpha$-independent" makes me suspicious.