99 lines
1.5 KiB
Markdown
99 lines
1.5 KiB
Markdown
# Goal
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## Goal
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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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- 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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$H_0$: 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 Moyal?
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The Landau and Moyal PDFs are really similar. Historically, the latter was
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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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{height=130pt}
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:::
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::: {.column width=33%}
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{height=130pt}
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:::
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::: {.column width=33%}
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{height=130pt}
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:::
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::::
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## Two similar distributions
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:::: {.columns}
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::: {.column width=50%}
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Landau PDF
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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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::: {.column width=50%}
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Moyal PDF
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$$
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M(x) = \frac{1}{\sqrt{2 \pi}} \exp \left[ - \frac{1}{2}
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\left( x + e^{- x} \right) \right]
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$$
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:::
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::::
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\vspace{10pt}
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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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## Two similar distributions
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## Statistical tests
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. . .
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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 empirical CDF
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. . .
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- **Trapani test**:
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test for finite or infinite moments
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