Suppose $X_1, . . . , X_n$ are i.i.d. with pdf $f(·)$.
We want to test the hypotheses \begin{align} H_0 &: f(x) = 2x , \;\, \text{ for } 0 \le x \le 1, \text{ against}, \\ H_1 &: f(x) = 3x^2 , \text{ for } 0\le x \le 1. \end{align}

Show that the level-$\alpha$ likelihood ratio test is of the form: $$ \text{reject }H_0 \text{ if } T_n \le c_{n,α}, $$ where $T_n$ has a $\chi^2$ distribution with $2n$ degrees of freedom.

The likelihood ratio is $$ \lambda = \frac{\prod f_0(x_i)}{\prod f_1(x_i)} = \frac{(2/3)^n}{t^n}, $$ with $t=\prod x_i$. Then I should show $$ T_n = -2\ln \lambda \sim \chi^2(2n), $$ I guess asymptotically.
How can the degree of freedom be $2n$?
I don't see any parameter in $H_0$ and $H_1$, to apply the theorem which says $$ -2\ln \lambda \sim \chi^2(\#\Theta_1-\#\Theta_0). $$

I asked on mathSE, but the question is still unresolved.

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    $\begingroup$ Wait ... there are no estimated parameters under either hypothesis? To apply the LRT don't you need the null to be a special case of the alternative? $\endgroup$ – Glen_b Mar 4 '13 at 1:27
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    $\begingroup$ @Glen: No. These are both point hypotheses. $\endgroup$ – cardinal Mar 4 '13 at 2:48
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    $\begingroup$ Hint: What is the distribution under the null of $-\log X_i$? $\endgroup$ – cardinal Mar 4 '13 at 2:51
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    $\begingroup$ To add to cardinal's hint, an exponential random variable $X$ is proportional to a $\chi^2$ random variable with two degrees of freedom, and that's where the question-setter is getting the $2n$ degrees of freedom. A couple of other comments: your likelihood ratio should be $\frac{(2/3)^n}{t}$, not $\frac{(2/3)^n}{t^n}$ where $t = \prod_i x_i$, or better yet, $\frac{t}{(2/3)^n}$ since the likelihood ratio is usually defined as $$\frac{f_1(x)}{f_0(x)}, ~~\text{not as}~ \frac{f_0(x)}{f_1(x)}$$ the way you have it. $\endgroup$ – Dilip Sarwate Mar 4 '13 at 3:28
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    $\begingroup$ Actually, the LR test is also of the form $U_n \le c'_{n,\alpha}$ where $U_n$ has any continuous distribution you please: merely apply an appropriate strictly monotonic increasing function to $T_n$ and adjust $c'_{n,\alpha}$ accordingly. In this sense the question is almost without content. $\endgroup$ – whuber Mar 4 '13 at 21:05

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