From 15b6519a7703b79e89d506b1c002ba8663b6f21d Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Tue, 9 Jun 2020 14:53:16 +0000
Subject: [PATCH] slides: write about the Trapani test

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+# Trapani test
+
+
+## Finite/infinite momenta
+
+For a Landau PDF:
+\begin{align*}
+  E_L[x] &\longrightarrow + \infty  \\
+  V_L[x] &\longrightarrow + \infty
+\end{align*}
+
+. . .
+
+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*}
+
+
+## Finite/infinite momenta
+
+- Check whether a momentum 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 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}
+  \definecolor{cyclamen}{RGB}{146,24,43}
+  % 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:
+$$
+  \sum_j \zeta_j (u) \quad \text{follows} \quad
+    G \left( \frac{r}{2}, \frac{r}{4} \right)
+  \thus \vartheta (u) \quad \text{follows} \quad
+    G \left( 0, 1 \right)
+$$
+
+. . .
+
+- Test statistic:
+$$
+  \chi^2 = \int_{\underbar{u}}^{\bar{u}} du \vartheta^2 (u)
+$$
+
+
+## Trapani test
+
+According to L. Trapani (10.1016/j.jeconom.2015.08.006):
+
+- $r = o(N)$
+- $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
+
+. . .
+
+$\mu_k$ must be scale invariant for $k > 1$:
+
+$$
+  \tilde{\mu_k} = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
+  \with \phi \in (0, k)
+$$
+
+
+# Samples results
+
+
+## Samples results
+
+$N = 50000$ sampled points
+
+. . .
+
+Landau sample:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $D = 0.004$
+  - $p = 0.379$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Compatible!}
+  $$
+:::
+::::
+
+\vspace{10pt}
+
+. . .
+
+Moyal sample:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $D = 0.153$
+  - $p = 0.000$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Not compatible!}
+  $$
+:::
+::::