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Suppose I have a sequence of iid random variables $X_1, \ldots, X_n$ following the pdf:

$$ f_\theta (x) = \theta x^{\theta-1} $$

for $\theta >0$ and $0 <x<1$.

I would like to obtain a level-$\alpha$ likelihood ratio test for the null hypothesis $H_0: \theta = \theta_0$ versus the two-sided alternative $H_1: \theta \neq \theta_0$ where $\theta_0$ is a known constant.

MY ATTEMPT:

I first construct the ratio:

\begin{align} \lambda(x) &= \frac{\sup_{\theta=\theta_0}L(\theta\mid X)}{\sup_{\theta\neq\theta_0}L(\theta\mid X)} \\[10pt] &= \frac{\theta_0^n \left(e^{\sum \log x_i}\right)^{\theta_0-1}}{\left(\frac n {-\sum \log x_i}\right)^n \left(e^{\sum \log x_i}\right)^{\left(\frac n {-\sum \log x_i} - 1\right)}} \\[10pt] &= \left(\frac{-\theta_0 \sum \log x_i} n \right)^n e^{n+\theta_0 \sum \log x_i} \end{align}

The denominator is calculated using the MLE of $\theta$ which is $\theta^\text{MLE} = \frac n {-\sum \log x_i}$. Now, I'd like to find the likelihood ratio test, in that I would like to choose a constant $c$ such that:

$$ \alpha = \sup_{\theta = \theta_0} P_\theta (\lambda(x) \leq c) $$

Now, $-\sum \log x_i \sim \operatorname{Gamma}(n, \theta)$ but I CANNOT isolate the above equation due to the $\log$ form. What is the right answer here? Thanks!

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  • $\begingroup$ This distribution is $Beta(\theta, 1);$ you may want to look at the Wikipedia article on 'beta distribution'. Without going through the details, I'm guessing that your likelihood ratio test criterion simplifies to something involving a sample 'geometric mean'. $\endgroup$
    – BruceET
    Dec 31 '16 at 7:21
  • $\begingroup$ @BruceET : That the likelihood ratio test statistic depends on the data only through the geometric mean of the sample is perfectly clear, but it doesn't seem to answer the question. $\endgroup$ Jan 1 '17 at 20:00
  • $\begingroup$ Cross-post: stats.stackexchange.com/questions/253889/…. $\endgroup$ Nov 5 '18 at 20:26
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Let $T=-\log(X_1 \cdots X_n).$ This is positive with probability $1$. The likelihood-ratio test statistic as derived in the question is an increasing function of $T^n \exp(-\theta_0 T).$ This has its largest possible value when $T = n/\theta_0 >0$ and it decreases as $T$ moves away from $n/\theta_0$ in either direction, approaching $0$ as $T\downarrow 0$ or $T\to\infty.$ Thus you reject $H_0$ if $T$ is either too big or too small. If you find $c_1$ and $c_2$ so that $\Pr(T<c_1) = \alpha/2 = \Pr(T>c_2),$ then you have a test at the level that you want.

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