Let $W_1,\ldots, W_n$, $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ be independent samples. They have the laplace density: $$f_W(x)=\frac{1}{2}\exp\lbrace -|x-\theta_1| \rbrace$$ $$f_Y(x)=\frac{1}{2}\exp\lbrace -|x-\theta_2| \rbrace$$ $$f_X(x)=\frac{1}{2}\exp\lbrace -|x-\theta_3| \rbrace$$
Derive the approximative Power for the likelihood ratio test with $\alpha=0.05$ for $H_0: \theta_1=\theta_2=\theta_3$
I want to find the answer in terms of a non-central $\chi^2$ distribution, where I need to identify the degrees of freedome and the non centrality parameter.
So first of all I need to find the $2\log \lambda = 2(\ell(\hat{\theta})-\ell(\tilde{\theta}))$ then find the power of the test. Any help is greatly appreciated :)