# Unknown variance hypothesis testing for normal random variables

Let $X_1,...,X_n$ be iid random variables each with a $N(\mu_0,\sigma^2)$ distribution, where $\mu_0$ is known and $\sigma^2$ is unknown. Find the best (most powerful) test of size at most $\alpha$ for testing $H_0:\sigma^2=\sigma_0^2$ against $H_1:\sigma^2=\sigma_1^2$ for known $\sigma_0^2$ and $\sigma_1^2 > \sigma_0^2$.

### Progress

These are simple hypotheses, so I should be able to use the Neyman-Pearson lemma to find the best test size $\leq \alpha$. However, when I carry out the calculation, I am finding it difficult to simply the likelihood ratio, or to show that it is strictly increasingl (so that I can, for example, just consider the distribution of $\bar X$. The likelihood ratio is

$$\Lambda_{\underline{x}}(H_0;H_1)=\left(\frac{\sigma_0}{\sigma_1}\right)^n \exp\left(\sum(x_i-\mu_0)^2 \left[\frac{1}{2\sigma_0^2}-\frac{1}{2\sigma_1^2}\right]\right)$$

I understand what the Neyman-Pearson lemma states, I just don't understand how we apply it to this particular problem, as it seems that the likelihood ratio I have is too complicated to use: I think there must be a way to distill the essential information from it, perhaps by using monotonicity, but I'm not sure how to implement this in practice.

• Is there a Lemma that you can use if the conditions of the lemma are satisfied? What type of Hypotheses are these (simple or composite)? The type of hypotheses determines if you can use the lemma or the theorem. – user61752 Mar 12 '13 at 1:46
• These are simple hypotheses, so I should be able to use the Neyman-Pearson lemma to find the best test size $\leq \alpha$. However, when I carry out the calculation, I am finding it difficult to simply the likelihood ratio, or to show that it is strictly increasingly (so that I can, for example, just consider the distribution of $\bar(X)$ – user55225 Mar 12 '13 at 1:49
• What do you get as your LRT? Since $\mu$ is known, are they the same regardless if you put $\sigma_{1}^{2}$ or $\sigma_{0}^{2}$ into your equation? – user61752 Mar 12 '13 at 2:06
• $\Lambda_{\underline{x}}(H_0;H_1)=(\frac{\sigma_0}{\sigma_1})^n exp(\sum(x_i-\mu_0)^2 [\frac{1}{2\sigma_0^2}-\frac{1}{2\sigma_1^2}])$ – user55225 Mar 12 '13 at 2:11
• Okie dokie. So I found a great pdf, found [here][1] that might help. The help starts around page 4 of the pdf, which is pertinent to your question. [1]:pages.stern.nyu.edu/~churvich/Regress/Handouts/Chapt7.pdf – user61752 Mar 12 '13 at 3:17