182 lines
4.0 KiB
Markdown
182 lines
4.0 KiB
Markdown
# Trapani test
|
|
|
|
|
|
## A pathological distribution
|
|
|
|
Because of its fat tail:
|
|
\begin{align*}
|
|
\mu_1 &= \text{E}\left[|x|\right] \longrightarrow + \infty \\
|
|
\mu_2 &= \text{E}\left[|x|^2\right] \longrightarrow + \infty
|
|
\end{align*}
|
|
|
|
. . .
|
|
|
|
No closed form for parameters $\thus$ numerical estimations
|
|
|
|
. . .
|
|
|
|
For a Moyal PDF:
|
|
\begin{align*}
|
|
E_M[x] &= \mu + \sigma [ \gamma + \ln(2) ] \\
|
|
V_M[x] &= \frac{\pi^2 \sigma^2}{2}
|
|
\end{align*}
|
|
|
|
|
|
## Infinite moments
|
|
|
|
- Check whether a moment is finite or infinite
|
|
\begin{align*}
|
|
\text{infinite} &\thus Landau \\
|
|
\text{finite} &\thus Moyal
|
|
\end{align*}
|
|
|
|
. . .
|
|
|
|
|
|
# Trapani test
|
|
|
|
|
|
## Trapani test
|
|
|
|
::: incremental
|
|
|
|
- Random variable $\left\{ x_i \right\}$ sampled from a distribution $f$
|
|
- Sample moments according to $f$ moments
|
|
- $H_0$: $\mu_k \longrightarrow + \infty$
|
|
- Statistic with 1 dof chi-squared distribution
|
|
|
|
:::
|
|
|
|
|
|
## Trapani test
|
|
|
|
- Start with $\left\{ x_i \right\}^N$ and compute $\mu_k$ as:
|
|
$$
|
|
\mu_k = \frac{1}{N} \sum_{i = 1}^N |x_i|^k
|
|
$$
|
|
|
|
. . .
|
|
|
|
- Generate $r$ points $\left\{ \xi_j\right\}^r$ according to $G(0, 1)$ and define
|
|
$\left\{ a_j \right\}^r$ as:
|
|
$$
|
|
a_j = \sqrt{e^{\mu_k}} \cdot \xi_j
|
|
\thus G'\left( 0, \sqrt{e^{\mu_k}} \right)
|
|
$$
|
|
|
|
. . .
|
|
|
|
The greater $\mu^k$, the 'larger' $G'$
|
|
|
|
- if $\mu_k \longrightarrow + \infty \thus a_j$ distributed uniformly
|
|
|
|
|
|
## Trapani test
|
|
|
|
- Define the sequence: $\left\{ \zeta_j (u) \right\}^r$ as:
|
|
$$
|
|
\zeta_j (u) = \theta( u - a_j) \with \theta - \text{Heaviside}
|
|
$$
|
|
|
|
. . .
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
% line
|
|
\draw [line width=3, ->, cyclamen] (0,0) -- (10,0);
|
|
\node [right] at (10,0) {$u$};
|
|
% tic
|
|
\draw [thick] (5,-0.3) -- (5,0.3);
|
|
\node [above] at (5,0.3) {$u_0$};
|
|
% aj tics
|
|
\draw [thick, cyclamen] (1,-0.2) -- (1,0.2);
|
|
\node [below right, cyclamen] at (1,-0.2) {$a_{j+2}$};
|
|
\draw [thick, cyclamen] (2,-0.2) -- (2,0.2);
|
|
\node [below right, cyclamen] at (2,-0.2) {$a_j$};
|
|
\draw [thick, cyclamen] (5.2,-0.2) -- (5.2,0.2);
|
|
\node [below right, cyclamen] at (5.2,-0.2) {$a_{j+2}$};
|
|
\draw [thick, cyclamen] (6,-0.2) -- (6,0.2);
|
|
\node [below right, cyclamen] at (6,-0.2) {$a_{j+3}$};
|
|
\draw [thick, cyclamen] (8.5,-0.2) -- (8.5,0.2);
|
|
\node [below right, cyclamen] at (8.5,-0.2) {$a_{j+4}$};
|
|
% notes
|
|
\node [below] at (1,-1) {0};
|
|
\node [below] at (2,-1) {0};
|
|
\node [below] at (5.2,-1) {1};
|
|
\node [below] at (6,-1) {1};
|
|
\node [below] at (8.5,-1) {1};
|
|
\draw [thick, ->] (1,-0.5) -- (1,-1);
|
|
\draw [thick, ->] (2,-0.5) -- (2,-1);
|
|
\draw [thick, ->] (5.2,-0.5) -- (5.2,-1);
|
|
\draw [thick, ->] (6,-0.5) -- (6,-1);
|
|
\draw [thick, ->] (8.5,-0.5) -- (8.5,-1);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
. . .
|
|
|
|
If $a_j$ uniformly distributed and $N \rightarrow + \infty$:
|
|
|
|
- $\zeta_j (u)$ Bernoulli PDF with $P(\zeta_j (u) = 1) = \frac{1}{2}$
|
|
|
|
|
|
## Trapani test
|
|
|
|
- Define the function $\vartheta (u)$ as:
|
|
$$
|
|
\vartheta (u) = \frac{2}{\sqrt{r}}
|
|
\left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
|
|
$$
|
|
|
|
. . .
|
|
|
|
If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT:
|
|
$$
|
|
\sum_j \zeta_j (u) \hence
|
|
G \left( \frac{r}{2}, \frac{r}{4} \right)
|
|
\thus \vartheta (u) \hence
|
|
G \left( 0, 1 \right)
|
|
$$
|
|
|
|
. . .
|
|
|
|
- Test statistic:
|
|
$$
|
|
\Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u)
|
|
$$
|
|
|
|
|
|
## Trapani test
|
|
|
|
According to L. Trapani [@trapani15]:
|
|
|
|
- $r = o(N) \hence r = N^{0.75}$
|
|
- $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
|
|
- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$
|
|
|
|
. . .
|
|
|
|
$\mu_k$ must be scale invariant for $k > 1$:
|
|
|
|
$$
|
|
\tilde{\mu_k} = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
|
|
\with \phi \in (0, k)
|
|
$$
|
|
|
|
|
|
## Trapani test
|
|
|
|
If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed
|
|
\vspace{20pt}
|
|
Rewriting:
|
|
$$
|
|
\vartheta (u) = \frac{2}{\sqrt{r}}
|
|
\left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
|
|
= \frac{2}{\sqrt{r}}
|
|
\sum_{j} \left[ \zeta_j (u) - \frac{1}{2} \right]
|
|
$$
|
|
|
|
\vspace{20pt}
|
|
|
|
Residues become very large $\hence$ $p$-values decreases.
|