Suppose that $X$ is a continuous random variable with p.d.f $f(x) = \theta x^{\theta-1}I\{x ∈ (0; 1)\}$. Derive the generalized likelihood ratio test for testing $H_0 : \theta = \theta_0$ versus $H_1 : \theta \neq \theta_0$ ̸based on a random sample of size $n$ from $X$.

So I have gotten to the point of finding that you reject $H_0$ for $$\lambda(x)=\left( \frac{-\theta_0 \sum \ln (x_i)}{n}\right)^n\left(\prod_{i=1}^n x_i \right)^{\theta_0+\frac{n}{\sum \ln(x_i)}}<c$$ but now I have to find what $c$ is. The solutions I am given say that $\lambda(x)<c$ implies that $\ln (\lambda (x))<\ln(c)$ and then rewrites the above inequality as a sum of logarithms. After that, though, he writes that to find $c$, "One can let $T(X)=-\sum \ln(X_i)$ and create a function of $T(X)$ and solve graphically". What does that really mean as far as finding c, and how would I go about doing that??


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