# Likelihood ratio test Weibull distributed r.v -- problem calculating the critical region

Let $$X$$ be a r.v which is Weibull distributed with $$k=4$$ and $$\lambda>0$$. I'm looking at the following test:

\begin{align*} H_0:\lambda &= 1\\ H_1: \lambda&\not = 1. \end{align*}

I have already done the calculations for the actual likelihood ratio, it is \begin{align*} \lambda_n(\tilde{x}) &= \dfrac{L(1,\tilde{x})}{L(\hat{\lambda},\tilde{x}) }\\ &=\left(\dfrac{1}{n}\sum_{i\leq n}x_i^4\right)^n\exp\left(\sum_{i\leq n}x_i^4+n\right)\\ &=A^ne^{-n(A-1)}, \end{align*} where $$A =A(\tilde{x}) = \frac1n\sum_{i\leq n}x^4_i$$. We define the critical region as $$R = \left\{ \tilde{x} \mid \lambda_n(\tilde{x}) \leq \lambda_0 \right\}$$, with $$\lambda_0$$ such that $$P(\Lambda_n(\tilde{X})\leq \lambda_0) = 0.05,$$ and we reject the null hypothesis iff $$\tilde{x} \in R$$.

I know how to calculate the critical region if I know the correct value for $$\lambda_0$$. But I don't know how to find that value, since it seems I need to know how $$\Lambda_n$$ is distributed, which I don't.

• What does "$\Lambda_n(\tilde X)$" refer to? Presumably it's a test statistic, so you can compute its distribution under the null hypothesis and thereby solve for $\lambda_0.$ Why, then, don't you know its distribution?
– whuber
Aug 9 '20 at 14:56
• @whuber $\Lambda(\tilde{X}_n)= \lambda(\tilde{X}_n)$ is the likelihood ratio for the likelihood ratio test. Finding the distribution of this r.v is precisely what I don't know how to do. Also, it has to be exact, so I can't use any asymptotic results. Aug 9 '20 at 16:00