t distribution and F distribution

Definition 2.17 t distribution

• $n \in \mathbb{N}$,
• $\delta \in \mathbb{R}$,
• $Y \sim \chi^{2}(n)$,
• $Z \sim N(\delta, 1)$,
• $Y, Z$,
  • independent

\[X := \frac{ Z }{ \sqrt{Y/n} } .\]

The distribution of $X$ is said to be t distribution with $n$ degree of freedom and noncentrality parameter $\delta$. The p.d.f. of this distribution is given by

\[f(x; n, \delta) := \frac{ 1 }{ \sqrt{n\pi} } e^{-\frac{\delta^{2}}{2}} \sum_{r=0}^{\infty} \frac{ 2^{\frac{r}{2}} }{ r! } \frac{ \Gamma ( \frac{ n + r + 1 }{ 2 } ) }{ \Gamma(n/2) } \left( \frac{ \delta x }{ \sqrt{n} } \right)^{r} \left( 1 + \frac{ x^{2} }{ n } \right)^{\frac{-(n + r + 1)}{2}} .\]

In particular, t distribution with $n$ degree of freedom and denote $t(k) := t(k, 0)$ when $\delta = 0$. In this case, the p.d.f. of $\chi^{2}(k)$ is given by

\[\begin{eqnarray} f(x; n, 0) & = & \frac{ 1 }{ \sqrt{n\pi} } \sum_{r=0}^{\infty} \frac{ 2^{\frac{0}{2}} }{ r! } \frac{ \Gamma ( \frac{ n + r + 1 }{ 2 } ) }{ \Gamma(n/2) } \left( \frac{ 0 x }{ \sqrt{n} } \right)^{r} \left( 1 + \frac{ x^{2} }{ n } \right)^{\frac{-(n + r + 1)}{2}} \nonumber \\ & = & \frac{ 1 }{ \sqrt{n\pi} } \frac{ \Gamma ( \frac{ n + 1 }{ 2 } ) }{ \Gamma(n/2) } \left( 1 + \frac{ x^{2} }{ n } \right)^{\frac{-(n + 1)}{2}} \end{eqnarray}\]

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Example 2.1

• $X_{1}, \ldots, X_{n} \sim \mathrm{N}(\mu, \sigma^{2})$,
  • i.i.d.
• $U \in \mathbb{R}^{n \times n}$,
  • orthogonal matrix such that

\[U := \left( \begin{array}{ccccc} \frac{1}{\sqrt{n}} & \cdots & & & \frac{1}{\sqrt{n}} \\ \frac{1}{\sqrt{(2 - 1)2}} & -\frac{2 - 1}{\sqrt{(2 - 1)2}} & \cdots & & 0 \\ \frac{1}{\sqrt{(3 - 1)3}} & \frac{1}{\sqrt{(3 - 1)3}} & -\frac{3 - 1}{\sqrt{(3 - 1)3}} & \cdots & 0 \\ & \vdots & \ddots & \ddots & \\ \frac{1}{\sqrt{(n - 1)n}} & \cdots & & \frac{1}{\sqrt{(n - 1)n}} & -\frac{n - 1}{\sqrt{(n - 1)n}} \end{array} \right)\]

To check orthogonality,

\[\begin{eqnarray} i = 2, \ldots, n, \quad j, k = 1, \ldots, i - 1, \quad U_{i}^{j} & = & U_{i}^{k} \nonumber \\ i = 2, \ldots, n, \quad U_{i}^{i} & = & - (i - 1) U_{i}^{1} \nonumber \\ \|U_{1}\| & = & \left( \sum_{j=1}^{n} \frac{ 1 }{ n } \right)^{1/2} = 1 \nonumber \\ i = 2, \ldots, n, \quad \|U_{i}\| & = & \left( \sum_{j=1}^{i-1} \frac{ 1 }{ (i - 1)i } + \frac{ (i - 1)^{2} }{ (i - 1)i } \right)^{1/2} \nonumber \\ & = & \left( \frac{ i - 1 }{ (i - 1)i } + \frac{ i - 1 }{ i } \right)^{1/2} \nonumber \\ & = & \left( \frac{ 1 }{ i } + \frac{ i - 1 }{ i } \right)^{1/2} \nonumber \\ & = & 1 \nonumber \end{eqnarray}\]

• $Y := UX$,
  • $Y_{i} := \sum_{j=1}^{n} U_{i}^{j}X_{j} \ (i = 1, \ldots, n)$,
    • independent
  • $Y_{1} \sim \mathrm{N}(\sqrt{n}\mu, \sigma^{2})$,
  • $Y_{i} \sim \mathrm{N}(0, \sigma^{2}) \ (i = 2, \ldots, n)$,

\[\begin{eqnarray} Y_{1} & = & \frac{1}{\sqrt{n}} \sum_{j=1}^{n} X_{j} \nonumber \\ i = 2, \ldots, n, \quad Y_{i} & = & \frac{1}{\sqrt{i - 1}i} \sum_{j=1}^{i-1} X_{j} - \frac{i - 1}{\sqrt{i - 1}i} X_{i} \end{eqnarray}\]

Sample mean $\bar{X}$ and sample variance $S$ are

\[\begin{eqnarray} \bar{X} & := & \frac{1}{n} \sum_{j=1}^{n} X_{j} \nonumber \\ S & := & \frac{1}{n} \sum_{j=1}^{n} (X_{j} - \bar{X})^{2} \nonumber \\ & = & \frac{1}{n} \sum_{j=1}^{n} \left( X_{j}^{2} - 2\bar{X}X_{j} + \bar{X}^{2} \right) \nonumber \\ & = & \frac{1}{n} \sum_{j=1}^{n} X_{j}^{2} - 2\bar{X}^{2} + \bar{X}^{2} \nonumber \\ & = & \frac{1}{n} \sum_{j=1}^{n} X_{j}^{2} - \bar{X}^{2} \end{eqnarray}\]

$U$ is orthogonal matrix so that \(\|X\|^{2} = \|UY\|^{2}\), that is,

\[\sum_{j=1}^{n} X_{j}^{2} = \sum_{j=1}^{n} Y_{j}^{2} .\]

Sample variable satisfies

\[\begin{eqnarray} S & = & \frac{1}{n} \sum_{j=1}^{n} X_{j}^{2} - \bar{X}^{2} \nonumber \\ & = & \frac{1}{n} \sum_{j=1}^{n} Y_{j}^{2} - \frac{1}{n} \bar{Y}_{1}^{2} \nonumber \\ & = & \frac{1}{n} \sum_{j=2}^{n} Y_{j}^{2} \nonumber \end{eqnarray}\]

Hence $S^{2}$ and $Y_{1}$ are independent. Moreover,

\[\begin{eqnarray} \frac{ Y_{1} - \sqrt{n}\mu }{ \sigma } & \sim & \mathrm{N}(0, 1) \nonumber \\ \frac{ nS }{ \sigma^{2}(n - 1) } = \frac{ 1 }{ (n - 1) } \sum_{j=2}^{n} \left( \frac{Y_{j}}{\sigma} \right)^{2} & \sim & \chi^{2}(n - 1) . \nonumber \end{eqnarray}\]

Therefore,

\[\begin{eqnarray} \frac{ \frac{ Y_{1} - \sqrt{n}\mu }{ \sigma } }{ \sqrt{ \frac{ n S }{ \sigma^{2}(n - 1) } } } & = & \frac{ \frac{ Y_{1} - \sqrt{n}\mu }{ \sigma } }{ \frac{ \sqrt{n} \sqrt{S} }{ \sigma \sqrt{n - 1} } } \nonumber \\ & = & \frac{ \sqrt{n - 1} ( Y_{1} - \sqrt{n}\mu ) }{ \sqrt{n} \sqrt{S} } \nonumber \\ & = & \frac{ \sqrt{n - 1} ( \frac{1}{\sqrt{n}} Y_{1} - \mu ) }{ \sqrt{S} } \nonumber \\ & = & \frac{ \sqrt{n - 1} ( \bar{X} - \mu ) }{ \sqrt{S} } \sim t(n - 1) \nonumber \end{eqnarray}\]

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