# The Reliability of Stored Water behind Dams Using the Multi-Component Stress-Strength System

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## Abstract

**:**

## 1. Introduction

## 2. Reliability Function

## 3. Maximum-Likelihood Estimation

#### Fisher Information Matrix

## 4. Bayesian Estimation

#### Gibbs Sampling

- Start with initial guess $\left({\alpha}_{1}^{\left(0\right)},{\alpha}_{2}^{\left(0\right)},{\beta}_{1}^{\left(0\right)},{\beta}_{2}^{\left(0\right)}\right).$
- Set $l=1.$
- Using the following M–H algorithm, generate ${\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{1}^{\left(l\right)}$ and ${\beta}_{2}^{\left(l\right)}\phantom{\rule{3.33333pt}{0ex}}$ from${\pi}_{1}^{*}\left({\alpha}_{1}^{\left(l\right)}\mid {\alpha}_{2}^{(l-1)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right),\phantom{\rule{3.33333pt}{0ex}}{\pi}_{2}^{*}\left({\alpha}_{2}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)$$,\phantom{\rule{3.33333pt}{0ex}}{\pi}_{3}^{*}\left({\beta}_{1}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)\phantom{\rule{3.33333pt}{0ex}}$ and ${\pi}_{4}^{*}\left({\beta}_{2}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{1}^{\left(l\right)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)\phantom{\rule{3.33333pt}{0ex}}$ with the normal proposal distributions$$N\left({\alpha}_{1}^{(l-1)},Var\left({\alpha}_{1}\right)\right),\phantom{\rule{3.33333pt}{0ex}}N\left({\alpha}_{2}^{(l-1)},Var\left({\alpha}_{2}\right)\right),\phantom{\rule{3.33333pt}{0ex}}N\left({\beta}_{1}^{(l-1)},Var\left({\beta}_{1}\right)\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}N\left({\beta}_{2}^{(l-1)},Var\left({\beta}_{2}\right)\right),$$
- Generate a proposal ${\alpha}_{1}^{*}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\alpha}_{1}^{(l-1)},Var\left({\alpha}_{1}\right)\right),\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{2}^{*}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\alpha}_{2}^{(l-1)},Var\left({\alpha}_{2}\right)\right),\phantom{\rule{3.33333pt}{0ex}}{\beta}_{1}^{*}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\beta}_{1}^{(l-1)},Var\left({\beta}_{1}\right)\right)\phantom{\rule{3.33333pt}{0ex}}$ and ${\beta}_{2}^{*}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\beta}_{2}^{(l-1)},Var\left({\beta}_{2}\right)\right).$

- (i)
- Evaluate the acceptance probabilities$$\left.\begin{array}{c}{\eta}_{{\alpha}_{1}}=min\left[1,\frac{{\pi}_{1}^{*}\left({\alpha}_{1}^{*}\mid {\alpha}_{2}^{(l-1)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}{{\pi}_{1}^{*}\left({\alpha}_{1}^{\left(l\right)}\mid {\alpha}_{2}^{(l-1)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}\right],\hfill \\ \\ {\eta}_{{\alpha}_{2}}=min\left[1,\frac{{\pi}_{2}^{*}\left({\alpha}_{2}^{*}\mid {\alpha}_{1}^{\left(l\right)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}{{\pi}_{2}^{*}\left({\alpha}_{2}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\beta}_{1}^{(l-1)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}\right]\hfill \\ \\ {\eta}_{{\beta}_{1}}=min\left[1,\frac{{\pi}_{3}^{*}\left({\beta}_{1}^{*}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}{{\pi}_{3}^{*}\left({\beta}_{1}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{2}^{(l-1)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}\right],\hfill \\ \\ {\eta}_{{\beta}_{2}}=min\left[1,\frac{{\pi}_{4}^{*}\left({\beta}_{2}^{*}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{1}^{\left(l\right)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}{{\pi}_{4}^{*}\left({\beta}_{2}^{\left(l\right)}\mid {\alpha}_{1}^{\left(l\right)},{\alpha}_{2}^{\left(l\right)},{\beta}_{1}^{\left(l\right)},\mathrm{x}\u0320,\phantom{\rule{4.pt}{0ex}}\mathrm{y}\u0320\right)}\right].\hfill \end{array}\right\}$$
- (ii)
- Generate a ${u}_{1}$, ${u}_{2},{u}_{3}\phantom{\rule{3.33333pt}{0ex}}$ and ${u}_{4}$ from a uniform $(0,1)$ distribution.
- (iii)
- If ${u}_{1}<{\eta}_{{\alpha}_{1}}$, accept the proposal and set ${\alpha}_{1}^{\left(l\right)}={\alpha}_{1}^{*}$, else set ${\alpha}_{1}^{\left(l\right)}={\alpha}_{1}^{(l-1)}$.
- (iv)
- If ${u}_{2}<{\eta}_{{\alpha}_{2}}$, accept the proposal and set ${\alpha}_{2}^{\left(l\right)}={\alpha}_{2}^{*}$, else set ${\alpha}_{2}^{\left(l\right)}={\alpha}_{2}^{(l-1)}$.
- (v)
- If ${u}_{3}$$<{\eta}_{{\beta}_{1}}$, accept the proposal and set ${\beta}_{1}^{\left(l\right)}={\beta}_{1}^{*}$, else set ${\beta}_{1}^{\left(l\right)}={\beta}_{1}^{(l-1)}$.
- (vi)
- If ${u}_{4}<{\eta}_{{\beta}_{2}}$,accept the proposal and set ${\beta}_{2}^{\left(l\right)}={\beta}_{2}^{*}$, else set ${\beta}_{2}^{\left(l\right)}={\beta}_{2}^{(l-1)}.$

- 5.
- Compute ${R}_{s,k}^{\left(l\right)}$ at (${\alpha}_{1}^{\left(l\right)},\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{2}^{\left(l\right)},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{1}^{\left(l\right)},{\beta}_{2}^{\left(l\right)}$).
- 6.
- Set $l=l+1.$
- 7.
- Repeat Steps $\left(3\right)-\left(6\right),$ N times and obtain ${\alpha}_{1}^{\left(l\right)},\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{2}^{\left(l\right)},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{1}^{\left(l\right)},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2}^{\left(l\right)}\phantom{\rule{3.33333pt}{0ex}}$ and ${R}_{s,k}^{\left(l\right)},l=1,2,\cdots ,N.$
- 8.
- To compute the CRIs of ${\alpha}_{1},{\alpha}_{2},$${\beta}_{1}$, ${\beta}_{2}$ and ${R}_{s,k},$${\psi}_{k}^{\left(l\right)},k=1,2,3,4,5,\phantom{\rule{3.33333pt}{0ex}}\left({\psi}_{1},{\psi}_{2},{\psi}_{3},{\psi}_{4},{\psi}_{5}\right)=\left({\alpha}_{1},{\alpha}_{2},{\beta}_{1},{\beta}_{2},{R}_{s,k}\right)$ as ${\psi}_{k}^{\left(1\right)}<{\psi}_{k}^{\left(2\right)}\cdots <{\psi}_{k}^{\left(N\right)},$ then the $100(1-\gamma )\%\phantom{\rule{3.33333pt}{0ex}}$ CRIs of $\phantom{\rule{3.33333pt}{0ex}}{\psi}_{k}$ is

## 5. Real Data Analysis

## 6. Simulation Study

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Convergence for the estimated parameters ${\alpha}_{1}$, ${\beta}_{1}$, ${\alpha}_{1}$, ${\beta}_{2}$ and R.

**Table 1.**The point estimates for ${\alpha}_{1}$, ${\beta}_{1}$, ${\alpha}_{2}$, ${\beta}_{2}$ and R.

${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{R}}_{2,4}$ | |
---|---|---|---|---|---|

MLE | 0.303858 | 0.402204 | 0.00569851 | 238.391 | 0.00789931 |

Bayes | 0.469386 | 0.393857 | 0.00850074 | 237.782 | 0.00824543 |

**Table 2.**The MLE and Bayesian estimates for the parameters $({\alpha}_{1},{\beta}_{1},{\alpha}_{2},{\beta}_{2})=(0.2,0.5,0.05,200)$ with the associated MSE between parenthesis.

MLE | Bayesian | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$({\mathbf{\alpha}}_{\mathbf{1}},{\mathbf{\beta}}_{\mathbf{1}},{\mathbf{\alpha}}_{\mathbf{2}},{\mathbf{\beta}}_{\mathbf{2}})$ | $(\mathit{s},\mathit{k})$ | $\mathit{n}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | |

(0.2, 0.5, 0.05, 200) | (3, 5) | 30 | 0.20146 | 0.50203 | 0.04999 | 199.837 | 0.1808 | 0.45054 | 0.04486 | 198.838 | |

(0.00017) | (0.00296) | (0.000) | (32.8054) | (0.0002) | (0.0034) | (0.000) | (33.0127) | ||||

40 | 0.2014 | 0.50013 | 0.05001 | 199.969 | 0.18074 | 0.44884 | 0.04488 | 198.969 | |||

(0.00013) | (0.00225) | (0.000) | (31.3774) | (0.00014) | (0.00271) | (0.000) | (33.9823) | ||||

50 | 0.2011 | 0.50103 | 0.04998 | 199.897 | 0.18047 | 0.44964 | 0.04485 | 198.898 | |||

(0.0001) | (0.00176) | (0.000) | (32.0521) | (0.00012) | (0.002) | (0.000) | (37.8047) | ||||

100 | 0.20071 | 0.49831 | 0.04993 | 199.688 | 0.18012 | 0.4472 | 0.04481 | 198.689 | |||

(0.00005) | (0.0009) | (0.000) | (31.9973) | (0.00005) | (0.00116) | (0.000) | (39.0693) | ||||

150 | 0.20037 | 0.5002 | 0.04999 | 200.017 | 0.17982 | 0.4489 | 0.04486 | 199.017 | |||

(0.00003) | (0.00059) | (0.000) | (33.7554) | (0.00004) | (0.00059) | (0.000) | (40.6417) | ||||

(5, 5) | 30 | 0.20222 | 0.49926 | 0.04997 | 199.657 | 0.18148 | 0.44805 | 0.04484 | 198.659 | ||

(0.00017) | (0.00292) | (0.000) | (33.13) | (0.0002) | (0.00331) | (0.000) | (35.7178) | ||||

40 | 0.20177 | 0.50036 | 0.04999 | 199.715 | 0.18107 | 0.44904 | 0.04486 | 198.716 | |||

(0.00012) | (0.00205) | (0.000) | (32.5364) | (0.00013) | (0.00232) | (0.000) | (38.4116) | ||||

50 | 0.20087 | 0.50181 | 0.04995 | 199.765 | 0.18027 | 0.45035 | 0.04483 | 198.767 | |||

(0.0001) | (0.00176) | (0.000) | (34.8049) | (0.00012) | (0.00227) | (0.000) | (35.433) | ||||

100 | 0.20051 | 0.5015 | 0.04999 | 199.824 | 0.17994 | 0.45006 | 0.04486 | 198.824 | |||

(0.00005) | (0.00088) | (0.000) | (31.978) | (0.00005) | (0.00114) | (0.000) | (36.2068) | ||||

150 | 0.20041 | 0.50021 | 0.05002 | 199.727 | 0.17986 | 0.44891 | 0.04489 | 198.728 | |||

(0.00003) | (0.00058) | (0.000) | (32.943) | (0.00004) | (0.00058) | (0.000) | (34.9308) | ||||

(5, 6) | 30 | 0.20152 | 0.50146 | 0.05001 | 199.8 | 0.18085 | 0.45003 | 0.04488 | 198.801 | ||

(0.00014) | (0.0022) | (0.000) | (32.295) | (0.00015) | (0.00232) | (0.000) | (33.4603) | ||||

40 | 0.20155 | 0.50179 | 0.05005 | 199.89 | 0.18088 | 0.45032 | 0.04492 | 198.891 | |||

(0.00011) | (0.0018) | (0.000) | (34.2618) | (0.00013) | (0.00204) | (0.000) | (34.2076) | ||||

50 | 0.20141 | 0.49875 | 0.05 | 199.765 | 0.18075 | 0.44759 | 0.04487 | 198.766 | |||

(0.00009) | (0.00149) | (0.000) | (30.8775) | (0.0001) | (0.00175) | (0.000) | (32.8875) | ||||

100 | 0.20038 | 0.5005 | 0.04995 | 200.074 | 0.17983 | 0.44916 | 0.04483 | 199.074 | |||

(0.00004) | (0.00067) | (0.000) | (31.4818) | (0.00005) | (0.00075) | (0.000) | (32.6242) | ||||

150 | 0.20014 | 0.50009 | 0.04999 | 200.067 | 0.17962 | 0.4488 | 0.04486 | 199.067 | |||

(0.00003) | (0.00052) | (0.000) | (33.1109) | (0.00004) | (0.00064) | (0.000) | (36.4121) |

**Table 3.**The MLE and Bayesian estimates for the parameters $({\alpha}_{1},{\beta}_{1},{\alpha}_{2},{\beta}_{2})=(0.3,0.4,0.05,200)$ with the associated MSE between parenthesis.

MLE | Bayesian | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$({\mathbf{\alpha}}_{\mathbf{1}},{\mathbf{\beta}}_{\mathbf{1}},{\mathbf{\alpha}}_{\mathbf{2}},{\mathbf{\beta}}_{\mathbf{2}})$ | $(\mathit{s},\mathit{k})$ | $\mathit{n}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | |

(0.3, 0.4, 0.05, 200) | (3, 5) | 30 | 0.3025 | 0.39763 | 0.05 | 199.822 | 0.27147 | 0.35685 | 0.04487 | 198.822 | |

(0.00036) | (0.00226) | (0.000) | (33.8999) | (0.00038) | (0.00234) | (0.000) | (40.3491) | ||||

40 | 0.30229 | 0.39954 | 0.04984 | 199.835 | 0.27129 | 0.35856 | 0.04473 | 198.836 | |||

(0.00026) | (0.0016) | (0.000) | (34.5172) | (0.00026) | (0.00166) | (0.000) | (35.5571) | ||||

50 | 0.30102 | 0.40185 | 0.05003 | 199.753 | 0.27014 | 0.36064 | 0.0449 | 198.754 | |||

(0.00021) | (0.00123) | (0.000) | (34.6565) | (0.00025) | (0.00138) | (0.000) | (42.4399) | ||||

100 | 0.30021 | 0.40196 | 0.05007 | 200.059 | 0.26942 | 0.36074 | 0.04493 | 199.059 | |||

(0.00012) | (0.0007) | (0.000) | (32.4818) | (0.00013) | (0.0007) | (0.000) | (32.5149) | ||||

150 | 0.30078 | 0.39987 | 0.04999 | 199.555 | 0.26993 | 0.35886 | 0.04486 | 198.558 | |||

(0.00007) | (0.00043) | (0.000) | (33.0983) | (0.00009) | (0.00052) | (0.000) | (38.4439) | ||||

(5, 5) | 30 | 0.30285 | 0.4029 | 0.04994 | 199.654 | 0.27179 | 0.36157 | 0.04482 | 198.656 | ||

(0.00038) | (0.00225) | (0.000) | (33.433) | (0.00049) | (0.00238) | (0.000) | (40.4265) | ||||

40 | 0.30167 | 0.40134 | 0.04994 | 200.091 | 0.27073 | 0.36017 | 0.04482 | 199.09 | |||

(0.00026) | (0.00166) | (0.000) | (34.0351) | (0.00032) | (0.00196) | (0.000) | (33.7554) | ||||

50 | 0.30159 | 0.40057 | 0.04993 | 200.15 | 0.27065 | 0.35949 | 0.04481 | 199.149 | |||

(0.00022) | (0.00129) | (0.000) | (34.2804) | (0.00026) | (0.00164) | (0.000) | (43.0006) | ||||

100 | 0.30051 | 0.40013 | 0.04995 | 199.983 | 0.26969 | 0.35909 | 0.04483 | 198.983 | |||

(0.00011) | (0.00068) | (0.000) | (32.9482) | (0.00011) | (0.00069) | (0.000) | (33.3591) | ||||

150 | 0.30005 | 0.40084 | 0.04996 | 199.626 | 0.26928 | 0.35973 | 0.04484 | 198.628 | |||

(0.00007) | (0.00044) | (0.000) | (33.8825) | (0.00008) | (0.00051) | (0.000) | (35.5814) | ||||

(5, 6) | 30 | 0.30217 | 0.40028 | 0.04992 | 200.064 | 0.27118 | 0.35923 | 0.0448 | 199.063 | ||

(0.00031) | (0.00158) | (0.000) | (33.822) | (0.0004) | (0.00236) | (0.000) | (36.4273) | ||||

40 | 0.30233 | 0.39868 | 0.05001 | 199.671 | 0.27132 | 0.35779 | 0.04488 | 198.672 | |||

(0.00025) | (0.00129) | (0.000) | (32.1462) | (0.0003) | (0.00168) | (0.000) | (37.4052) | ||||

50 | 0.30195 | 0.39879 | 0.05 | 200.079 | 0.27098 | 0.35789 | 0.04487 | 199.079 | |||

(0.0002) | (0.00109) | (0.000) | (33.8499) | (0.00021) | (0.00116) | (0.000) | (40.087) | ||||

100 | 0.30038 | 0.40072 | 0.05001 | 199.938 | 0.26957 | 0.35962 | 0.04489 | 198.938 | |||

(0.00009) | (0.00055) | (0.000) | (32.5754) | (0.00011) | (0.00064) | (0.000) | (36.7374) | ||||

150 | 0.30054 | 0.40001 | 0.04998 | 200.075 | 0.26971 | 0.35899 | 0.04486 | 199.075 | |||

(0.00006) | (0.00036) | (0.000) | (33.7145) | (0.00006) | (0.0004) | (0.000) | (38.2541) |

**Table 4.**ACIs and CRIs for $({\alpha}_{1},{\beta}_{1},{\alpha}_{2},{\beta}_{2})=(0.2,0.5,0.05,200)$ with their corresponding coverage probabilities between parenthesis.

MLE | Bayesian | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$({\mathbf{\alpha}}_{\mathbf{1}},{\mathbf{\beta}}_{\mathbf{1}},{\mathbf{\alpha}}_{\mathbf{2}},{\mathbf{\beta}}_{\mathbf{2}})$ | $(\mathit{s},\mathit{k})$ | $\mathit{n}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | |

(0.2, 0.5, 0.05, 200) | (3, 5) | 30 | 0.0507 | 0.2109 | 0.0281 | 567.508 | 0.0304 | 0.1265 | 0.0169 | 29.5104 | |

(0.95) | (0.951) | (1.000) | (1.000) | (0.9949) | (0.9951) | (0.995) | (0.995) | ||||

40 | 0.0436 | 0.1833 | 0.0244 | 491.674 | 0.0261 | 0.11 | 0.0146 | 25.567 | |||

(0.944) | (0.944) | (1.000) | (1.000) | (0.995) | (0.995) | (0.995) | (0.9949) | ||||

50 | 0.0389 | 0.1643 | 0.0218 | 438.494 | 0.0233 | 0.0986 | 0.0131 | 22.8017 | |||

(0.951) | (0.948) | (1.000) | (1.000) | (0.9951) | (0.9951) | (0.9951) | (0.9949) | ||||

100 | 0.0275 | 0.1159 | 0.0153 | 308.949 | 0.0165 | 0.0695 | 0.0092 | 16.0653 | |||

(0.954) | (0.935) | (1.000) | (1.000) | (0.9949) | (0.995) | (0.9949) | (0.995) | ||||

150 | 0.0224 | 0.0948 | 0.0125 | 252.359 | 0.0134 | 0.0569 | 0.0075 | 13.1227 | |||

(0.962) | (0.948) | (1.000) | (1.000) | (0.9949) | (0.9951) | (0.9949) | (0.995) | ||||

(5, 5) | 30 | 0.0506 | 0.2115 | 0.0283 | 570.071 | 0.0303 | 0.1269 | 0.017 | 29.6437 | ||

(0.951) | (0.933) | (1.000) | (1.000) | (0.9949) | (0.995) | (0.9952) | (0.9949) | ||||

40 | 0.0435 | 0.1839 | 0.0245 | 492.382 | 0.0261 | 0.1103 | 0.0147 | 25.6038 | |||

(0.949) | (0.945) | (1.000) | (1.000) | (0.9949) | (0.9949) | (0.995) | (0.995) | ||||

50 | 0.039 | 0.164 | 0.0218 | 439.355 | 0.0234 | 0.0984 | 0.0131 | 22.8464 | |||

(0.964) | (0.956) | (1.000) | (1.000) | (0.995) | (0.9951) | (0.9949) | (0.9951) | ||||

100 | 0.0274 | 0.1165 | 0.0154 | 309.768 | 0.0164 | 0.0699 | 0.0092 | 16.1079 | |||

(0.954) | (0.941) | (1.000) | (1.000) | (0.995) | (0.995) | (0.9951) | (0.995) | ||||

150 | 0.0224 | 0.0949 | 0.0125 | 252.428 | 0.0134 | 0.057 | 0.0075 | 13.1262 | |||

(0.953) | (0.952) | (1.000) | (1.000) | (0.9951) | (0.9951) | (0.9951) | (0.995) | ||||

(5, 6) | 30 | 0.046 | 0.1931 | 0.0282 | 567.143 | 0.0276 | 0.1159 | 0.0169 | 29.4914 | ||

(0.946) | (0.951) | (1.000) | (1.000) | (0.9951) | (0.995) | (0.995) | (0.995) | ||||

40 | 0.0397 | 0.168 | 0.0245 | 493.172 | 0.0238 | 0.1008 | 0.0147 | 25.6449 | |||

(0.945) | (0.951) | (1.000) | (1.000) | (0.9949) | (0.9952) | (0.995) | (0.995) | ||||

50 | 0.0356 | 0.1499 | 0.0218 | 439.107 | 0.0213 | 0.09 | 0.0131 | 22.8335 | |||

(0.938) | (0.941) | (1.000) | (1.000) | (0.9949) | (0.995) | (0.9951) | (0.9948) | ||||

100 | 0.025 | 0.106 | 0.0153 | 308.724 | 0.015 | 0.0636 | 0.0092 | 16.0537 | |||

(0.957) | (0.958) | (1.000) | (1.000) | (0.9951) | (0.9951) | (0.9949) | (0.9949) | ||||

150 | 0.0204 | 0.0866 | 0.0125 | 251.642 | 0.0123 | 0.052 | 0.0075 | 13.0854 | |||

(0.955) | (0.941) | (1.000) | (1.000) | (0.9949) | (0.9951) | (0.995) | (0.995) |

**Table 5.**ACIs and CRIs for $({\alpha}_{1},{\beta}_{1},{\alpha}_{2},{\beta}_{2})=(0.3,0.4,0.05,200)$ with their corresponding coverage probabilities between parenthesis.

MLE | Bayesian | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$(\mathit{s},\mathit{k})$ | $\mathit{n}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{1}}}$ | $\widehat{{\mathbf{\alpha}}_{\mathbf{2}}}$ | $\widehat{{\mathbf{\beta}}_{\mathbf{2}}}$ | ||

(0.3, 0.4, 0.05, 200) | (3, 5) | 30 | 0.0759 | 0.1848 | 0.0283 | 571.737 | 0.0455 | 0.1109 | 0.017 | 29.7303 | |

(0.952) | (0.938) | (1.000) | (1.000) | (0.9951) | (0.995) | (0.995) | (0.995) | ||||

40 | 0.0656 | 0.1594 | 0.0244 | 492.21 | 0.0394 | 0.0957 | 0.0147 | 25.5949 | |||

(0.959) | (0.949) | (1.000) | (1.000) | (0.995) | (0.995) | (0.9951) | (0.9949) | ||||

50 | 0.0582 | 0.1438 | 0.0217 | 437.88 | 0.0349 | 0.0863 | 0.013 | 22.7697 | |||

(0.954) | (0.953) | (1.000) | (1.000) | (0.9948) | (0.9948) | (0.9951) | (0.995) | ||||

100 | 0.0412 | 0.1013 | 0.0154 | 309.34 | 0.0247 | 0.0608 | 0.0092 | 16.0857 | |||

(0.96) | (0.95) | (1.000) | (1.000) | (0.995) | (0.9951) | (0.9949) | (0.995) | ||||

150 | 0.0335 | 0.0827 | 0.0125 | 251.705 | 0.0201 | 0.0496 | 0.0075 | 13.0887 | |||

(0.946) | (0.946) | (1.000) | (1.000) | (0.995) | (0.995) | (0.995) | (0.9949) | ||||

(5, 5) | 30 | 0.0759 | 0.1844 | 0.0282 | 568.27 | 0.0456 | 0.1106 | 0.0169 | 29.55 | ||

(0.94) | (0.95) | (1.000) | (1.000) | (0.995) | (0.995) | (0.9951) | (0.9952) | ||||

40 | 0.0655 | 0.1599 | 0.0245 | 493.459 | 0.0393 | 0.0959 | 0.0147 | 25.6599 | |||

(0.95) | (0.942) | (1.000) | (1.000) | (0.9951) | (0.9948) | (0.995) | (0.995) | ||||

50 | 0.0584 | 0.1429 | 0.0218 | 438.597 | 0.035 | 0.0857 | 0.0131 | 22.807 | |||

(0.947) | (0.94) | (1.000) | (1.000) | (0.9951) | (0.9951) | (0.9949) | (0.9949) | ||||

100 | 0.0411 | 0.1015 | 0.0153 | 309.374 | 0.0247 | 0.0609 | 0.0092 | 16.0874 | |||

(0.954) | (0.95) | (1.000) | (1.000) | (0.995) | (0.9951) | (0.9951) | (0.995) | ||||

150 | 0.0336 | 0.0827 | 0.0125 | 252.025 | 0.0201 | 0.0496 | 0.0075 | 13.1053 | |||

(0.953) | (0.964) | (1.000) | (1.000) | (0.9952) | (0.9951) | (0.995) | (0.9949) | ||||

(5, 6) | 30 | 0.0689 | 0.1689 | 0.0283 | 570.784 | 0.0413 | 0.1014 | 0.017 | 29.6808 | ||

(0.956) | (0.945) | (1.000) | (1.000) | (0.995) | (0.9951) | (0.995) | (0.9951) | ||||

40 | 0.0597 | 0.1463 | 0.0245 | 494.035 | 0.0358 | 0.0878 | 0.0147 | 25.6898 | |||

(0.953) | (0.948) | (1.000) | (1.000) | (0.995) | (0.9951) | (0.995) | (0.995) | ||||

50 | 0.0533 | 0.1305 | 0.0217 | 437.669 | 0.032 | 0.0783 | 0.013 | 22.7588 | |||

(0.944) | (0.945) | (1.000) | (1.000) | (0.9951) | (0.995) | (0.995) | (0.9953) | ||||

100 | 0.0376 | 0.0925 | 0.0153 | 308.903 | 0.0225 | 0.0555 | 0.0092 | 16.063 | |||

(0.949) | (0.954) | (1.000) | (1.000) | (0.9951) | (0.9951) | (0.995) | (0.9949) | ||||

150 | 0.0306 | 0.0756 | 0.0125 | 252.177 | 0.0184 | 0.0453 | 0.0075 | 13.1132 | |||

(0.957) | (0.952) | (1.000) | (1.000) | (0.9951) | (0.9947) | (0.995) | (0.995) |

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## Share and Cite

**MDPI and ACS Style**

Haj Ahmad, H.; Ramadan, D.A.; Mansour, M.M.M.; Aboshady, M.S.
The Reliability of Stored Water behind Dams Using the Multi-Component Stress-Strength System. *Symmetry* **2023**, *15*, 766.
https://doi.org/10.3390/sym15030766

**AMA Style**

Haj Ahmad H, Ramadan DA, Mansour MMM, Aboshady MS.
The Reliability of Stored Water behind Dams Using the Multi-Component Stress-Strength System. *Symmetry*. 2023; 15(3):766.
https://doi.org/10.3390/sym15030766

**Chicago/Turabian Style**

Haj Ahmad, Hanan, Dina A. Ramadan, Mahmoud M. M. Mansour, and Mohamed S. Aboshady.
2023. "The Reliability of Stored Water behind Dams Using the Multi-Component Stress-Strength System" *Symmetry* 15, no. 3: 766.
https://doi.org/10.3390/sym15030766