sample of size n = 1 from p.d.f.
$f(x| \theta) = 1 +\theta^2 (0.5-x)$ if $0<x<1, 0 $ o/w
where $-1 ≤ \theta ≤ 1$
Derive the MP test for testing $$H_0 : \theta = 0$$ $$H_A : \theta = \theta_1$$ at level significance $\alpha$
Derive the power of this test at $\theta = \theta_1$
So we have been given this question but only covered it briefly so was wondering if someone could help please.
My thoughts are as follows: The first part I'm not too sure how to do, it is a bit challenging but i have thoughts of using likelihood ratio test?
The second part we just use the definition maybe?
I am pretty new to this topic but would like to get better + if someone could help it would be really helpful, thank you :)
Using Neyman Pearson's lemma you get
$$\frac{L(\theta_0|\mathbf{x})}{L(\theta_1|\mathbf{x})}\leq c$$
which leads immediately to the following critial region
$$x\leq c^*$$
In fact, simply substituting, you get
$$\frac{1}{1+\theta_1^2(0.5-x)}\leq c$$
that is very easy to be solved w.r.t. $x$ finding
$$x\leq c^*$$
In the constant $c^*$ all the expressions not depending on $x$ are included...but it is not a problem as $c^*$ for you is only a point at which evaluate an integral...
thus you can reject $H_0$ if
$$\mathbb{P}[X\leq c^*|\theta=0]=\alpha$$
that is
$$\int_0^{c^*} dx=\alpha$$
$$c^*=\alpha$$
to get the power, simply using the definition you get
$$\mathbb{P}[X\leq \alpha|\theta=\theta_1]=\gamma$$
that is
$$\int_0^{\alpha}[1+\theta^2(0.5-x)]dx=\left(1+\frac{\theta^2}{2}\right)\alpha-\frac{\theta^2}{2}\alpha^2$$
As you can see (fixing a certain $\alpha$) the power increases as $|\theta|$ increases...
this is the graphic of your power function w.r.t. $\theta$
(for this graphic I set $\alpha=5\%$)