Where he used the information that $\mu \leq \mu_0$?

Question

This example is from casella_statistical inference book.

Where he used the information that $\mu \leq \mu_0$? What if $\mu \geq \mu_0$?

Thanks.

Answer 1

I have gone through the problem like this:

The LRT statistic is

\begin{align} \lambda(\mathbf x)=\frac{\sup\limits_{H_0}L(\mu,\sigma^2\mid \mathbf x)}{\sup\limits_{H_0\cup H_1}L(\mu,\sigma^2\mid \mathbf x)} &=\frac{\sup\limits_{\mu\le \mu_0,\sigma^2}L(\mu,\sigma^2\mid \mathbf x)}{\sup\limits_{\mu,\sigma^2}L(\mu,\sigma^2\mid \mathbf x)} \\&=\frac{L\left(\hat{\hat\mu},\hat{\hat\sigma}^2\mid \mathbf x\right)}{L(\hat \mu,\hat\sigma^2\mid \mathbf x)} \end{align}

, where $(\hat{\hat\mu},\hat{\hat\sigma}^2)$ is the restricted MLE of $(\mu,\sigma^2)$ when $\mu\le \mu_0$ and $(\hat\mu,\hat\sigma^2)$ is the unrestricted MLE of the same. It can be verified that

$$\hat{\hat\mu}=\begin{cases}\hat\mu&,\text{ if }\mu\le \mu_0 \\ \mu_0 &,\text{ if }\mu>\mu_0\end{cases}$$

And that $$\hat{\hat\sigma}^2=\begin{cases} \hat\sigma^2&,\text{ if }\mu\le \mu_0\\ \frac{1}{n}\sum (x_i-\mu_0)^2 &,\text{ if }\mu>\mu_0\end{cases}$$

So the ratio becomes

\begin{align} \lambda(\mathbf x)&=\begin{cases}1&,\text{ if }\mu\le \mu_0 \\ \\ \frac{L\left(\mu_0,\hat{\hat\sigma}^2\mid \mathbf x\right)}{L(\hat\mu,\hat\sigma^2\mid \mathbf x)}&,\text{ if }\mu>\mu_0\end{cases} \end{align}

The case $\mu\ge \mu_0$ can be studied in exactly the same manner. For simply $\mu\ne \mu_0$, the ratio $\lambda(\cdot)$ no longer remains a piecewise function.