F-tests

Definition

F-distribution

• $N, M \in \mathbb{N}$,
• $U_{1} \sim \chi^{2}(N)$,
• $U_{2} \sim \chi^{2}(M)$,
• $U_{1}, U_{2}$ are independant

\[F := \frac{ U_{1}/N }{ U_{2}/M } .\]

Distribution of $F$ is said to be $F$-distribution with $N, M$ degree of freedom. We denote $F(N, M)$.

Two sample F test for population variances

$F$検定では、真の分布が正規分布の2つの分散が一致するか検定する。 $F$分布は$N$及び$M$によってのみ決まる。 つまり、正規分布の平均によらず、分散の検定ができる。

• $X \sim \mathrm{N}(\mu_{X}, \sigma_{X}^{2})$,
• $Y \sim \mathrm{N}(\mu_{Y}, \sigma_{Y}^{2})$,
• null hypothesis is that variances of $X$ and $Y$ are equal
• $X$, $Y$ are independent
• $\sigma_{X}$, $\sigma_{Y}$ are unkown

Example

• $n \in \mathbb{N}$,
  • the sample size for $X$
• $m \in \mathbb{N}$,
  • the sample size for $Y$
• $\alpha \in (0, 1)$,
  • significance level
• $s_{X}^{2}$,
  • sample variance of $X$
• $s_{Y}^{2}$,
  • sample variance of $Y$
• (1) State the hypothesis
  • null hypothesis
    • $H_{0}:\sigma_{X} = \sigma_{Y} $
  • alternative hypothesis
    • (a) $H_{0}:\sigma_{X} \neq \sigma_{Y}$
    • (b) $H_{0}:\sigma_{X} > \sigma_{Y}$
    • (c) $H_{0}:\sigma_{X} < \sigma_{Y}$
• (2) compute the test statistic

\[\begin{eqnarray} f = \frac{ s_{X}^{2} }{ s_{Y}^{2} } \end{eqnarray}\]

• (3) Compute $p$ value
  • $F \sim \mathrm{F}(n - 1, m - 1)$,
  • (a) \(p := 2P(F \ge f)\)
  • (b) $p := P(F \ge f)$
  • (c) $p := P(F \ge f)$
• (4)
  • If $p < \alpha$, reject $H_{0}$,
  • otherwise, fail to reject $H_{0}$,

Theorem 25

• $X \sim \mathrm{N}(\mu_{X}, \sigma^{2})$,
• $Y \sim \mathrm{N}(\mu_{Y}, \sigma^{2})$,
• \(X_{1}, \ldots, X_{N}\),
  • $X$のi.i.d
• \(Y_{1}, \ldots, Y_{M}\),
  • $Y$のi.i.d

Then

\[F := \frac{ V_{N}(X_{1}, \ldots, X_{N}) }{ V_{M}(Y_{1}, \ldots, Y_{M}) } \sim F(N - 1, M - 1) .\]

proof.

Let

\[\begin{eqnarray} U_{X} & := & \frac{ (N-1) V_{N}(X_{1}, \ldots, X_{N}) }{ \sigma^{2} }, \nonumber \\ U_{Y} & := & \frac{ (M-1) V_{M}(Y_{1}, \ldots, Y_{N}) }{ \sigma^{2} } . \end{eqnarray}\]

\(U_{X} \sim \chi^{2}(N-1)\), \(U_{Y} \sim \chi^{2}(M-1)\). Thus,

\[\begin{eqnarray} F & = & \frac{ V_{N}(X_{1}, \ldots, X_{N}) }{ V_{M}(Y_{1}, \ldots, Y_{M}) } \nonumber \\ & = & \frac{ V_{N}(X_{1}, \ldots, X_{N}) }{ V_{M}(Y_{1}, \ldots, Y_{M}) } \frac{ \frac{ 1 }{ \sigma^{2} } }{ \frac{ 1 }{ \sigma^{2} } } \frac{ \frac{ (N - 1) }{ (N - 1) } }{ \frac{ (M - 1) }{ (M - 1) } } \nonumber \\ & = & \frac{ \frac{ (N - 1) V_{N}(X_{1}, \ldots, X_{N}) }{ \sigma^{2} } }{ \frac{ (M - 1) V_{M}(Y_{1}, \ldots, Y_{M}) }{ \sigma^{2} } } \frac{ \frac{ 1 }{ (N - 1) } }{ \frac{ }{ (M - 1) } } \nonumber \\ & = & \frac{ \frac{ U_{X} }{ (N - 1) } }{ \frac{ U_{Y} }{ (M - 1) } } \nonumber \end{eqnarray}\]

The distribution of $F$ is $F(N - 1, M - 1)$.

$\Box$

Reference