slides: review Trapani section

slides/sections/6.md

@@ -76,13 +76,13 @@
 
 ## Trapani test
 
+::: incremental
+
 - 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:
   $$
@@ -90,9 +90,7 @@
     \thus G\left( 0, \sqrt{e^{\mu_k}} \right)
   $$
 
-. . .
+- The greater $\mu^k$, the 'larger' $G\left( 0, \sqrt{e^{\mu_k}} \right)$
-
-The greater $\mu^k$, the 'larger' $G\left( 0, \sqrt{e^{\mu_k}} \right)$
 $$
 \begin{cases}
   \mu_k \longrightarrow + \infty \\
@@ -101,6 +99,8 @@ $$
 \thus a_j \text{ distributed uniformly}
 $$
 
+:::
+
 
 ## Trapani test
 
@@ -146,24 +146,24 @@ $$
 
 . . .
 
-If $a_j$ uniformly distributed:
+- If $a_j$ uniformly distributed:
 
-- $\zeta_j (u)$ Bernoulli PDF with $P\left( \zeta_j (u) = 1 \right) = \frac{1}{2}$
+  $\zeta_j (u)$ Bernoulli PDF with $P\left( \zeta_j (u) = 1 \right) = \frac{1}{2}$
   $\hence \text{E}[\zeta_j]_j = \frac{1}{2}
    \quad \wedge \quad \text{Var}[\zeta_j]_j = \frac{1}{4}$
 
 
 ## Trapani test
 
+::: incremental
+
 - 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, for the CLT:
-
-If $a_j$ uniformly distributed, for the CLT:
 $$
   \sum_j \zeta_j (u) \hence
     G \left( \frac{r}{2}, \frac{\sqrt{r}}{2} \right)
@@ -171,48 +171,52 @@ $$
     G \left( 0, 1 \right)
 $$
 
-. . .
-
 - Test statistic:
 $$
   \Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u)
 $$
 
+:::
+
+
+## Trapani test
+
+**Under** $\bold{H_0}$: $\mu_k \to +\infty$
+
+- $\Theta \to \chi^2$ as $n,r \to +\infty$
+
+. . .
+
+**Under** $\bold{H_a}$: $\mu_k < + \infty$
+
+::: incremental
+
+- $\text{E}[\zeta_j] \neq \frac{1}{2}$
+
+- the residues
+  $\vartheta (u) = \frac{2}{\sqrt{r}}
+          \sum_{j} \left[ \zeta_j (u) - \frac{1}{2} \right]$
+  become large
+
+- $\Theta \to +\infty$ as $n,r \to +\infty$
+
+:::
+
 
 ## Trapani test
 
 According to L. Trapani [@trapani15]:
 
 - $r = o(N) \hence r = N^{0.75}$
+
 - $\underbar{u} = -1 \quad \wedge \quad \bar{u} = 1$
-- $\psi(u) = \frac{1}{\bar{u} - \underbar{u}} \, \chi_{[\underbar{u}, \bar{u}]}$
+
+- $\psi(u) = \frac{1}{2} \, \chi_{[\underbar{u}, \bar{u}]}$
 
 . . .
 
 $\mu_k$ must be scale invariant for $k > 1$:
-
 $$
   \mu_k^* = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
   \with \phi \in (0, k)
 $$
-
-
-## Trapani test
-
-If $\mu_k < + \infty \hence \left\{ a_j \right\}$ 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.