7.1: Zero way ANOVA (= Student $t$-distribution )
In the previous chapter, we introduced the statistical hypothesis testing for student $t$-distribution, which is characterized as "zero" way ANOVA (analysis of variance ). In this section, we review "zero" way ANOVA (analysis of variance ).
Consider the classical basic structure
\begin{align}
[ C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq B(L^2 (\Omega, \nu ))]
\end{align}
where
\begin{align}
\Omega = {\mathbb R} \times {\mathbb R}_+
=
\{ (\mu, \sigma )
\;|\;
\mu
\mbox{ is real,}
\sigma
\mbox{ is positive real}
\}
\end{align}
Consider the simultaneous normal measurement ${\mathsf M}_{L^\infty ({\mathbb R} \times {\mathbb R}_+)}$ $({\mathsf O}_G^n = ({\mathbb R}^n, {\mathcal B}_{\mathbb R}^n, {{{G}}^n}) ,$ $S_{[(\mu, \sigma)]})$ ( in $L^\infty({\mathbb R} \times {\mathbb R}_+)$). For completeness, recall that
\begin{align}
&
[{{{G}}}^n
({\mathop{\mbox{\Large $\times$}}}_{k=1}^n \Xi_k)]
(\omega)
=
{\mathop{\mbox{\Large $\times$}}}_{k=1}^n
[{{{G}}}(\Xi_k)](\omega)
\nonumber
\\
=
&
\frac{1}{({{\sqrt{2 \pi }\sigma{}}})^n}
\underset{{{\mathop{\mbox{\Large $\times$}}}_{k=1}^n \Xi_k }}{\int \cdots \int}
\exp[{}- \frac{\sum_{k=1}^n ({}{x_k} - {}{\mu} )^2
}
{2 \sigma^2} {}] d {}{x_1} d {}{x_2}\cdots dx_n
\tag{7.1}
\\
&
\qquad
(\forall \Xi_k \in {\cal B}_{{\mathbb R}{}}^{}
(k=1,2,\ldots, n),
\quad
\forall {}{\omega}=(\mu, \sigma ) \in \Omega = {\mathbb R}\times {\mathbb R}_+).
\nonumber
\end{align}
And recall the state space $\Omega = {\mathbb R} \times {\mathbb R}_+$, the measured value space $X={\mathbb R}^n$, the second state space(=parameter space) $\Theta={\mathbb R}$. Also, recall the estimator $E:X(={\mathbb R}^n) \to \Theta(={\mathbb R})$ defined by
\begin{align}
E(x)=E(x_1, x_2, \ldots , x_n )
=
\overline{\mu}(x)
=
\frac{x_1 + x_2 + \cdots + x_n}{n}
\tag{7.2}
\end{align}
and the system quantity $\pi:\Omega(={\mathbb R} \times {\mathbb R}_+)
\to
\Theta(={\mathbb R})$ defined by
\begin{align}
\Omega(={\mathbb R} \times {\mathbb R}_+)
\ni \omega
=
(\mu, \sigma )
\mapsto \pi (\mu, \sigma )
=
\mu
\in
\Theta(={\mathbb R})
\tag{7.3}
\end{align}
The essence of "studentized" is to define the semi-metric $d_\Theta^x (\forall x \in X)$ in the second state space $\Theta (={\mathbb R})$such that
\begin{align}
d_\Theta^x (\theta^{(1)}, \theta^{(2)})
=
\frac{|\theta^{(1)}-\theta^{(2)}|}{\sqrt{n}{\overline{\sigma}(x)}}
=
\frac{|\theta^{(1)}-\theta^{(2)}|}{\sqrt{\overline{SS}(x)}}
\quad
\qquad
(\forall x \in X={\mathbb R}^n,
\forall \theta^{(1)}, \theta^{(2)} \in \Theta={\mathbb R}
)
\tag{7.4}
\end{align}
where
\begin{align}
{{\overline{SS}}}
(x) =
{{\overline{SS}}}
(x_1,x_2,\ldots , x_n ) =
{\sum_{k=1}^n ( x_k -
\overline{\mu}
(x))^2}
\quad( \forall x=(x_1,x_2,\ldots , x_n ) \in {\mathbb R}^n)
\nonumber
\end{align}
Thus, as mentioned in the previous chapter, our problem is characterized as follows.
Problem 7.1 [The zero-way ANOVA]. Consider the simultaneous normal measurement ${\mathsf M}_{L^\infty ({\mathbb R} \times {\mathbb R}_+)}$ $({\mathsf O}_G^n = ({\mathbb R}^n, {\mathcal B}_{\mathbb R}^n, {{{G}}^n}) ,$ $S_{[(\mu, \sigma)]})$. Here, assume that
\begin{align}
\mu = \mu_0
\end{align}
That is, the null hypothesis $H_N$ is defined by $H_N=\{ \mu_0 \}$ $(\subseteq
\Theta=
{\mathbb R} )
)$. Consider $0 < \alpha \ll 1$.
Then, find the largest ${\widehat R}_{{H_N}}^{\alpha; \Theta}( \subseteq \Theta)$ (independent of $\sigma$) such that
| $(A_1):$ | the probability that a measured value $x(\in{\mathbb R}^n )$
(obtained by $
{\mathsf M}_{L^\infty ({\mathbb R} \times {\mathbb R}_+ )} ({\mathsf O}_G^{{{n}}} = (X(\equiv {\mathbb R}^{{{n}}}), {\mathcal B}_{\mathbb R}^{{{n}}}, {{{G}}^{{{n}}}} ),
S_{[(\mu_0, \sigma )]}
)
$) satisfies
\begin{align}
E(x) \in {\widehat R}_{{H_N}}^{\alpha; \Theta}
\tag{7.5}
\end{align}
is less than $\alpha$.
|
Answer. We see, for any $ \omega=(\mu_0, \sigma ) (\in \Omega= {\mathbb R} \times {\mathbb R}_+ )$,
\begin{align}
&
[G^n(\{ x \in X \;:\;
d^x_\Theta ( E(x) , \pi( \omega ) )
\ge \eta
\}
)](\omega )
\nonumber
\\
=&
[G^n(\{ x \in X \;:\;
\frac{
|\overline{\mu}(x)- \mu_0 |}{
{{{\sqrt{\overline{SS}(x)}}}}
}
\ge \eta
\}
)](\omega )
\nonumber
\\
=
&
\frac{1}{({{\sqrt{2 \pi }\sigma{}}})^n}
\underset{
\eta \sqrt{n-1}
\le
\frac{
|\overline{\mu}(x)- \mu_0 |}{
{\sqrt{\overline{SS}(x)}}
/\sqrt{n-1}
}
}{\int \cdots \int}
\exp[{}- \frac{\sum_{k=1}^n ({}{x_k} - {}{\mu_0} )^2
}
{2 \sigma^2} {}] d {}{x_1} d {}{x_2}\cdots dx_n
\nonumber
\\
=
&
\frac{1}{({{\sqrt{2 \pi }{}}})^n}
\underset{
\eta^2 n({n-1})
\le
\frac{
n(\overline{\mu}(x))^2
}{
{\overline{SS}(x)}/({n-1})
}
}
{\int \cdots \int}
\exp[{}- \frac{\sum_{k=1}^n ({}{x_k} {} )^2
}
{2 } {}] d {}{x_1} d {}{x_2}\cdots dx_n
\tag{7.6}
\end{align}
$(A_2)$: by the formula of Gauss integrals ( Formula 7.8 (A)( in $\S$7.4)), we see
\begin{align}
=
&
\int^{\infty}_{
\eta^2 n({n-1})
} p_{(1,{{n}}-1) }^F(t) dt =
\alpha \;\; (\mbox{ e.g., $\alpha=0.05$})
\tag{7.7}
\end{align}
where $p_{(1,{{n}}-1) }^F$ is the probability density function of $F$-distribution with $(1,n-1)$ degree of freedom.
Note that the probability density function $p_{(n_1,n_2)}^F(t)$ of $F$-distribution with $(n_1,n_2)$ degree of freedom is defined by
\begin{align}
p_{(n_1,n_2)}^F(t)
=
\frac{1}{B(n_1/2, n_2/2)}
\Big(\frac{n_1}{n_2} \Big)^{n_1/2}
\frac{t^{(n_1-2)/2}}{(1+n_1t/n_2)^{(n_1+n_2)/2}}
\qquad
(t \ge 0)
\tag{7.8}
\end{align}
where $B(\cdot, \cdot)$ is the Beta function.
The $\alpha$-point: $F_{n_1, \alpha}^{n_2}$ $( > 0)$ is defined by
\begin{align}
\int^{\infty}_{F_{n_1, \alpha}^{n_2} } p_{(n_1,n_2) }^F (t) dt =\alpha
\qquad
(0 < \alpha \ll 1. \mbox{ e.g., } \alpha=0.05)
\tag{7.9}
\end{align}
Thus, it suffices to solve the following equation:
\begin{align}
{\eta^2 n({{n}}-1) }{ }
={F_{n-1, \alpha}^{1} }
\tag{7.10}
\end{align}
Therefore,
\begin{align}
(\eta^\alpha_{\omega})^2 =
\frac{{F_{n-1, \alpha}^{1} }}{n(n-1)}
\tag{7.11}
\end{align}
Then, the rejection region${\widehat R}_{{H_N}}^{\alpha; \Theta}$( (or ${\widehat R}_{H_N}^{\alpha; X}$) is calculated as
\begin{align}
{\widehat R}_{{H_N}}^{\alpha; \Theta}
&
=
\bigcap_{\omega =(\mu, \sigma ) \in \Omega (={\mathbb R} \times {\mathbb R}_+) \mbox{ such that }
\pi(\omega)= \mu \in {H_N}(=\{\mu_0\})}
\{
E(x)
(\in
\Theta)
:
\;\;
d^x_\Theta (E(x),
\pi(\omega )
)
\ge
\eta^\alpha_{\omega }
\}
\nonumber
\\
&
=
\{\overline{\mu}(x) \in \Theta(={\mathbb R})
\;:\;
\frac{
|\overline{\mu}(x)- \mu_0 |}{
\sqrt{{\overline{SS}(x)}}
}
\ge
\eta_\omega^\alpha
\}
=
\{\overline{\mu}(x) \in \Theta(={\mathbb R})
\;:\;
\frac{
|\overline{\mu}(x)- \mu_0 |}{
\overline{\sigma}(x)
}
\ge
\eta_\omega^\alpha \sqrt{n}
\}
\nonumber
\\
&
=
\Big\{\overline{\mu}(x) \in \Theta(={\mathbb R})
\;:\;
\frac{
|\overline{\mu}(x)- \mu_0 |}{
\overline{\sigma}(x)
}
\ge
\sqrt{\frac{F_{n-1, \alpha}^1}{n-1}}
\;\;
\Big\}
\nonumber
\\
&
=
\Big\{\overline{\mu}(x) \in \Theta(={\mathbb R})
\;:\;
\mu_0 \le
\overline{\mu}(x)
-
{{\overline{\sigma}(x)}}
\sqrt{\frac{F_{n-1, \alpha}^1}{n-1}}
\mbox{ or }
\overline{\mu}(x)
+
{{\overline{\sigma}(x)}}
\sqrt{\frac{F_{n-1, \alpha}^1}{n-1}}
\le \mu_0
\Big\}
\tag{7.12}
\end{align}
and,
\begin{align}
{\widehat R}_{H_N}^{\alpha; X}
&=
E^{-1}({\widehat R}_{{H_N}}^{\alpha; \Theta})
\nonumber
\\
&
=
\Big\{x \in X(={\mathbb R}^n)
\;:\;
\mu_0 \le
\overline{\mu}(x)
-
{{\overline{\sigma}(x)}}
\sqrt{\frac{F_{n-1, \alpha}^1}{n-1}}
\mbox{ or }
\overline{\mu}(x)
+
{{\overline{\sigma}(x)}}
\sqrt{\frac{F_{n-1, \alpha}^1}{n-1}}
\le \mu_0
\Big\}
\tag{7.13}
\end{align}
QED
| $\fbox{Note 7.1}$ | (i): It should be noted that the mathematical part in the above argument is only the (A$_2$).
(ii): Also, note that
| $(\sharp):$ | $\;\;\; F$-distribution with $(1,n-1)$ degree of freedom
= the student $t$-distribution with $(n-1)$ degree of freedom
|
Thus, we conclude that
\begin{align}
(7.12)=(6.83)
\qquad
(7.13)=(6.84)
\end{align}
|
