slides: misc changes requested by rnhmjoj

slides/sections/0.md

@@ -45,6 +45,11 @@ header-includes: |
     \hspace{30pt} \Longrightarrow \hspace{30pt}
   }
 
+  % "thus" in text
+  \DeclareMathOperator{\hence}{%
+    \quad \longrightarrow \quad
+  }
+
   % "and" in formulas
   \DeclareMathOperator{\et}{%
     \hspace{30pt} \wedge \hspace{30pt}

slides/sections/1.md

@@ -84,10 +84,10 @@ utilized in the approximation of the former.
 
 . . .
 
-- Kolmogorov - Smirnov:
+- Kolmogorov - Smirnov test:
   - compatibility between expected and observed CDF
 
 . . .
 
 - Trapani test:
-  - compatibility between expected and observed momenta
+  - compatibility between expected and observed moments

slides/sections/2.md

@@ -35,7 +35,7 @@ $$
 
 ## Landau mode
 
-- Maximum $\quad \Longrightarrow \quad \partial_x L(\mu) = 0$
+- Maximum $\hence \partial_x L(\mu) = 0$
 
 . . .
 

slides/sections/4.md

@@ -8,7 +8,7 @@ how to estimate their median, mode and FWHM?
 
 . . .
 
-- Binning data $\quad \longrightarrow \quad$ result depending on bin-width
+- Binning data $\hence$ result depending on bin-width
 
 . . .
 

slides/sections/6.md

@@ -27,7 +27,7 @@ Median:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::
@@ -46,7 +46,7 @@ Mode:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::
@@ -65,7 +65,7 @@ FWHM:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::
@@ -89,7 +89,7 @@ $$
 
 reverse sampling
 
-- sampling $y$ uniformly in [0, 1] $\quad \longrightarrow \quad x = Q_M(y)$
+- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
 
 
 ## Compatibility results:
@@ -104,7 +104,7 @@ Median:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Not compatible!}
+  \hence \text{Not compatible!}
   $$
 :::
 ::::
@@ -123,7 +123,7 @@ Mode:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::
@@ -142,7 +142,7 @@ FWHM:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::

slides/sections/7.md

@@ -62,7 +62,7 @@ Landau sample:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Compatible!}
+  \hence \text{Compatible!}
   $$
 :::
 ::::
@@ -81,7 +81,7 @@ Moyal sample:
 
 :::  {.column width=50%}
   $$
-  \thus \text{Not compatible!}
+  \hence \text{Not compatible!}
   $$
 :::
 ::::

slides/sections/8.md

@@ -1,12 +1,12 @@
 # Trapani test
 
 
-## Finite/infinite momenta
+## Infinite moments
 
 For a Landau PDF:
 \begin{align*}
   E_L[x] &\longrightarrow + \infty  \\
-  V_L[x] &\longrightarrow + \infty
+  V_L[x] \text{undefined}
 \end{align*}
 
 . . .
@@ -18,9 +18,9 @@ For a Moyal PDF:
 \end{align*}
 
 
-## Finite/infinite momenta
+## Infinite moments
 
-- Check whether a momentum is finite or infinite
+- Check whether a moment is finite or infinite
 \begin{align*}
   \text{infinite} &\thus Landau \\
   \text{finite}   &\thus Moyal
@@ -39,7 +39,7 @@ For a Moyal PDF:
   - 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
+  - Statistic with 1 dof chi-squared distribution
 
 :::
 
@@ -129,9 +129,9 @@ $$
 
 If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT:
 $$
-  \sum_j \zeta_j (u) \quad \text{follows} \quad
+  \sum_j \zeta_j (u) \hence
     G \left( \frac{r}{2}, \frac{r}{4} \right)
-  \thus \vartheta (u) \quad \text{follows} \quad
+  \thus \vartheta (u) \hence
     G \left( 0, 1 \right)
 $$
 
@@ -139,7 +139,7 @@ $$
 
 - Test statistic:
 $$
-  \chi^2 = \int_{\underbar{u}}^{\bar{u}} du \vartheta^2 (u)
+  \Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u)
 $$
 
 
@@ -147,8 +147,9 @@ $$
 
 According to L. Trapani (10.1016/j.jeconom.2015.08.006):
 
-- $r = o(N)$
+- $r = o(N) \hence r = N^{0.75}$
 - $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
+- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$
 
 . . .
 
@@ -160,45 +161,90 @@ $$
 $$
 
 
+## 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.
+
+
 # Samples results
 
 
 ## Samples results
 
-$N = 50000$ sampled points
-
 . . .
 
 Landau sample:
 
 :::: {.columns}
-:::  {.column width=50%}
-  - $D = 0.004$
-  - $p = 0.379$
+:::  {.column width=33%}
+  $$
+  \mu_1
+  \begin{cases}
+    \Theta = 0.255 \\
+    p = 0.614
+  \end{cases}
+  $$
 :::
 
-:::  {.column width=50%}
+:::  {.column width=33% .c}
   $$
-  \thus \text{Compatible!}
+  \mu_2
+  \begin{cases}
+    \Theta = 0.432 \\
+    p = 0.511
+  \end{cases}
+  $$
+:::
+
+:::  {.column width=33% .c}
+  $$
+  \hence \text{Infinite!}
   $$
 :::
 ::::
 
-\vspace{10pt}
-
 . . .
 
+\vspace{20pt}
+
 Moyal sample:
 
 :::: {.columns}
-:::  {.column width=50%}
-  - $D = 0.153$
-  - $p = 0.000$
+:::  {.column width=33%}
+  $$
+  \mu_1
+  \begin{cases}
+    \Theta^2 = 106 \\
+    p = 0.000
+  \end{cases}
+  $$
 :::
 
-:::  {.column width=50%}
+:::  {.column width=33%}
   $$
-  \thus \text{Not compatible!}
+  \mu_2
+  \begin{cases}
+    \Theta^2 = 162 \\
+    p = 0.000
+  \end{cases}
+  $$
+:::
+
+:::  {.column width=33% .c}
+  $$
+  \hence \text{Finite!}
   $$
 :::
 ::::