sections: fix a lot of things
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@ -1,5 +1,5 @@
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---
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title: Title
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title: Testing for a Landau distribution
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date: \today
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author:
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- Giulia Marcer
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@ -16,9 +16,19 @@ fontsize: 12pt
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mainfont: Fira Sans
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mainfontoptions:
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- BoldFont=Fira Sans
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mathfont: FiraMath-Regular
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references:
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- type: article-journal
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id: trapani15
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author:
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family: Trapani
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given: Lorenzo
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title: testing for (in)finite moments
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container-title: Journal of Econometrix
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issued:
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year: 2015
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header-includes: |
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```{=latex}
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%% Colors
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@ -32,6 +42,11 @@ header-includes: |
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\definecolor{yellow}{HTML}{CFB017}
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\setbeamercolor{frametitle}{bg=mDarkRed}
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\definecolor{cyclamen}{RGB}{146,24,43}
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\usepackage{ulem}
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\newcommand\strike{\bgroup\markoverwith{%
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\textcolor{mDarkRed}{\rule[0.5ex]{2pt}{1pt}}}\ULon}
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% center images
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\LetLtxMacro{\oldIncludegraphics}{\includegraphics}
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@ -40,6 +55,7 @@ header-includes: |
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\oldIncludegraphics[#1]{#2}
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}
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% "thus" in formulas
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\DeclareMathOperator{\thus}{%
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\hspace{30pt} \Longrightarrow \hspace{30pt}
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@ -69,5 +85,9 @@ header-includes: |
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\DeclareMathOperator{\ob}{%
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^{\text{obs}}
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}
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\setbeamercovered{transparent}
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```
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csl: ../notes/docs/bibliography.csl
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...
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@ -3,24 +3,26 @@
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## Goal
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- Generate a sample $L$ of points from a Landau PDF
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- Generate a sample $M$ of points from a Moyal PDF
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Construct six statistical tests to assert whether a sample comes from a Landau
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distribution
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. . .
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- Implement a bunch of statistical tests
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- Generate a sample $L$ from a Landau PDF
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- Generate a sample $M$ from a Moyal PDF
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. . .
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- Check if they work:
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- the sample $L$ truly comes from a Landau PDF
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- the sample $M$ does not come from a Landau PDF
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$H_0$: sample following Landau PDF
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- can we accept $H_0$ for $L$?
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- can we reject $H_0$ for $M$?
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## Why?
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## Why Moyal?
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The Landau and Moyal PDFs are really similar. Historically, the latter was
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utilized in the approximation of the former.
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utilized as an approximation of the former.
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:::: {.columns}
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::: {.column width=33%}
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@ -79,15 +81,18 @@ utilized in the approximation of the former.
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. . .
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- Parameters comparison:
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- compatibility between expected and observed PDF parameters
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- **Properties test**:
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compatibility between expected and observed PDF properties
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. . .
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- Kolmogorov - Smirnov test:
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- compatibility between expected and observed CDF
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- **Kolmogorov - Smirnov test**:
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compatibility between expected and empirical CDF
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. . .
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- Trapani test:
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- compatibility between expected and observed moments
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- **Trapani test**:
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test for finite or infinite moments
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@ -1,18 +1,24 @@
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# Landau PDF
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## A pathological distribution
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## Landau PDF
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Because of its fat tail:
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:::: {.columns}
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::: {.column width=50% .c}
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$$
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L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
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dt \, e^{-t \ln(t) -xt} \sin (\pi t)
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$$
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:::
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\begin{align*}
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E[x] &\longrightarrow + \infty \\
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V[x] &\longrightarrow + \infty
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\end{align*}
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::: {.column width=50%}
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:::
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::::
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. . .
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No closed form for parameters $\thus$ numerical estimations
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No closed form for \textcolor{cyclamen}{ANYTHING}
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## Landau median
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@ -28,9 +34,17 @@ $$
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- CDF computed by numerical integration
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- QDF computed by numerical root-finding (Brent)
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$$
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m_L\ex = 1.3557804...
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$$
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture]
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\node at (0,0) (here) {$m_L\ex = 1.3557804...$};
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\pause
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\node [opacity=0.5, xscale=0.35, yscale=0.25 ] at (here) {\includegraphics{images/high.png}};
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## Landau mode
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@ -41,9 +55,17 @@ $$
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- Computed by numerical minimization (Brent)
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$$
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\mu_L\ex = − 0.22278...
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$$
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture]
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\node at (0,0) (here) {$\mu_L\ex = − 0.22278...$};
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\pause
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\node [opacity=0.5, xscale=0.32, yscale=0.25 ] at (here) {\includegraphics{images/high.png}};
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## Landau FWHM
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@ -62,6 +84,14 @@ $$
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- Computed by numerical root finding (Brent)
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$$
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w_L\ex = 4.018645...
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$$
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture]
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\node at (0,0) (here) {$w_L\ex = 4.018645...$};
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\pause
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\node [opacity=0.5, xscale=0.32, yscale=0.25 ] at (here) {\includegraphics{images/high.png}};
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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@ -121,12 +121,8 @@ $$
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## Moyal FWHM
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$$
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x_+ - x_- = W_0 \left( - \frac{1}{4 e} \right)
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- W_{-1} \left( - \frac{1}{4 e} \right)
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= 3.590806098...
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= a
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x_+ - x_- = 3.590806098... = a
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$$
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\begin{align*}
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M(z)
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&\thus w_M^{\text{exp}} = a \\
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@ -1,34 +1,46 @@
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# Sample parameters estimation
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# Sample statistics
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## Sample parameters estimation
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## Sample statistics
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Once the points are sampled,
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how to estimate their median, mode and FWHM?
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How to estimate sample median, mode and FWHM?
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. . .
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- Binning data $\hence$ result depending on bin-width
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- \only<3>\strike{Binning data $\hence$ depends wildly on bin-width}
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. . .
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- Alternative solutions
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- Robust estimators
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- Kernel density estimation
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## Sample median
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$$
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m = Q \left( \frac{1}{2} \right)
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$$
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:::: {.columns}
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::: {.column width=50% .c}
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$$
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F(m) = \frac{1}{2}
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$$
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. . .
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\vspace{20pt}
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- Sort points in ascending order
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. . .
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. . .
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- Sort points in ascending order
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- Middle element if odd
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- Average of the two central elements if even
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. . .
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- Middle element if odd
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Average of the two central elements if even
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:::
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::: {.column width=50%}
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:::
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::::
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## Sample mode
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@ -37,11 +49,78 @@ Most probable value
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. . .
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HSM
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Half Sample Mode
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- Iteratively identify the smallest interval containing half points
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- Once the sample is reduced to less than three points, take average
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. . .
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture]
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% line
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\draw [line width=3, ->, cyclamen] (-5,0) -- (5,0);
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\node [right] at (5,0) {$x$};
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% points
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\draw [blue, fill=blue] (-4.6,-0.1) rectangle (-4.8,0.1);
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\draw [blue, fill=blue] (-4,-0.1) rectangle (-4.2,0.1);
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\draw [blue, fill=blue] (-3.3,-0.1) rectangle (-3.5,0.1);
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\draw [blue, fill=blue] (-2.3,-0.1) rectangle (-2.5,0.1);
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\draw [blue, fill=blue] (-0.6,-0.1) rectangle (-0.8,0.1);
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\draw [blue, fill=blue] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw [blue, fill=blue] (1.1,-0.1) rectangle (1.3,0.1);
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\draw [blue, fill=blue] (2 ,-0.1) rectangle (2.2,0.1);
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\draw [blue, fill=blue] (2.7,-0.1) rectangle (2.9,0.1);
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\draw [blue, fill=blue] (4,-0.1) rectangle (4.2,0.1);
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% future nodes
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\node at (-1,-0.3) (1a) {};
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\node at (3.1,0.3) (1b) {};
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\node at (0.9,-0.3) (2a) {};
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\node at (1.8,-0.3) (3a) {};
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% result nodes
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\node at (2.45,-0.7) (f1) {};
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\node at (2.45,0.7) (f2) {};
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (1a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (2a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [orange, fill=orange, opacity=0.5] (3a) rectangle (1b);
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\end{tikzpicture}
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\end{center}
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. . .
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\begin{center}
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\begin{tikzpicture}[remember picture, overlay]
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% region
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\draw [cyclamen, ultra thick] (f1) -- (f2);
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\end{tikzpicture}
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\end{center}
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## Sample FWHM
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@ -49,9 +128,10 @@ $$
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\text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
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$$
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\setbeamercovered{transparent}
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. . .
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KDE
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Kernel Density Estimation
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- empirical PDF construction:
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@ -82,9 +162,9 @@ with:
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. . .
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\vspace{10pt}
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Numerical root finding (Brent)
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Numerical minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$
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Numerical root finding (Brent) for $\quad f_{\varepsilon}(x_{\pm}) =
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\frac{f_{\varepsilon_{\text{max}}}}{2}$
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## Sample FWHM
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@ -1,81 +1,43 @@
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# MC simulations
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# Kolmogorov - Smirnov test
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## In summary
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## KS
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-----------------------------------------------------
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Landau Moyal
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----------------- ----------------- -----------------
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median $m_L\ex$ $m_M\ex (μ, σ)$
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Quantify distance between expected and observed CDF
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mode $\mu_L\ex$ $\mu_M\ex (μ)$
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. . .
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FWHM $w_L\ex$ $w_M\ex (σ)$
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-----------------------------------------------------
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KS statistic:
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## Moyal parameters
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A $M(x)$ similar to $L(x)$ can be found by imposing:
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\vspace{15pt}
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- equal mode
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$$
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\mu_M\ex = \mu_L\ex \approx −0.22278298...
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D_N = \text{sup}_x |F_N(x) - F(x)|
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$$
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- $F(x)$ is the expected CDF
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- $F_N(x)$ is the empirical CDF of $N$ sampled points
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- sort points in ascending order
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- number of points preceding the point normalized by $N$
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## KS
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$H_0$: points sampled according to $F(x)$
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. . .
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If $H_0$ is true:
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- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
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Kolmogorov distribution with CDF:
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$$
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P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
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\sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
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$$
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. . .
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- equal width
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$$
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w_M\ex = w_L\ex = \sigma \cdot a
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$$
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a $p$-value can be computed
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$$
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\implies \sigma_M \approx 1.1191486...
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$$
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## Moyal parameters
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:::: {.columns}
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::: {.column width=50%}
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:::
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::: {.column width=50%}
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:::
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::::
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## Moyal parameters
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This leads to more different medians:
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\begin{align*}
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m_M = 0.787... \thus &m_M = 0.658... \\
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&m_L = 1.355...
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\end{align*}
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## Compatibility test
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Comparing results:
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$$
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p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
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t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
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$$
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- $x\ex$ and $x\ob$ are the expected and observed values
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- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
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. . .
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At 95% confidence level, the values are compatible if:
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$$
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p > 0.05
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$$
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- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$
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@ -1,148 +1,181 @@
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# Landau sample
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# Trapani test
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## Sample
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## A pathological distribution
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Sample N = 50'000 random points following $L(x)$
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Because of its fat tail:
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\begin{align*}
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\mu_1 &= \text{E}\left[|x|\right] \longrightarrow + \infty \\
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\mu_2 &= \text{E}\left[|x|^2\right] \longrightarrow + \infty
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\end{align*}
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. . .
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No closed form for parameters $\thus$ numerical estimations
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. . .
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For a Moyal PDF:
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\begin{align*}
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E_M[x] &= \mu + \sigma [ \gamma + \ln(2) ] \\
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V_M[x] &= \frac{\pi^2 \sigma^2}{2}
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\end{align*}
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## Infinite moments
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- Check whether a moment is finite or infinite
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\begin{align*}
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\text{infinite} &\thus Landau \\
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\text{finite} &\thus Moyal
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\end{align*}
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. . .
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# Trapani test
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## Trapani test
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::: incremental
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- Random variable $\left\{ x_i \right\}$ sampled from a distribution $f$
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- Sample moments according to $f$ moments
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- $H_0$: $\mu_k \longrightarrow + \infty$
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- Statistic with 1 dof chi-squared distribution
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||||
:::
|
||||
|
||||
|
||||
## 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:
|
||||
$$
|
||||
L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
|
||||
dt \, e^{-t \ln(t) -xt} \sin (\pi t)
|
||||
\vartheta (u) = \frac{2}{\sqrt{r}}
|
||||
\left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
gsl_ran_Landau(gsl_rng)
|
||||
|
||||
|
||||
## Compatibility results:
|
||||
|
||||
Median:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 0.761$
|
||||
- $p = 0.446$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
Mode:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 1.012$
|
||||
- $p = 0.311$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
FWHM:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t=1.338$
|
||||
- $p=0.181$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
|
||||
# Moyal sample
|
||||
|
||||
|
||||
## Sample
|
||||
|
||||
Sample N = 50'000 random points following $M_{\mu \sigma}(x)$
|
||||
|
||||
If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT:
|
||||
$$
|
||||
M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
|
||||
\left[ - \frac{1}{2} \left(
|
||||
\frac{x - \mu}{\sigma}
|
||||
+ e^{-\frac{x - \mu}{\sigma}} \right) \right]
|
||||
\sum_j \zeta_j (u) \hence
|
||||
G \left( \frac{r}{2}, \frac{r}{4} \right)
|
||||
\thus \vartheta (u) \hence
|
||||
G \left( 0, 1 \right)
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
reverse sampling
|
||||
|
||||
- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
|
||||
- Test statistic:
|
||||
$$
|
||||
\Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u)
|
||||
$$
|
||||
|
||||
|
||||
## Compatibility results:
|
||||
## Trapani test
|
||||
|
||||
Median:
|
||||
According to L. Trapani [@trapani15]:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 669.940$
|
||||
- $p = 0.000$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Not compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
- $r = o(N) \hence r = N^{0.75}$
|
||||
- $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
|
||||
- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$
|
||||
|
||||
. . .
|
||||
|
||||
Mode:
|
||||
$\mu_k$ must be scale invariant for $k > 1$:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 0.732$
|
||||
- $p = 0.464$
|
||||
:::
|
||||
$$
|
||||
\tilde{\mu_k} = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
|
||||
\with \phi \in (0, k)
|
||||
$$
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
## Trapani test
|
||||
|
||||
. . .
|
||||
If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed
|
||||
\vspace{20pt}
|
||||
Rewriting:
|
||||
$$
|
||||
\vartheta (u) = \frac{2}{\sqrt{r}}
|
||||
\left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
|
||||
= \frac{2}{\sqrt{r}}
|
||||
\sum_{j} \left[ \zeta_j (u) - \frac{1}{2} \right]
|
||||
$$
|
||||
|
||||
FWHM:
|
||||
\vspace{20pt}
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 1.329$
|
||||
- $p = 0.184$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
Residues become very large $\hence$ $p$-values decreases.
|
||||
|
||||
@ -1,87 +1,81 @@
|
||||
# Kolmogorov - Smirnov test
|
||||
# MC simulations
|
||||
|
||||
|
||||
## KS
|
||||
## In summary
|
||||
|
||||
Quantify distance between expected and observed CDF
|
||||
-----------------------------------------------------
|
||||
Landau Moyal
|
||||
----------------- ----------------- -----------------
|
||||
median $m_L\ex$ $m_M\ex (μ, σ)$
|
||||
|
||||
. . .
|
||||
mode $\mu_L\ex$ $\mu_M\ex (μ)$
|
||||
|
||||
KS statistic:
|
||||
FWHM $w_L\ex$ $w_M\ex (σ)$
|
||||
-----------------------------------------------------
|
||||
|
||||
|
||||
## Moyal parameters
|
||||
|
||||
A $M(x)$ similar to $L(x)$ can be found by imposing:
|
||||
|
||||
\vspace{15pt}
|
||||
|
||||
- equal mode
|
||||
$$
|
||||
D_N = \text{sup}_x |F_N(x) - F(x)|
|
||||
$$
|
||||
|
||||
- $F(x)$ is the expected CDF
|
||||
- $F_N(x)$ is the empirical CDF of $N$ sampled points
|
||||
- sort points in ascending order
|
||||
- number of points preceding the point normalized by $N$
|
||||
|
||||
|
||||
## KS
|
||||
|
||||
$H_0$: points sampled according to $F(x)$
|
||||
|
||||
. . .
|
||||
|
||||
If $H_0$ is true:
|
||||
|
||||
- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
|
||||
|
||||
Kolmogorov distribution with CDF:
|
||||
|
||||
$$
|
||||
P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
|
||||
\sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
|
||||
\mu_M\ex = \mu_L\ex \approx −0.22278298...
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
a $p$-value can be computed
|
||||
- equal width
|
||||
$$
|
||||
w_M\ex = w_L\ex = \sigma \cdot a
|
||||
$$
|
||||
|
||||
- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$
|
||||
$$
|
||||
\implies \sigma_M \approx 1.1191486...
|
||||
$$
|
||||
|
||||
|
||||
# Samples results
|
||||
|
||||
|
||||
## Samples results
|
||||
|
||||
$N = 50000$ sampled points
|
||||
|
||||
. . .
|
||||
|
||||
Landau sample:
|
||||
## Moyal parameters
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $D = 0.004$
|
||||
- $p = 0.379$
|
||||
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
## Moyal parameters
|
||||
|
||||
This leads to more different medians:
|
||||
|
||||
\begin{align*}
|
||||
m_M = 0.787... \thus &m_M = 0.658... \\
|
||||
&m_L = 1.355...
|
||||
\end{align*}
|
||||
|
||||
|
||||
## Compatibility test
|
||||
|
||||
Comparing results:
|
||||
|
||||
$$
|
||||
p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
|
||||
t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
|
||||
$$
|
||||
|
||||
- $x\ex$ and $x\ob$ are the expected and observed values
|
||||
- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
|
||||
|
||||
. . .
|
||||
|
||||
Moyal sample:
|
||||
At 95% confidence level, the values are compatible if:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $D = 0.153$
|
||||
- $p = 0.000$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Not compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
$$
|
||||
p > 0.05
|
||||
$$
|
||||
|
||||
@ -1,184 +1,198 @@
|
||||
# Trapani test
|
||||
# Landau sample
|
||||
|
||||
|
||||
## Infinite moments
|
||||
## Sample
|
||||
|
||||
For a Landau PDF:
|
||||
\begin{align*}
|
||||
E_L[x] &\longrightarrow + \infty \\
|
||||
V_L[x] \text{undefined}
|
||||
\end{align*}
|
||||
Sample N = 50'000 random points following $L(x)$
|
||||
|
||||
$$
|
||||
L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
|
||||
dt \, e^{-t \ln(t) -xt} \sin (\pi t)
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
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*}
|
||||
gsl_ran_Landau(gsl_rng)
|
||||
|
||||
|
||||
## Infinite moments
|
||||
## Compatibility results:
|
||||
|
||||
- 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
|
||||
Median:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 0.761$
|
||||
- $p = 0.446$
|
||||
:::
|
||||
|
||||
|
||||
## Trapani test
|
||||
|
||||
- Start with $\left\{ x_i \right\}^N$ and compute $\mu_k$ as:
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\mu_k = \frac{1}{N} \sum_{i = 1}^N |x_i|^k
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
- Generate $r$ points $\left\{ \xi_j\right\}^r$ according to $G(0, 1)$ and define
|
||||
$\left\{ a_j \right\}^r$ as:
|
||||
Mode:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 1.012$
|
||||
- $p = 0.311$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
a_j = \sqrt{e^{\mu_k}} \cdot \xi_j
|
||||
\thus G'\left( 0, \sqrt{e^{\mu_k}} \right)
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
The greater $\mu^k$, the 'larger' $G'$
|
||||
FWHM:
|
||||
|
||||
- if $\mu_k \longrightarrow + \infty \thus a_j$ distributed uniformly
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t=1.338$
|
||||
- $p=0.181$
|
||||
:::
|
||||
|
||||
|
||||
## Trapani test
|
||||
|
||||
- Define the sequence: $\left\{ \zeta_j (u) \right\}^r$ as:
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\zeta_j (u) = \theta( u - a_j) \with \theta - \text{Heaviside}
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
\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
|
||||
# Moyal sample
|
||||
|
||||
|
||||
## Sample
|
||||
|
||||
Sample N = 50'000 random points following $M_{\mu \sigma}(x)$
|
||||
|
||||
- Define the function $\vartheta (u)$ as:
|
||||
$$
|
||||
\vartheta (u) = \frac{2}{\sqrt{r}}
|
||||
\left[ \sum_{j} \zeta_j (u) - \frac{r}{2} \right]
|
||||
M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
|
||||
\left[ - \frac{1}{2} \left(
|
||||
\frac{x - \mu}{\sigma}
|
||||
+ e^{-\frac{x - \mu}{\sigma}} \right) \right]
|
||||
$$
|
||||
|
||||
. . .
|
||||
|
||||
If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT:
|
||||
$$
|
||||
\sum_j \zeta_j (u) \hence
|
||||
G \left( \frac{r}{2}, \frac{r}{4} \right)
|
||||
\thus \vartheta (u) \hence
|
||||
G \left( 0, 1 \right)
|
||||
$$
|
||||
reverse sampling
|
||||
|
||||
- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
|
||||
|
||||
|
||||
## Compatibility results:
|
||||
|
||||
Median:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 669.940$
|
||||
- $p = 0.000$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Not compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
- Test statistic:
|
||||
$$
|
||||
\Theta = \int_{\underbar{u}}^{\bar{u}} du \, \vartheta^2 (u) \psi(u)
|
||||
$$
|
||||
Mode:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 0.732$
|
||||
- $p = 0.464$
|
||||
:::
|
||||
|
||||
## Trapani test
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
According to L. Trapani (10.1016/j.jeconom.2015.08.006):
|
||||
|
||||
- $r = o(N) \hence r = N^{0.75}$
|
||||
- $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
|
||||
- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
$\mu_k$ must be scale invariant for $k > 1$:
|
||||
FWHM:
|
||||
|
||||
$$
|
||||
\tilde{\mu_k} = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
|
||||
\with \phi \in (0, k)
|
||||
$$
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $t = 1.329$
|
||||
- $p = 0.184$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
|
||||
## 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.
|
||||
# KS results
|
||||
|
||||
|
||||
# Samples results
|
||||
## Samples results
|
||||
|
||||
$N = 50000$ sampled points
|
||||
|
||||
. . .
|
||||
|
||||
Landau sample:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $D = 0.004$
|
||||
- $p = 0.379$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
\vspace{10pt}
|
||||
|
||||
. . .
|
||||
|
||||
Moyal sample:
|
||||
|
||||
:::: {.columns}
|
||||
::: {.column width=50%}
|
||||
- $D = 0.153$
|
||||
- $p = 0.000$
|
||||
:::
|
||||
|
||||
::: {.column width=50%}
|
||||
$$
|
||||
\hence \text{Not compatible!}
|
||||
$$
|
||||
:::
|
||||
::::
|
||||
|
||||
|
||||
# Trapani results
|
||||
|
||||
|
||||
## Samples results
|
||||
|
||||
Loading…
Reference in New Issue
Block a user