# MLR Test Solution clarification/expansion

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 $$\lambda(x)=\left( \frac{-\theta_0 \sum \ln (x_i)}{n}\right)^n\left(\prod_{i=1}^n x_i \right)^{\theta_0+\frac{n}{\sum \ln(x_i)}} but now I have to find what $$c$$ is. The solutions I am given say that $$\lambda(x) 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??

• May 10, 2020 at 8:22
• I believe it does. Thank you May 10, 2020 at 18:04