Designing a Uniformly Most Powerful Test

[Artusartos]

Homework Statement

Let $X_1, ... , X_n$ be iid normally distributed random variables $N(\mu, \sigma^2)$, $\mu \in \Bbb{R}$, $\sigma^2 > 0$.

a) Design a uniformly most powerful test with significance level $\alpha$ for testing $H_0: \sigma^2 = \sigma^2_0$ vs $H_1: \sigma^2 > \sigma^2_0$.

b) Give a formula for the power of the test.

Homework Equations

The Attempt at a Solution

Let $\sigma^2_1 > \sigma^2_0$.

$\Lambda = \frac{L(\sigma^2_0)}{L(\sigma^2_1)}$ = $\frac{(\frac{1}{\sqrt{2\pi}})^n (1/\sigma^2_0)^{n/2} e^{-\sum (X_i - \mu)^2/\sigma^2_0}}{(\frac{1}{\sqrt{2\pi}})^n (1/\sigma^2_1)^{n/2} e^{-\sum (X_i - \mu)^2/\sigma^2_1}} \leq k$ for some $k < 1$.

$\implies e^{\frac{-\sum (X_i - \mu)^2}{2\sigma^2_0} + \frac{\sum (X_i - \mu)^2}{2\sigma^2_1}} \leq (\frac{\sigma^2_0}{\sigma^2_1})^{n/2}k$

Let $k_1 = (\frac{\sigma^2_0}{\sigma^2_1})^{n/2}k$.

We have $\frac{-\sum (X_i - \mu)^2}{2\sigma^2_0} + \frac{\sum (X_i - \mu)^2}{2\sigma^2_1} \leq \ln(k_1)$

$\implies \sum (X_i - \mu)^2[\frac{1}{2\sigma^2_1} - \frac{1}{2\sigma^2_0}] \leq \ln(k_1)$

$\implies \sum (X_i - \mu)^2 \geq \ln(k_1)[\frac{1}{2\sigma^2_1} - \frac{1}{2\sigma^2_0}]^{-1}$

$\implies \frac{\sum (X_i - \mu)^2}{\sigma^2_0} \geq \ln(k_1)[\frac{1}{2\sigma^2_1} - \frac{1}{2\sigma^2_0}]^{-1}(\frac{1}{\sigma^2_0})$.

Let $k_2 = \ln(k_1)[\frac{1}{2\sigma^2_1} - \frac{1}{2\sigma^2_0}]^{-1}(\frac{1}{\sigma^2_0})$.

We know that $\frac{\sum (X_i - \mu)^2}{\sigma^2_0}$ has chi-square with n degrees of freedom.

So we choose $k_2$ such that $P_{H_0}( \frac{\sum (X_i - \mu)^2}{\sigma^2_0} \geq k_2) = \alpha$

b) $P_{H_1}( \frac{\sum (X_i - \mu)^2}{\sigma^2_0} \geq k_2)$ $= 1 - \int^{k_2}_0$ $(\frac{1}{\Gamma(n/2)2^{n/2}})x^{n/2-1}e^{-x/2} dx$.

Are my answers correct?

Thanks in advance

 

[mighty2000]

for your help.

I would say that your answers appear to be correct. However, it would be important to double check your calculations and make sure that your assumptions and equations are accurate. It is also important to clearly explain and justify your reasoning in each step.

Additionally, it would be helpful to provide some context or background information on the significance of this problem and how your proposed solution fits into the broader scientific context. This will help readers understand the relevance and importance of your work.