Kyudan/arXiv_latex · Datasets at Hugging Face

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\widehat{\vartheta_{l}}\geq t_{2}
l_{2}(\eta,\widehat{\eta})\propto e^{c(\widehat{\eta}-\eta)}-c(\widehat{\eta}-% \eta)-1,\quad c\neq 0.
E(\widehat{\vartheta}_{MLE})=\left(1-v^{n}\right)^{-1}{\left[\sum_{\mathcal{D}% =1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\left(\vartheta+T_{k,% \mathcal{D}}\right)+\vartheta\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}[]{l}\gamma-1\\ k-1\end{array}\right)\left(\vartheta+T_{k,\gamma}\right)\right],
\vartheta_{U}
P_{a}=\frac{p_{a}}{1-p_{c}}
L(x,\vartheta)=\frac{1}{\vartheta^{\gamma}}exp\left\{-\frac{1}{\vartheta}\sum% \limits_{i=1}^{\mathcal{D}}x_{i,n}-\frac{(n-\gamma)}{\vartheta}\mathcal{T}^{*}% \right\},
L(\vartheta,x)=-\gamma\operatorname{log}\vartheta-\frac{1}{\vartheta}\sum% \limits_{i=1}^{\mathcal{D}}x_{i,n}-\frac{(n-\gamma)}{\vartheta}\mathcal{T}^{*},
\widehat{\vartheta_{s}}<t_{1}.
\widehat{g}_{s}=E(g\mid Data)=\frac{\int_{\vartheta}g(\vartheta)L(x,\vartheta)% \rho(\vartheta)d\vartheta}{\int_{\vartheta}L(x,\vartheta)\rho(\vartheta)d% \vartheta}.
h(\vartheta\mid Data)=\frac{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a\right)^% {(\mathcal{D}+b)}}{\Gamma(\mathcal{D}+b)}\vartheta^{-\left(\mathcal{D}+b+1% \right)}\operatorname{exp}\left\{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a% \right)/\vartheta\right\}
\mathcal{D}\geq 1
\hat{\vartheta}_{MLE}
\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{\gamma,n}\}
\frac{1}{1-p_{c}}\ E\left(\widehat{\vartheta_{l}}\right)
\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{l}}\right)}{1-p_{c}}\right)\text{at}\ % \vartheta_{A}
t_{2}=2157
\vartheta_{A}
P_{1}(n,t_{1},t_{2})
\displaystyle\left(P_{a}\mid\vartheta=\vartheta_{U}\right)\leq\beta,
n,t_{1},
\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{l}}\right)}{1-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]}\right)\text{at}\ \vartheta_{A}
P_{r}=\frac{p_{r}}{1-p_{c}}
\displaystyle\left(\frac{P\left[Z\geq\frac{t_{2}-E\left(\widehat{\vartheta_{l}% }\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}\right)}}\right]}{1-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]}\mid\vartheta=\vartheta_{U}% \right)\leq\beta,
\mathcal{T}=50
\widehat{\vartheta_{l}}=\widehat{\vartheta}_{MLE}-\frac{1}{c}\operatorname{ln}% \left[1+\frac{c}{2\mathcal{D}}\left(c\widehat{\vartheta}_{MLE}^{2}-2a+2% \widehat{\vartheta}_{MLE}(b-1)\right)\right].
\displaystyle=\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).
p_{a}=P\left(\widehat{\vartheta_{s}}\geq t_{2}\right)=P\left[Z\geq\frac{t_{2}-% E\left(\widehat{\vartheta_{s}}\right)}{\sqrt{V\left(\widehat{\vartheta_{s}}% \right)}}\right],
p_{c}=P\left(t_{1}\leq\widehat{\vartheta_{s}}<t_{2}\right)=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].
g=g(\vartheta)
P_{1}\left(n,t_{1},t_{2}\right)
\displaystyle\left(\frac{P\left[Z\geq\frac{t_{2}-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_{U}% \right)\leq\beta,
\text{ETC}=\frac{C\ E\left(\widehat{\vartheta_{l}}\right)}{1-p_{c}},
\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-p_{c}}\right)\text{at}\ % \vartheta_{A}
\widehat{\vartheta_{l}}>t_{2}
\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{\gamma}\}
\widehat{\vartheta_{s}}
\displaystyle\text{Data type I}:
\displaystyle=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],
t_{1}\leq\widehat{\vartheta_{l}}<t_{2}
P_{2}(n,t_{1},t_{2})
\mathcal{T}=2000
E\left(\widehat{\vartheta_{s}}\right)=U_{1}\left(E\left(\widehat{\vartheta}_{% MLE}\right)\right)=\frac{\mathcal{D}\ E\left(\hat{\vartheta}_{MLE}\right)+a}{% \mathcal{D}+b-1},
\displaystyle P_{2}(n,t_{1},t_{2}):\qquad
\frac{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a\right)}{\vartheta}
\displaystyle=\left(\frac{\mathcal{D}}{\mathcal{D}+b-1}\right)^{2}\sigma^{2}% \left({\widehat{\vartheta}_{MLE}}\right).
t_{2}=2065
X_{11}=1594
\displaystyle\left\{X_{1,n}<X_{2,n}<\cdots<X_{\gamma,n}\right\}\quad\text{if}% \quad\mathcal{T}^{*}=X_{\gamma},
n,t_{1},t_{2}
f(x,\vartheta)=\frac{1}{\vartheta}exp\left\{-\frac{-x}{\vartheta}\right\}% \qquad x\geq 0,\vartheta>0
\widehat{\vartheta_{s}}=U_{1}\left(\widehat{\vartheta}_{MLE}\right)
\widehat{\vartheta_{l}}<t_{1}.
\vartheta_{A}=500
c=-0.5
\displaystyle P\left(\text{Lot is rejected}\mid\vartheta=\vartheta_{A}\right)
\mathcal{T}^{*}=X_{\gamma}
\displaystyle\leq\alpha,
E\left(\widehat{\vartheta}_{MLE}^{2}\right)=\left(1-v^{n}\right)^{-1}\left[% \sum_{\mathcal{D}=1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\left% \{\frac{\vartheta^{2}}{\mathcal{D}}(1+\mathcal{D})+2T_{k,\mathcal{D}}\vartheta% +\left(T_{k,\mathcal{D}}\right)^{2}\right\}\right.\\ +\frac{\vartheta^{2}}{\gamma}(1+\gamma)+\gamma\left(\begin{array}[]{l}n\\ \gamma\end{array}\right)\left.\cdot\sum_{k=1}^{\gamma}\frac{(-1)^{k}v^{n-% \gamma+k}}{n-\gamma+k}\left(\begin{array}[]{l}\gamma-1\\ k-1\end{array}\right)\left\{\frac{\vartheta^{2}}{\gamma}(1+\gamma)+2T_{k,% \gamma}\vartheta+\left(T_{k,\gamma}\right)^{2}\right\}\right].
\widehat{\vartheta_{s}}\geq t_{2}
E\left(\widehat{\vartheta_{s}}\right)=\frac{\mathcal{D}\ E\left(\hat{\vartheta% }_{MLE}\right)+a}{\mathcal{D}+b-1},
\displaystyle V\left(\widehat{\vartheta_{l}}\right)
0<\alpha,\beta<1
\widehat{\vartheta_{l}}=2883.2339
\chi^{2}_{2\left(b+a\right)}/2
\widehat{\vartheta_{s}}=\frac{\mathcal{D}\ \hat{\vartheta}_{MLE}+a}{\mathcal{D% }+b-1}.
\displaystyle=U_{2}\left(E\left(\widehat{\vartheta}_{MLE}\right)\right)
\vartheta_{A}=200
\widehat{\vartheta}_{MLE}
\begin{aligned} &11,35,49,170,329,381,708,958,1062,1167,1594,1925,1990,2223,23% 27,2400,2451,2471,2551,2565,2568,\\ &2694,2702,2761,2831,3034,3059,3112,3214,3478,3504,4329,6367,6976,7846,13403.% \end{aligned}
\displaystyle\left(\frac{P\left[Z<\frac{t_{1}-E\left(\widehat{\vartheta_{l}}% \right)}{\sqrt{V\left(\widehat{\vartheta_{l}}\right)}}\right]}{1-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]}\mid\vartheta=\vartheta_{A}% \right)\leq\alpha,
\displaystyle V\left(\widehat{\vartheta_{s}}\right)
\widehat{g}_{l}=-\frac{1}{c}\ln E\left(e^{-cg}\mid Data\right),
\rho(\vartheta)=\frac{a^{b}}{\Gamma(b)}\vartheta^{-(b+1)}e^{-a/\vartheta},% \quad\vartheta>0,\ a>0,\ \text{and}\ b>0.
\displaystyle\leq\beta,
\widehat{\vartheta_{s}}=2577.9286
\gamma=11
\displaystyle\left(\frac{p_{a}}{1-p_{c}}\mid\vartheta=\vartheta_{U}\right)\leq\beta,
\frac{1}{1-p_{c}}

\widehat{\vartheta_{l}}\geq t_{2}

l_{2}(\eta,\widehat{\eta})\propto e^{c(\widehat{\eta}-\eta)}-c(\widehat{\eta}-% \eta)-1,\quad c\neq 0.

E(\widehat{\vartheta}_{MLE})=\left(1-v^{n}\right)^{-1}{\left[\sum_{\mathcal{D}% =1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\left(\vartheta+T_{k,% \mathcal{D}}\right)+\vartheta\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}[]{l}\gamma-1\\ k-1\end{array}\right)\left(\vartheta+T_{k,\gamma}\right)\right],

\vartheta_{U}

P_{a}=\frac{p_{a}}{1-p_{c}}

L(x,\vartheta)=\frac{1}{\vartheta^{\gamma}}exp\left\{-\frac{1}{\vartheta}\sum% \limits_{i=1}^{\mathcal{D}}x_{i,n}-\frac{(n-\gamma)}{\vartheta}\mathcal{T}^{*}% \right\},

L(\vartheta,x)=-\gamma\operatorname{log}\vartheta-\frac{1}{\vartheta}\sum% \limits_{i=1}^{\mathcal{D}}x_{i,n}-\frac{(n-\gamma)}{\vartheta}\mathcal{T}^{*},

\widehat{\vartheta_{s}}<t_{1}.

\widehat{g}_{s}=E(g\mid Data)=\frac{\int_{\vartheta}g(\vartheta)L(x,\vartheta)% \rho(\vartheta)d\vartheta}{\int_{\vartheta}L(x,\vartheta)\rho(\vartheta)d% \vartheta}.

h(\vartheta\mid Data)=\frac{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a\right)^% {(\mathcal{D}+b)}}{\Gamma(\mathcal{D}+b)}\vartheta^{-\left(\mathcal{D}+b+1% \right)}\operatorname{exp}\left\{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a% \right)/\vartheta\right\}

\mathcal{D}\geq 1

\hat{\vartheta}_{MLE}

\mathcal{T}^{*}=\operatorname{Min}\{\mathcal{T},X_{\gamma,n}\}

\frac{1}{1-p_{c}}\ E\left(\widehat{\vartheta_{l}}\right)

\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{l}}\right)}{1-p_{c}}\right)\text{at}\ % \vartheta_{A}

t_{2}=2157

\vartheta_{A}

P_{1}(n,t_{1},t_{2})

\displaystyle\left(P_{a}\mid\vartheta=\vartheta_{U}\right)\leq\beta,

n,t_{1},

\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{l}}\right)}{1-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]}\right)\text{at}\ \vartheta_{A}

P_{r}=\frac{p_{r}}{1-p_{c}}

\displaystyle\left(\frac{P\left[Z\geq\frac{t_{2}-E\left(\widehat{\vartheta_{l}% }\right)}{\sqrt{V\left(\widehat{\vartheta_{l}}\right)}}\right]}{1-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]}\mid\vartheta=\vartheta_{U}% \right)\leq\beta,

\mathcal{T}=50

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

\displaystyle=\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).

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

p_{c}=P\left(t_{1}\leq\widehat{\vartheta_{s}}<t_{2}\right)=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].

g=g(\vartheta)

P_{1}\left(n,t_{1},t_{2}\right)

\displaystyle\left(\frac{P\left[Z\geq\frac{t_{2}-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_{U}% \right)\leq\beta,

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

\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(% \frac{C\ E\left(\widehat{\vartheta_{s}}\right)}{1-p_{c}}\right)\text{at}\ % \vartheta_{A}

\widehat{\vartheta_{l}}>t_{2}

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

\widehat{\vartheta_{s}}

\displaystyle\text{Data type I}:

\displaystyle=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],

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

P_{2}(n,t_{1},t_{2})

\mathcal{T}=2000

E\left(\widehat{\vartheta_{s}}\right)=U_{1}\left(E\left(\widehat{\vartheta}_{% MLE}\right)\right)=\frac{\mathcal{D}\ E\left(\hat{\vartheta}_{MLE}\right)+a}{% \mathcal{D}+b-1},

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

\frac{\left(\mathcal{D}\ \hat{\vartheta}_{MLE}+a\right)}{\vartheta}

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

t_{2}=2065

X_{11}=1594

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

n,t_{1},t_{2}

f(x,\vartheta)=\frac{1}{\vartheta}exp\left\{-\frac{-x}{\vartheta}\right\}% \qquad x\geq 0,\vartheta>0

\widehat{\vartheta_{s}}=U_{1}\left(\widehat{\vartheta}_{MLE}\right)

\widehat{\vartheta_{l}}<t_{1}.

\vartheta_{A}=500

c=-0.5

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

\mathcal{T}^{*}=X_{\gamma}

\displaystyle\leq\alpha,

E\left(\widehat{\vartheta}_{MLE}^{2}\right)=\left(1-v^{n}\right)^{-1}\left[% \sum_{\mathcal{D}=1}^{\gamma-1}\sum_{k=0}^{\mathcal{D}}A_{k,\mathcal{D}}\left% \{\frac{\vartheta^{2}}{\mathcal{D}}(1+\mathcal{D})+2T_{k,\mathcal{D}}\vartheta% +\left(T_{k,\mathcal{D}}\right)^{2}\right\}\right.\\ +\frac{\vartheta^{2}}{\gamma}(1+\gamma)+\gamma\left(\begin{array}[]{l}n\\ \gamma\end{array}\right)\left.\cdot\sum_{k=1}^{\gamma}\frac{(-1)^{k}v^{n-% \gamma+k}}{n-\gamma+k}\left(\begin{array}[]{l}\gamma-1\\ k-1\end{array}\right)\left\{\frac{\vartheta^{2}}{\gamma}(1+\gamma)+2T_{k,% \gamma}\vartheta+\left(T_{k,\gamma}\right)^{2}\right\}\right].

\widehat{\vartheta_{s}}\geq t_{2}

E\left(\widehat{\vartheta_{s}}\right)=\frac{\mathcal{D}\ E\left(\hat{\vartheta% }_{MLE}\right)+a}{\mathcal{D}+b-1},

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

0<\alpha,\beta<1

\widehat{\vartheta_{l}}=2883.2339

\chi^{2}_{2\left(b+a\right)}/2

\widehat{\vartheta_{s}}=\frac{\mathcal{D}\ \hat{\vartheta}_{MLE}+a}{\mathcal{D% }+b-1}.

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

\vartheta_{A}=200

\widehat{\vartheta}_{MLE}

\begin{aligned} &11,35,49,170,329,381,708,958,1062,1167,1594,1925,1990,2223,23% 27,2400,2451,2471,2551,2565,2568,\\ &2694,2702,2761,2831,3034,3059,3112,3214,3478,3504,4329,6367,6976,7846,13403.% \end{aligned}

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

\displaystyle V\left(\widehat{\vartheta_{s}}\right)

\widehat{g}_{l}=-\frac{1}{c}\ln E\left(e^{-cg}\mid Data\right),

\rho(\vartheta)=\frac{a^{b}}{\Gamma(b)}\vartheta^{-(b+1)}e^{-a/\vartheta},% \quad\vartheta>0,\ a>0,\ \text{and}\ b>0.

\displaystyle\leq\beta,

\widehat{\vartheta_{s}}=2577.9286

\gamma=11

\displaystyle\left(\frac{p_{a}}{1-p_{c}}\mid\vartheta=\vartheta_{U}\right)\leq\beta,

\frac{1}{1-p_{c}}