Kyudan/arXiv_latex · Datasets at Hugging Face

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x_{i}=x_{min}
\xi^{\prime}_{n}(\vec{x},\vec{y})=\frac{\xi_{n}(\vec{x},\vec{y})}{\xi_{n}(\vec% {y},\vec{y})}
x_{1}\leq x_{2}\leq\ldots\leq x_{n}
x_{min}=x_{max}
x_{i}=x_{max}
f(x)=\sin(x)
X\sim\mbox{binom}(1,p)
m=cn^{\alpha}
q(1-\alpha/2)
\tau_{n}^{2}=n
\log\operatorname{Var}(\xi_{n})=\log V+\gamma\log n
2(x_{i^{\prime}}-x_{j^{\prime}})>0
d\mu(t)=f(t)dt
d\mu(t)=f_{Y}(t)dt
\displaystyle\int P(Y\!\geq\!t)d\mu(t)
\boldsymbol{Y=XZ}
f(t)=-\frac{d}{dt}P(Y\geq t)
\xi_{m}^{*}
Y\sim\mbox{unif}(-1,1)
\varepsilon\sim\mbox{norm}(0,\sigma^{2})
\xi\gtrsim 0.4
r_{min}
m=3,p=0.5
\|\xi^{\prime}_{n}\|\geq\|\xi_{n}\|
\displaystyle=-\frac{1}{3}P(Y\!\geq\!t)^{3}\Big{\|}_{-\infty}^{\infty}=\frac{1}% {3}
P(1)=p
\mbox{dnorm}(0,\sigma^{2})(x)=\varphi(z/\sigma)/\sigma
Y\sim\mbox{equal}(m^{\prime},-1,1)
\xi_{n}(\vec{x},\vec{y})=1-\frac{3\sum_{i=1}^{n-1}\|r_{i+1}-r_{i}\|}{n^{2}-1}
\displaystyle=\mbox{dunif(a,b)}*\mbox{dnorm}(0,\sigma^{2})(y)
1_{\{Y\geq t\}}
x_{1}=x_{2}=\ldots=x_{i}
\displaystyle P(Y\!\geq\!t\|X\!=\!x)
P(Y\!\geq\!t\|X=x)^{2}=P(Y\!\geq\!t)^{2}
\displaystyle\quad\mbox{for }x=1
Y\sim X^{2}+\varepsilon
\mbox{Var}(Z)=E(Z^{2})-E(Z)^{2}
f(x)<t
X,Y\sim\operatorname{equal}(5,-1,1)
m=2\sqrt{n}
\displaystyle=\Phi\left(\frac{f(x)-t}{\sigma}\right)
\xi(X,Y)=1
x_{min}=x_{1}\leq x_{2}\leq\ldots\leq x_{n}=x_{max}
\sum_{k=1}^{n-1}\|x_{k+1}-x_{k}\|=\sum_{k=1}^{n-1}(x_{k+1}-x_{k})=x_{n}-x_{1}
\displaystyle P(Y\geq 1\|X\!=\!x)
E(1_{\{Y\geq t\}}\|X)=1_{\{Y\geq t\}}
\gamma<n
\displaystyle\text{Data type II}:
l_{1}(\eta,\widehat{\eta})=(\widehat{\eta}-\eta)^{2}
\displaystyle\left(\frac{p_{r}}{1-p_{c}}\mid\vartheta=\vartheta_{A}\right)\leq\alpha,
\rho(\vartheta)
V\left(\widehat{\vartheta_{l}}\right)=\left(1-\frac{2cE\left(\widehat{% \vartheta}_{MLE}\right)+2b-2}{2\mathcal{D}+c\left(cE\left(\widehat{\vartheta}_% {MLE}\right)^{2}-2a+2E\left(\widehat{\vartheta}_{MLE}\right)(b-1)\right)}% \right)^{2}\ \sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right).
g(\vartheta)
\text{ETC}=\frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-p_{c}},
\vartheta_{U}=200
\displaystyle\left(P_{r}\mid\vartheta=\vartheta_{A}\right)\leq\alpha,
v=\operatorname{exp}\left\{-\frac{\mathcal{T}}{\vartheta}\right\},\quad T_{k,% \mathcal{D}}=\frac{(n-\mathcal{D}+k)\mathcal{T}}{\mathcal{D}},
q(x;p,t)=\begin{cases}\frac{p^{t}}{\Gamma(t)}x^{t-1}e^{-px},&x>0\\ 0,&\text{ otherwise. }\end{cases}
X_{9}=1062
A_{k,\mathcal{D}}=(-1)^{k}\left(\begin{array}[]{l}n\\ \mathcal{D}\end{array}\right)\left(\begin{array}[]{l}\mathcal{D}\\ k\end{array}\right)u^{n-\mathcal{D}+k},
\displaystyle=\left(U_{2}^{\prime}\left(E\left(\widehat{\vartheta}_{MLE}\right% )\right)\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)
E(\widehat{\vartheta}_{MLE}^{2})
X_{\gamma,n}
E\left(\widehat{\vartheta_{l}}\right)=E\left(\widehat{\vartheta}_{MLE}\right)-% \frac{1}{c}\operatorname{ln}\left[1+\frac{c}{2\mathcal{D}}\left(cE\left(% \widehat{\vartheta}_{MLE}\right)^{2}-2a+2E\left(\widehat{\vartheta}_{MLE}% \right)(b-1)\right)\right],
p_{a}=P\left(\widehat{\vartheta_{l}}\geq t_{2}\right)=P\left[Z\geq\frac{t_{2}-% E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}% \right)}}\right],
t_{1}=2156
\widehat{\vartheta_{l}}=-\frac{1}{c}\ln E\left(e^{-c\vartheta}\mid Data\right).
f_{\widehat{\vartheta}_{MLE}}(x)=\left(1-v^{n}\right)^{-1}\left[\sum_{\mathcal% {D}=1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\ q\left(x-\mathcal{% T}_{k,\mathcal{D}};\frac{\mathcal{D}}{\vartheta},\mathcal{D}\right)+q\left(x;% \frac{\gamma}{\vartheta},\gamma\right)\right.\\ +\left.\gamma\left(\begin{array}[]{l}n\\ \gamma\end{array}\right)\sum_{k=1}^{\gamma}\frac{(-1)^{k}v^{n-\gamma+k}}{n-% \gamma+k}\left(\begin{array}[]{c}\gamma-1\\ k-1\end{array}\right)q\left(x-T_{k,\gamma};\frac{\gamma}{\vartheta},\gamma% \right)\right],\ 0<x<n\mathcal{T}
p_{c}=P\left(t_{1}\leq\widehat{\vartheta_{l}}<t_{2}\right)=P\left[\frac{t_{1}-% E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}% \right)}}\leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V% \left(\widehat{\vartheta_{l}}\right)}}\right].
E\left(e^{-cg}\mid Data\right)=\frac{\int_{\vartheta}e^{-cg(\vartheta)}L(x,% \vartheta)\rho(\vartheta)d\vartheta}{\int_{\vartheta}L(x,\vartheta)\rho(% \vartheta)d\vartheta}.
\vartheta_{U}=100
\vartheta_{A}=3000
X_{1,n},X_{2,n},...,X_{n,n}
\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{11}\}=1594
\vartheta_{U}=600
\hat{\vartheta}_{MLE}=\begin{cases}\frac{1}{\mathcal{D}}\left[\sum\limits_{i=1% }^{\mathcal{D}}x_{i,n}+(n-\mathcal{D})\mathcal{T}\right]&\text{if}\quad 1\leq% \mathcal{D}\leq\gamma-1\\ \frac{1}{\gamma}\left[\sum\limits_{i=1}^{\gamma}x_{i,n}+(n-\gamma)X_{\gamma,n}% \right]&\text{if}\quad\mathcal{D}=\gamma\\ n\mathcal{T}&\text{if}\quad\mathcal{D}=0.\end{cases}
p_{a},p_{r}
\displaystyle P_{1}(n,t_{1},t_{2}):\qquad
\displaystyle=\left(U_{1}^{\prime}\left(E\left(\widehat{\vartheta}_{MLE}\right% )\right)\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)
\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)
V\left(\widehat{\vartheta_{s}}\right)=\left(\frac{\mathcal{D}}{\mathcal{D}+b-1% }\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right).
t_{1}=2064
g(\vartheta)=\vartheta
b=2.5
\mathcal{D}+b>1
\mathcal{D}=\gamma
\widehat{\vartheta_{l}}
p_{r}=P\left(\widehat{\vartheta_{l}}<t_{1}\right)=P\left[Z<\frac{t_{1}-E\left(% \widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}\right)}}% \right],
\displaystyle\left(\frac{P\left[Z<\frac{t_{1}-E\left(\widehat{\vartheta_{s}}% \right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}\right]}{1-P\left[\frac{% t_{1}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{% s}}\right)}}\leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V% \left(\widehat{\vartheta_{s}}\right)}}\right]}\mid\vartheta=\vartheta_{A}% \right)\leq\alpha,
\displaystyle P\left(\text{Lot is accepted}\mid\vartheta=\vartheta_{U}\right)
U_{2}\left(\widehat{\vartheta}_{MLE}\right)
\widehat{\vartheta}_{l}
p_{r}=P\left(\widehat{\vartheta_{s}}<t_{1}\right)=P\left[Z<\frac{t_{1}-E\left(% \widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}% \right],
\displaystyle\left\{X_{1,n}<X_{2,n}<\cdots<X_{\mathcal{D},n}\right\}\quad\text% {if}\quad\mathcal{T}^{*}=\mathcal{T},
\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{9}\}=1062
\displaystyle E\left(\widehat{\vartheta_{l}}\right)
\gamma=9
n,t_{1}
\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-P\left[\frac{t_{1}-E\left(% \widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}% \leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V\left(% \widehat{\vartheta_{s}}\right)}}\right]}\right)\text{at}\ \vartheta_{A}
t_{1}\leq\widehat{\vartheta_{s}}<t_{2}

x_{i}=x_{min}

\xi^{\prime}_{n}(\vec{x},\vec{y})=\frac{\xi_{n}(\vec{x},\vec{y})}{\xi_{n}(\vec% {y},\vec{y})}

x_{1}\leq x_{2}\leq\ldots\leq x_{n}

x_{min}=x_{max}

x_{i}=x_{max}

f(x)=\sin(x)

X\sim\mbox{binom}(1,p)

m=cn^{\alpha}

q(1-\alpha/2)

\tau_{n}^{2}=n

\log\operatorname{Var}(\xi_{n})=\log V+\gamma\log n

2(x_{i^{\prime}}-x_{j^{\prime}})>0

d\mu(t)=f(t)dt

d\mu(t)=f_{Y}(t)dt

\displaystyle\int P(Y\!\geq\!t)d\mu(t)

\boldsymbol{Y=XZ}

f(t)=-\frac{d}{dt}P(Y\geq t)

\xi_{m}^{*}

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

\varepsilon\sim\mbox{norm}(0,\sigma^{2})

\xi\gtrsim 0.4

r_{min}

m=3,p=0.5

|\xi^{\prime}_{n}|\geq|\xi_{n}|

\displaystyle=-\frac{1}{3}P(Y\!\geq\!t)^{3}\Big{|}_{-\infty}^{\infty}=\frac{1}% {3}

P(1)=p

\mbox{dnorm}(0,\sigma^{2})(x)=\varphi(z/\sigma)/\sigma

Y\sim\mbox{equal}(m^{\prime},-1,1)

\xi_{n}(\vec{x},\vec{y})=1-\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}-1}

\displaystyle=\mbox{dunif(a,b)}*\mbox{dnorm}(0,\sigma^{2})(y)

1_{\{Y\geq t\}}

x_{1}=x_{2}=\ldots=x_{i}

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

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

\displaystyle\quad\mbox{for }x=1

Y\sim X^{2}+\varepsilon

\mbox{Var}(Z)=E(Z^{2})-E(Z)^{2}

f(x)<t

X,Y\sim\operatorname{equal}(5,-1,1)

m=2\sqrt{n}

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

\xi(X,Y)=1

x_{min}=x_{1}\leq x_{2}\leq\ldots\leq x_{n}=x_{max}

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

\displaystyle P(Y\geq 1|X\!=\!x)

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

\gamma<n

\displaystyle\text{Data type II}:

l_{1}(\eta,\widehat{\eta})=(\widehat{\eta}-\eta)^{2}

\displaystyle\left(\frac{p_{r}}{1-p_{c}}\mid\vartheta=\vartheta_{A}\right)\leq\alpha,

\rho(\vartheta)

V\left(\widehat{\vartheta_{l}}\right)=\left(1-\frac{2cE\left(\widehat{% \vartheta}_{MLE}\right)+2b-2}{2\mathcal{D}+c\left(cE\left(\widehat{\vartheta}_% {MLE}\right)^{2}-2a+2E\left(\widehat{\vartheta}_{MLE}\right)(b-1)\right)}% \right)^{2}\ \sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right).

g(\vartheta)

\text{ETC}=\frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-p_{c}},

\vartheta_{U}=200

\displaystyle\left(P_{r}\mid\vartheta=\vartheta_{A}\right)\leq\alpha,

v=\operatorname{exp}\left\{-\frac{\mathcal{T}}{\vartheta}\right\},\quad T_{k,% \mathcal{D}}=\frac{(n-\mathcal{D}+k)\mathcal{T}}{\mathcal{D}},

q(x;p,t)=\begin{cases}\frac{p^{t}}{\Gamma(t)}x^{t-1}e^{-px},&x>0\\ 0,&\text{ otherwise. }\end{cases}

X_{9}=1062

A_{k,\mathcal{D}}=(-1)^{k}\left(\begin{array}[]{l}n\\ \mathcal{D}\end{array}\right)\left(\begin{array}[]{l}\mathcal{D}\\ k\end{array}\right)u^{n-\mathcal{D}+k},

\displaystyle=\left(U_{2}^{\prime}\left(E\left(\widehat{\vartheta}_{MLE}\right% )\right)\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)

E(\widehat{\vartheta}_{MLE}^{2})

X_{\gamma,n}

E\left(\widehat{\vartheta_{l}}\right)=E\left(\widehat{\vartheta}_{MLE}\right)-% \frac{1}{c}\operatorname{ln}\left[1+\frac{c}{2\mathcal{D}}\left(cE\left(% \widehat{\vartheta}_{MLE}\right)^{2}-2a+2E\left(\widehat{\vartheta}_{MLE}% \right)(b-1)\right)\right],

p_{a}=P\left(\widehat{\vartheta_{l}}\geq t_{2}\right)=P\left[Z\geq\frac{t_{2}-% E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}% \right)}}\right],

t_{1}=2156

\widehat{\vartheta_{l}}=-\frac{1}{c}\ln E\left(e^{-c\vartheta}\mid Data\right).

f_{\widehat{\vartheta}_{MLE}}(x)=\left(1-v^{n}\right)^{-1}\left[\sum_{\mathcal% {D}=1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\ q\left(x-\mathcal{% T}_{k,\mathcal{D}};\frac{\mathcal{D}}{\vartheta},\mathcal{D}\right)+q\left(x;% \frac{\gamma}{\vartheta},\gamma\right)\right.\\ +\left.\gamma\left(\begin{array}[]{l}n\\ \gamma\end{array}\right)\sum_{k=1}^{\gamma}\frac{(-1)^{k}v^{n-\gamma+k}}{n-% \gamma+k}\left(\begin{array}[]{c}\gamma-1\\ k-1\end{array}\right)q\left(x-T_{k,\gamma};\frac{\gamma}{\vartheta},\gamma% \right)\right],\ 0<x<n\mathcal{T}

p_{c}=P\left(t_{1}\leq\widehat{\vartheta_{l}}<t_{2}\right)=P\left[\frac{t_{1}-% E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}% \right)}}\leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{l}}\right)}{\sqrt{V% \left(\widehat{\vartheta_{l}}\right)}}\right].

E\left(e^{-cg}\mid Data\right)=\frac{\int_{\vartheta}e^{-cg(\vartheta)}L(x,% \vartheta)\rho(\vartheta)d\vartheta}{\int_{\vartheta}L(x,\vartheta)\rho(% \vartheta)d\vartheta}.

\vartheta_{U}=100

\vartheta_{A}=3000

X_{1,n},X_{2,n},...,X_{n,n}

\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{11}\}=1594

\vartheta_{U}=600

\hat{\vartheta}_{MLE}=\begin{cases}\frac{1}{\mathcal{D}}\left[\sum\limits_{i=1% }^{\mathcal{D}}x_{i,n}+(n-\mathcal{D})\mathcal{T}\right]&\text{if}\quad 1\leq% \mathcal{D}\leq\gamma-1\\ \frac{1}{\gamma}\left[\sum\limits_{i=1}^{\gamma}x_{i,n}+(n-\gamma)X_{\gamma,n}% \right]&\text{if}\quad\mathcal{D}=\gamma\\ n\mathcal{T}&\text{if}\quad\mathcal{D}=0.\end{cases}

p_{a},p_{r}

\displaystyle P_{1}(n,t_{1},t_{2}):\qquad

\displaystyle=\left(U_{1}^{\prime}\left(E\left(\widehat{\vartheta}_{MLE}\right% )\right)\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)

\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right)

V\left(\widehat{\vartheta_{s}}\right)=\left(\frac{\mathcal{D}}{\mathcal{D}+b-1% }\right)^{2}\sigma^{2}\left({\widehat{\vartheta}_{MLE}}\right).

t_{1}=2064

g(\vartheta)=\vartheta

b=2.5

\mathcal{D}+b>1

\mathcal{D}=\gamma

\widehat{\vartheta_{l}}

p_{r}=P\left(\widehat{\vartheta_{l}}<t_{1}\right)=P\left[Z<\frac{t_{1}-E\left(% \widehat{\vartheta_{l}}\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}\right)}}% \right],

\displaystyle\left(\frac{P\left[Z<\frac{t_{1}-E\left(\widehat{\vartheta_{s}}% \right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}\right]}{1-P\left[\frac{% t_{1}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{% s}}\right)}}\leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V% \left(\widehat{\vartheta_{s}}\right)}}\right]}\mid\vartheta=\vartheta_{A}% \right)\leq\alpha,

\displaystyle P\left(\text{Lot is accepted}\mid\vartheta=\vartheta_{U}\right)

U_{2}\left(\widehat{\vartheta}_{MLE}\right)

\widehat{\vartheta}_{l}

p_{r}=P\left(\widehat{\vartheta_{s}}<t_{1}\right)=P\left[Z<\frac{t_{1}-E\left(% \widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}% \right],

\displaystyle\left\{X_{1,n}<X_{2,n}<\cdots<X_{\mathcal{D},n}\right\}\quad\text% {if}\quad\mathcal{T}^{*}=\mathcal{T},

\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{9}\}=1062

\displaystyle E\left(\widehat{\vartheta_{l}}\right)

\gamma=9

n,t_{1}

\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-P\left[\frac{t_{1}-E\left(% \widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{s}}\right)}}% \leq Z<\frac{t_{2}-E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V\left(% \widehat{\vartheta_{s}}\right)}}\right]}\right)\text{at}\ \vartheta_{A}

t_{1}\leq\widehat{\vartheta_{s}}<t_{2}