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I have a random sample of ${X_1,...,X_n}$ from the following pdf: $${\theta \beta^ \theta \over {x^{\theta+1}}}$$ where $\theta>0$, $\beta>0$, $x\ge\beta$

I want to find the LRT to test $H_o:\beta=1$ vs $H_a:\beta\neq1$, using $\alpha$=0.05 to find the critical values of the test.

I know the MLEs are $$\hat\beta = x_{(1)}=min_{(i)}\{X_i\}'s$$ $$\hat\theta = {n \over \sum_{i=1}^n ln(x_i)-nln(x_{(1)})}$$

which should give the LRT under the null of $$\Lambda (\mathbf x)= {\hat\theta^n\prod_{I=1}^n {1 \over x_i^{\hat\theta+1}} \over \hat\theta^n x_{(1)} ^{\hat\theta(n)} \prod_{I=1}^n {1 \over x_i^{\hat\theta+1}} }$$

But I am stuck on simplifying and am not sure what distribution this statistic would follow...a guess would be to transform it with Wilks' Theorem for a chi-sq approximation.

Rosalie
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1 Answers1

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Yes, without the restriction of assuming the null hypothesis, the MLEs are $\hat{\beta} = x_{(1)}$ and $ \hat{\theta} = \frac{n}{\sum_i(\log x_i - \log x_{(1)})}. $ However, when you are assuming the null, there is a different estimate for $\theta$: $$ \hat{\theta}_0 = \frac{n}{\sum_i\log x_i }. $$

This means your expression for the LR test statistic is actually supposed to be $$ \Lambda (\mathbf x) = \frac{\hat{\theta}_0^n (\prod_i x_i)^{-(\hat{\theta}_0 + 1)} }{\hat{\theta}^n\hat{\beta}^{n\hat{\theta}} (\prod_i x_i)^{-(\hat{\theta} + 1)}}. $$ Then you can use Wilks' theorem to find the distribution of this under the null.

Taylor
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  • Hi thank you. Yes the test is of the form:

    $nln[1-T(X)]$ where T(X)=$\frac{nln(x_{(1)})}{\sum_i=1^nln(x_{i})}$

    Which could be easily approximated for a critical value~ $\chi_2^2$.

    However what if expressing the fraction in terms of the exponential normalized spacings yields... $\frac{U}{U+V}$~$\beta$(1,n-1) where U~Gamma(1,$\theta$) independent of V~Gamma(n-1,$\theta$)?

    What is preferred?

     – Rosalie Sep 15 '18 at 04:52