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

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

Some tasks

  • 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.