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I am having a hard time understanding what this following question is asking. I do not have any clue pn how to start it. Any help would be great. Thanks in advance. Let $X_1,X_2,...,X_n$ be a random sample, $X=(X_1,X_2,...,X_n)^T$, $X=x \in B \subset R^n$ be the observed sample set. Let $f(x|\theta)$ be the joint pdf of the given sample $X$, and let $\Theta$ be an entire parameter set in $R$. For $\Theta \neq \phi$, define $$\lambda(x)=\frac{\text{sup}_{\Theta_0}L(\theta|x)}{\text{sup}_{\Theta}L(\theta|x)}$$ 1) Show that $0\leq \lambda(x) \leq 1$ for all $x\in B$.
2) As $\text{sup}_ \Theta$ $L(\theta|x)$ increases, $\lambda(x)$ decreases.
3) $$\lambda(x)=\frac{L(\hat{\theta}_{0}|x)}{L(\hat{\theta}|x)}$$
4) $$\lambda(x)=\frac{\text{sup}_{\Theta_0}g(T(x)|\theta)}{\text{sup}_{\Theta}g(T(x)|\theta)}=\lambda^{*}(x)$$ where $T(X)$ is a sufficient statistics and $f(x|\theta)=g(T(x)|\theta)h(x).$
5) As $\text{sup}_{\Theta}L(\theta|x)$ decreases, $\lambda(x)$ increases.
6) Under what conditions on $\Theta_{0}$ , $\lambda(X)$ is called the likelihood Ratio test statistic?
7) Assuming that $\lambda(X)$ is a Likelihood ratio test statistic, what can you say about $\{x: \lambda(x) \leq c \}$ for any number $c$ satisfying $0 \leq c \leq 1$?

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You might want to define the set $\Theta_0$. – Did Jul 12 '11 at 7:49
    
Instead of $\sup_\Theta L(\theta\mid x)$, I'd have been more explicit and written $\sup_{\theta\in\Theta} L(\theta\mid x)$. – Michael Hardy Mar 15 '12 at 13:28

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