Marc Lavielle,
July 16, 2015

### Preliminary definitions

#### One-sided test

Let $$X_1$$, $$X_2$$, \ldots, $$X_n$$, $$n$$ random variables independent and identically distributed, with unknown mean $$m$$.

Assuming that the $$X_i$$'s are normally distributed, we want to test the hypothesis $$H_0$$: “$$m=0$$” v.s. $$H_1$$: “$$m>0$$”

The statistic is $$T = \bar{X}_n$$ and the rejection region, that leads to rejection of $$H_0$$, has the form $$\{T > S\}$$, where $$S$$ is the critical value of the test.

By definition, the level $$\alpha$$ and the power $$\eta$$ of the test are, respectively, defined by

$\alpha = P_{m=0}(T >S)$ $\eta(\mu) = P_{m=\mu}(T >S)$

Known variance

Let us assume first that the variance of $$X_i$$ is known: $$\sigma^2$$ represents the variance of $$X_i$$.

Then, $\frac{\sqrt{n}(T-m)}{\sigma} \sim {\cal N}(0,1)$ and

\begin{aligned} \alpha &= P_{m=0}( \frac{\sqrt{n}}{\sigma} \, T >\frac{\sqrt{n}}{\sigma} \, S) \\ & = 1 - \Phi(\frac{\sqrt{n}}{\sigma} \, S) \\ \eta(\mu) &= P_{m=\mu}(\frac{\sqrt{n}}{\sigma} (T-\mu) >\frac{\sqrt{n}}{\sigma}(S-\mu)) \\ &= 1 - \Phi(\frac{\sqrt{n}}{\sigma} (S-\mu)) \end{aligned} where $$\Phi$$ is the cumulative distribution function of a $${\cal N}(0,1)$$ random variable.

Unknown variance

The variance of $$X_i$$ is unknown: $$\sigma^2$$ represents now the empirical estimate of the variance of $$X_i$$.

Then, $\frac{\sqrt{n}(T-m)}{\sigma} \sim t_{n-1}$ where $$t_\nu$$ is the $$t$$-distribution with $$\nu$$ degrees of freedom. Then,

\begin{aligned} \alpha & = 1 - \Phi_{n-1}(\frac{\sqrt{n}}{\sigma} \, S) \\ \eta(\mu) &= 1 - \Phi_{n-1}(\frac{\sqrt{n}}{\sigma} (S-\mu)) \end{aligned} where $$\Phi_\nu$$ is the cdf of a $$t$$-distribution with $$\nu$$ degrees of freedom.

#### Two-sided test

We now want to test the hypothesis $$H_0$$: “$$m=0$$” v.s. $$H_1$$: “$$m \neq 0$$”

The statistic is still $$T = \bar{X}_n$$ but the rejection region has now the form $$\{|T| > S\}$$ where $$S\geq 0$$.

\begin{aligned} \alpha &= P_{m=0}(|T| >S) \\ \eta(\mu) &= P_{m=\mu}(|T| >S)\end{aligned}

Known variance

Here, $$\alpha = P_{m=0}( \frac{\sqrt{n}}{\sigma} \, |T| >\frac{\sqrt{n}}{\sigma} \, S)$$. Then,

$\frac{\alpha}{2} = P_{m=0}( \frac{\sqrt{n}}{\sigma} \, T >\frac{\sqrt{n}}{\sigma} \, S)$ and $\alpha = 2(1 - \Phi(\frac{\sqrt{n}}{\sigma} \, S))$

On the other hand, \begin{aligned} \eta(\mu) &= P_{m=\mu}(T>S) + P_{m=\mu}(T<-S) \\\ &= P_{m=\mu}( \frac{\sqrt{n}}{\sigma} (T-\mu) >\frac{\sqrt{n}}{\sigma} (S-\mu)) + P_{m=\mu}( \frac{\sqrt{n}}{\sigma} (T-\mu) <\frac{\sqrt{n}}{\sigma} (-S-\mu)) \\\ &= 1 - \Phi(\frac{\sqrt{n}}{\sigma} (S-\mu)) + \Phi(\frac{\sqrt{n}}{\sigma} (-S-\mu)) \end{aligned}

Unknown variance

\begin{aligned} \alpha &= 2(1 - \Phi_{n-1}(\frac{\sqrt{n}}{\sigma} S)) \\ \eta(\mu) &= 1 - \Phi_{n-1}(\frac{\sqrt{n}}{\sigma}(S-\mu)) + \Phi_{n-1}(\frac{\sqrt{n}}{\sigma}(-S-\mu)) \end{aligned}

• Display the power $$\eta$$ as a function of $$\mu$$, $$n$$, $$\sigma$$
• Compare the power when $$\sigma$$ is known or estimated. What happens when $$n$$ incerases?
• Display the power $$\eta$$ as a function of the level $$\alpha$$. Change the values of $$n$$, $$\mu$$ and $$\sigma$$. What are the optimal experimental conditions for such test?
• Display the sample size $$n$$ as a function of the power of the test. Comment on this graph. Assume that $$\sigma=2$$. Given a significance level $$\alpha=0.05$$, what is the required sample size to yield a power $$\eta=0.75$$ for detecting a mean $$m>1$$ ? $$m>0.5$$ ? What happens when the level increases?
• Display the sample size $$n$$ as a function of \$\mum. Comment on this graph.