Let $X_1,...,X_n$ be some data observed lying in the unit interval. We want to test the null hypothesis $H_0$ that $X_1,...,X_n$ are i.i.d uniform on the unit interval, versus $H_1$, that $X_1,...,X_n$ are i.i.d with distribution $f(x) = 6x(1-x)$ with $x \in [0,1]$. How would I compute the likelihood ratio for this?
I am at a bit of a loss as to how to come up with a likelihood, as it seems we are no longer estimating a single parameter $\theta$ but now instead estimating a distribution, which seems markedly more complicated.
Here, you are performing what is called a simple hypothesis test, which involves comparing two models that are fully specified a priori (i.e., not fitted to the data, which would require you to take into account model complexity).
Let $\mathcal{L}_1$ be the likelihood of the uniform model, and $\mathcal{L}_2$ be the likelihood of the parabolic model, then:
$$\mathcal{L}_1 = \prod_1^n X_i,\; \;\rm{and}\;\; \mathcal{L}_2 = \prod_1^n 6X_i(1-X_i)$$
You then form the likelihood ratio:
$$\Lambda = \frac{\mathcal{L}_1}{\mathcal{L}_2}$$
You need to pick a value $c \in \mathbb{R^+}$ so that you reject the uniform model in favor of the quadratic model if $\Lambda < c$ and do not reject otherwise.
You will probably need to do some Monte Carlo simulation studies to calculate the null distribution of $\Lambda$ assuming the uniform model is true, so you can set $c$ to give you the desired significance level.