Existence of a Uniformly Most Powerful Test

Question

let $X_1$ be a single observation from the density: $$f(x;\theta)=\theta x^{\theta-1}\boldsymbol{1}\{x\in(0,1)\}, \text{ where } \theta>0.$$

Is there a UMP size-$\alpha$ test of $H_0:\theta\geq 2$ versus $H_1:\theta<2$?

I have the hint to show the likelihood ratio is monotone, but I don't see why the likelihood ratio is such.

Answer 1

The likelihood ratio for all $\theta_i$ such that $\theta_2>\theta_1(>0)$ is

\begin{align} r(x)&=\frac{f(x;\theta_2)}{f(x;\theta_1)} \\&=\left(\frac{\theta_2}{\theta_1}\right)x^{\theta_2-\theta_1}\quad,\,0<x<1 \end{align}

The above is clearly a non-decreasing function of $x$ for all $x\in(0,1)$.

In other words, $f(x;\theta)$ has monotone likelihood ratio (MLR) in $T(x)=x$.

Now you can use the Karlin-Rubin theorem to answer the question.