Kyudan/arXiv_latex · Datasets at Hugging Face

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H_{0}:\xi=0
x_{min}=\min\{x_{1},\ldots,x_{n}\}
\xi_{n}(\vec{x},\vec{y})=1-\frac{n\sum_{i=1}^{n-1}\|r_{i+1}-r_{i}\|}{2\sum_{i=1}% ^{n}l_{i}(n-l_{i})}
\xi_{n}(\vec{y},\vec{y})
\displaystyle\geq\sum_{k=i}^{j-1}\|x_{k+1}-x_{k}\|\geq x_{max}-x_{min}
\displaystyle\quad+\underbrace{\sum_{k=i^{\prime}}^{j^{\prime}-1}\|x_{k+1}-x_{k% }\|}_{\geq\;x_{i^{\prime}}-x_{j^{\prime}}}+\underbrace{\sum_{k=j^{\prime}}^{j-1% }\|x_{k+1}-x_{k}\|}_{\geq\;x_{max}-x_{j^{\prime}}}
(a,b,n)
\vec{y}
Y\sim\mbox{binom}(m^{\prime},p^{\prime})
\displaystyle P(Y\geq 0\|X\!=\!x)
X\sim\mbox{equal}(a,b,n)
Y\sim XY
q(\alpha/2)
50\leq n\leq 5000
\xi(X,Y)=6\int\!\!\!\int\!P(Y\!\geq\!t\|X\!=\!x)^{2}d\lambda(x)\,d\mu(t)-2
f(x)\geq t
x_{i^{\prime}}>x_{j^{\prime}}
\xi_{n}(\vec{x},\vec{y})\leq\xi_{n}(\vec{y},\vec{y})
\varphi(z)=e^{-z^{2}/2}/\sqrt{2\pi}
\displaystyle\quad+(x_{i-1}-x_{i-2})\ldots+(x_{j+1}-x_{j})\|
\tau_{m}(\xi_{m}^{*}-\xi_{n})
\displaystyle\frac{\int\!\Big{(}\!\int\!P(Y\!\geq\!t\|X\!=\!x)^{2}d\lambda(x)-P% (Y\!\geq\!t)^{2}\Big{)}d\mu(t)}{\int P(Y\!\geq\!t)\Big{(}1-P(Y\!\geq\!t)\Big{)% }\,d\mu(t)}
\boldsymbol{Y=f(X)+\varepsilon}
\displaystyle=\int\limits_{t}^{\infty}\frac{1}{\sigma}\,\varphi\left(\frac{y-x% }{\sigma}\right)dy
\left[\xi_{n}-\frac{q(1-\alpha/2)}{\tau_{n}},\;\xi_{n}-\frac{q(\alpha/2)}{\tau% _{n}}\right]
\sum_{k=1}^{n-1}\|x_{k+1}-x_{k}\|\geq x_{max}-x_{min}
E(1_{\{Y\geq t\}}\|X)
X,Y\sim\operatorname{unif}(-1,1)
m=c\cdot n^{\alpha}
500\leq n\leq 5000
m^{\prime}=6,p^{\prime}=0.3
\frac{1}{b-a}\int\limits_{a}^{b}P(Y\!\geq\!t\|X\!=\!x)^{2}dx=\frac{\sigma}{b-a}% \bigg{[}z\Phi(z)^{2}\\ +2\Phi(z)\varphi(z)-\frac{1}{\sqrt{\pi}}\Phi(z\sqrt{2})\bigg{]}_{\frac{a-t}{% \sigma}}^{\frac{b-t}{\sigma}}
x_{max}-x_{min}
\operatorname{Var}(\xi_{n})
\displaystyle\leq\sum_{k=j}^{i-1}\|x_{k+1}-x_{k}\|\leq\sum_{k=1}^{n-1}\|x_{k+1}-x% _{k}\|
\displaystyle=\Phi\left(\frac{x-t}{\sigma}\right)
\displaystyle\quad\mbox{for }x=0
\boldsymbol{Y=X+\varepsilon}
\vec{y}=(y_{1},\ldots,y_{n})
\tau_{n}^{2}\operatorname{Var}(\xi_{n})
\tau_{n}=\sqrt{n}
i\leq i^{\prime}<j^{\prime}\leq j
X\sim\mbox{binom}(m,p)
(\vec{x},\vec{y})^{*}
x_{j}=x_{j+1}=\ldots=x_{n}
j>j
d\lambda(x)=\mbox{dunif}(a,b)(x)\,dx
\alpha\in\{1/2,2/3,3/4\}
r_{max}-r_{min}=n-r_{min}
i^{\prime},j^{\prime}
(n-2)/(n+1)
\xi^{\prime}_{n}
\mbox{dunif}(a,b)(x)=1/(b-a)
P(Y=1)=pp^{\prime}\quad\mbox{and}\quad P(Y=0)=1-pp^{\prime}
\boldsymbol{Y=X^{2}+\varepsilon}
Y\sim\sin(2\pi X)+\varepsilon
X\sim\mbox{unif}(a,b)
\displaystyle x_{min}=\sum_{k=i}^{j-1}\|x_{k+1}-x_{k}\|=\underbrace{\sum_{k=i}^{% i^{\prime}-1}\|x_{k+1}-x_{k}\|}_{\geq\;x_{i^{\prime}}-x_{min}}
X\sim\mbox{equal}(m,-1,1)
m^{\prime}=2
k\to n-k
\displaystyle\int P(Y\!\geq\!t)^{2}d\mu(t)
\xi=\frac{(1-p)p^{\prime}}{1-pp^{\prime}}
\xi_{n}(\vec{y},\vec{y})=1-\frac{n(n-r_{min})}{2\sum_{i=1}^{n}l_{i}(n-l_{i})}
Y\sim X+\varepsilon
\boldsymbol{Y,X}
X\sim\mbox{bernoulli}(p)
\displaystyle P(Y\geq t\|X\!=\!x)
\displaystyle=-\frac{1}{2}P(Y\!\geq\!t)^{2}\Big{\|}_{-\infty}^{\infty}=\frac{1}% {2}
\xi_{n}^{\prime}
\varepsilon\sim-\sigma\sqrt{m}+\frac{2\sigma}{\sqrt{m}}\mbox{binom}(m,0.5)
\displaystyle x_{min}=\|x_{i}-x_{j}\|=\|(x_{i}-x_{i-1})
\boldsymbol{Y=\sin(2\pi X)+\varepsilon}
\displaystyle=\int\limits_{t}^{\infty}\frac{1}{\sigma}\,\varphi\left(\frac{y-f% (x)}{\sigma}\right)dy
\displaystyle=\frac{1}{b-a}\left(\Phi\left(\frac{y-a}{\sigma}\right)-\Phi\left% (\frac{y-b}{\sigma}\right)\right)
Z\sim\mbox{bernoulli}(p^{\prime})
x_{j}=x_{max}
\operatorname{Var}(\xi_{n})\sim n^{-1}
P(Y\geq t)
\xi(X,Y)=\frac{\int\operatorname{Var}\Big{(}E(1_{\{Y\geq t\}}\|X)\Big{)}d\mu(t)% }{\int\operatorname{Var}\Big{(}1_{\{Y\geq t\}}\Big{)}d\mu(t)}
\xi_{n}(\vec{x},\vec{y})
n(n^{2}-1)/3
\displaystyle x_{max}-
\displaystyle=\sum_{k=1}^{n-1}\|x_{k+1}-x_{k}\|
\sum_{i}\|r_{i+1}-r_{i}\|
x_{max}=\max\{x_{1},\ldots,x_{n}\}
1_{\{Y\geq t\}}=\left\{\begin{array}[]{ll}1&\mbox{ with probability }P(Y\geq t% )\\ 0&\mbox{ with probability }1-P(Y\geq t)\end{array}\right.
P(Y\geq t)=\int P(Y\geq t\mid X=x)\,d\lambda(x)
X\sim\mbox{unif}(-1,1)
P(Y\geq t\|X=x)
\displaystyle(X,Y)=
\displaystyle\quad\mbox{for both }x
2/5\sqrt{n}
P(Y\!\geq\!t\|X=x)^{2}
\varepsilon\sim-\sigma\sqrt{m^{\prime}}+\frac{2\sigma}{\sqrt{m^{\prime}}}\mbox% {binom}(m^{\prime},0.5)
Z\sim\mbox{binom}(1,p^{\prime})
\displaystyle\geq x_{max}-x_{min}+2(x_{i^{\prime}}-x_{j^{\prime}})
\displaystyle=p^{\prime}
p^{\prime}=0.5
x_{j}=x_{min}

H_{0}:\xi=0

x_{min}=\min\{x_{1},\ldots,x_{n}\}

\xi_{n}(\vec{x},\vec{y})=1-\frac{n\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{2\sum_{i=1}% ^{n}l_{i}(n-l_{i})}

\xi_{n}(\vec{y},\vec{y})

\displaystyle\geq\sum_{k=i}^{j-1}|x_{k+1}-x_{k}|\geq x_{max}-x_{min}

\displaystyle\quad+\underbrace{\sum_{k=i^{\prime}}^{j^{\prime}-1}|x_{k+1}-x_{k% }|}_{\geq\;x_{i^{\prime}}-x_{j^{\prime}}}+\underbrace{\sum_{k=j^{\prime}}^{j-1% }|x_{k+1}-x_{k}|}_{\geq\;x_{max}-x_{j^{\prime}}}

(a,b,n)

\vec{y}

Y\sim\mbox{binom}(m^{\prime},p^{\prime})

\displaystyle P(Y\geq 0|X\!=\!x)

X\sim\mbox{equal}(a,b,n)

Y\sim XY

q(\alpha/2)

50\leq n\leq 5000

\xi(X,Y)=6\int\!\!\!\int\!P(Y\!\geq\!t|X\!=\!x)^{2}d\lambda(x)\,d\mu(t)-2

f(x)\geq t

x_{i^{\prime}}>x_{j^{\prime}}

\xi_{n}(\vec{x},\vec{y})\leq\xi_{n}(\vec{y},\vec{y})

\varphi(z)=e^{-z^{2}/2}/\sqrt{2\pi}

\displaystyle\quad+(x_{i-1}-x_{i-2})\ldots+(x_{j+1}-x_{j})|

\tau_{m}(\xi_{m}^{*}-\xi_{n})

\displaystyle\frac{\int\!\Big{(}\!\int\!P(Y\!\geq\!t|X\!=\!x)^{2}d\lambda(x)-P% (Y\!\geq\!t)^{2}\Big{)}d\mu(t)}{\int P(Y\!\geq\!t)\Big{(}1-P(Y\!\geq\!t)\Big{)% }\,d\mu(t)}

\boldsymbol{Y=f(X)+\varepsilon}

\displaystyle=\int\limits_{t}^{\infty}\frac{1}{\sigma}\,\varphi\left(\frac{y-x% }{\sigma}\right)dy

\left[\xi_{n}-\frac{q(1-\alpha/2)}{\tau_{n}},\;\xi_{n}-\frac{q(\alpha/2)}{\tau% _{n}}\right]

\sum_{k=1}^{n-1}|x_{k+1}-x_{k}|\geq x_{max}-x_{min}

E(1_{\{Y\geq t\}}|X)

X,Y\sim\operatorname{unif}(-1,1)

m=c\cdot n^{\alpha}

500\leq n\leq 5000

m^{\prime}=6,p^{\prime}=0.3

\frac{1}{b-a}\int\limits_{a}^{b}P(Y\!\geq\!t|X\!=\!x)^{2}dx=\frac{\sigma}{b-a}% \bigg{[}z\Phi(z)^{2}\\ +2\Phi(z)\varphi(z)-\frac{1}{\sqrt{\pi}}\Phi(z\sqrt{2})\bigg{]}_{\frac{a-t}{% \sigma}}^{\frac{b-t}{\sigma}}

x_{max}-x_{min}

\operatorname{Var}(\xi_{n})

\displaystyle\leq\sum_{k=j}^{i-1}|x_{k+1}-x_{k}|\leq\sum_{k=1}^{n-1}|x_{k+1}-x% _{k}|

\displaystyle=\Phi\left(\frac{x-t}{\sigma}\right)

\displaystyle\quad\mbox{for }x=0

\boldsymbol{Y=X+\varepsilon}

\vec{y}=(y_{1},\ldots,y_{n})

\tau_{n}^{2}\operatorname{Var}(\xi_{n})

\tau_{n}=\sqrt{n}

i\leq i^{\prime}<j^{\prime}\leq j

X\sim\mbox{binom}(m,p)

(\vec{x},\vec{y})^{*}

x_{j}=x_{j+1}=\ldots=x_{n}

j>j

d\lambda(x)=\mbox{dunif}(a,b)(x)\,dx

\alpha\in\{1/2,2/3,3/4\}

r_{max}-r_{min}=n-r_{min}

i^{\prime},j^{\prime}

(n-2)/(n+1)

\xi^{\prime}_{n}

\mbox{dunif}(a,b)(x)=1/(b-a)

P(Y=1)=pp^{\prime}\quad\mbox{and}\quad P(Y=0)=1-pp^{\prime}

\boldsymbol{Y=X^{2}+\varepsilon}

Y\sim\sin(2\pi X)+\varepsilon

X\sim\mbox{unif}(a,b)

\displaystyle x_{min}=\sum_{k=i}^{j-1}|x_{k+1}-x_{k}|=\underbrace{\sum_{k=i}^{% i^{\prime}-1}|x_{k+1}-x_{k}|}_{\geq\;x_{i^{\prime}}-x_{min}}

X\sim\mbox{equal}(m,-1,1)

m^{\prime}=2

k\to n-k

\displaystyle\int P(Y\!\geq\!t)^{2}d\mu(t)

\xi=\frac{(1-p)p^{\prime}}{1-pp^{\prime}}

\xi_{n}(\vec{y},\vec{y})=1-\frac{n(n-r_{min})}{2\sum_{i=1}^{n}l_{i}(n-l_{i})}

Y\sim X+\varepsilon

\boldsymbol{Y,X}

X\sim\mbox{bernoulli}(p)

\displaystyle P(Y\geq t|X\!=\!x)

\displaystyle=-\frac{1}{2}P(Y\!\geq\!t)^{2}\Big{|}_{-\infty}^{\infty}=\frac{1}% {2}

\xi_{n}^{\prime}

\varepsilon\sim-\sigma\sqrt{m}+\frac{2\sigma}{\sqrt{m}}\mbox{binom}(m,0.5)

\displaystyle x_{min}=|x_{i}-x_{j}|=|(x_{i}-x_{i-1})

\boldsymbol{Y=\sin(2\pi X)+\varepsilon}

\displaystyle=\int\limits_{t}^{\infty}\frac{1}{\sigma}\,\varphi\left(\frac{y-f% (x)}{\sigma}\right)dy

\displaystyle=\frac{1}{b-a}\left(\Phi\left(\frac{y-a}{\sigma}\right)-\Phi\left% (\frac{y-b}{\sigma}\right)\right)

Z\sim\mbox{bernoulli}(p^{\prime})

x_{j}=x_{max}

\operatorname{Var}(\xi_{n})\sim n^{-1}

P(Y\geq t)

\xi(X,Y)=\frac{\int\operatorname{Var}\Big{(}E(1_{\{Y\geq t\}}|X)\Big{)}d\mu(t)% }{\int\operatorname{Var}\Big{(}1_{\{Y\geq t\}}\Big{)}d\mu(t)}

\xi_{n}(\vec{x},\vec{y})

n(n^{2}-1)/3

\displaystyle x_{max}-

\displaystyle=\sum_{k=1}^{n-1}|x_{k+1}-x_{k}|

\sum_{i}|r_{i+1}-r_{i}|

x_{max}=\max\{x_{1},\ldots,x_{n}\}

1_{\{Y\geq t\}}=\left\{\begin{array}[]{ll}1&\mbox{ with probability }P(Y\geq t% )\\ 0&\mbox{ with probability }1-P(Y\geq t)\end{array}\right.

P(Y\geq t)=\int P(Y\geq t\mid X=x)\,d\lambda(x)

X\sim\mbox{unif}(-1,1)

P(Y\geq t|X=x)

\displaystyle(X,Y)=

\displaystyle\quad\mbox{for both }x

2/5\sqrt{n}

P(Y\!\geq\!t|X=x)^{2}

\varepsilon\sim-\sigma\sqrt{m^{\prime}}+\frac{2\sigma}{\sqrt{m^{\prime}}}\mbox% {binom}(m^{\prime},0.5)

Z\sim\mbox{binom}(1,p^{\prime})

\displaystyle\geq x_{max}-x_{min}+2(x_{i^{\prime}}-x_{j^{\prime}})

\displaystyle=p^{\prime}

p^{\prime}=0.5

x_{j}=x_{min}