# Utilizing Empirical Bayes Estimation to Assess Reliability in Inverted Exponentiated Rayleigh Distribution with Progressive Hybrid Censored Medical Data

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## Abstract

**:**

## 1. Introduction

- Case 1: ${X}_{1:m:n}<{X}_{2:m:n}<\dots <{X}_{m:m:n}$, if ${X}_{m:m:n}<T$.
- Case 2: ${X}_{1:m:n}<{X}_{2:m:n}<\dots <{X}_{k:m:n}$, if ${X}_{k:m:n}<T<{X}_{k+1:m:n}$.

## 2. Maximum Likelihood Estimation

## 3. Bayes Estimation

## 4. Empirical Bayes Estimation

## 5. Application of IER Distribution to Real Data

## 6. Simulation

- For a given value of the prior parameter, $a(=0.5)$, generate the shape parameter $\eta (=2.0)$ from (12).
- Choose the values of $n(=50,100),m(=25,40,50,80),T(=1.5,3.0)$, and three different CSs, $({R}_{1},\cdots ,{R}_{m})$, presented in Table 1.
- The MLE of the prior parameter a is calculated and then the empirical Bayes estimates based on BSE and BLINEX (with $\Delta =0.4$ and $c=-0.3,3.0$) loss functions of the $\eta $, $S(x)$, and $h(x)$ (at time $x=1.2$) are calculated as shown in Section 4.
- Steps 3 to 5 are repeated $\mathcal{M}$ = 10,000 times and the average of estimates $(\overline{\widehat{\rho}}=\frac{1}{\mathcal{M}}{\sum}_{i=1}^{\mathcal{M}}{\widehat{\rho}}_{i})$, mean squared error $(\mathrm{MSE}(\widehat{\rho})=\frac{1}{\mathcal{M}}{\sum}_{i=1}^{\mathcal{M}}{({\widehat{\rho}}_{i}-\rho )}^{2})$, and relative absolute bias $(\mathrm{RAB}(\widehat{\rho})=\frac{1}{\mathcal{M}}{\sum}_{i=1}^{\mathcal{M}}\frac{|{\widehat{\rho}}_{i}-\rho |}{\rho})$ are calculated where $\widehat{\rho}$ is an estimate of $\rho $ (= $\eta $, $S(x)$, and $h(x)$ (at $x=1.2$)).
- The computational results of the ML, Bayes, and empirical Bayes estimates are presented in Table A1 and Table A2; see Appendix B.

#### Discussion

- The MSEs and RABs of the three methods of estimation decrease by increasing m for fixed T and n.
- The MSEs and RABs of the three methods of estimation decrease by increasing n for fixed T and m.
- The MSEs and RABs of the three methods of estimation decrease by increasing T for fixed n and m.
- The Bayes estimates of $\eta $, $S(x)$, and $h(x)$ (at time $x=1.2$) are better than the ML and empirical Bayes estimates by means of the MSEs and RABs.
- The empirical Bayes estimates of $\eta $, $S(x)$, and $h(x)$ (at time $x=1.2$) are better than the MLEs by means of the MSEs and RABs.
- The Bayes (empirical Bayes) estimates based on BLINEX (with $c=3.0$) loss function of $\eta $, $S(x)$, and $h(x)$ (at time $x=1.2$) are better than the Bayes (empirical Bayes) estimates based on BSE loss function by means of the MSEs and RABs.

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

**Table A1.**MLEs and Bayes estimates of $\eta $, $S\left(x\right)$, and $h\left(x\right)$ (at time $x=1.2$) with their MSEs and RABs.

Bayes Estimate | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

BLINEX | |||||||||||||||

MLE | BSE | $\mathit{c}=-\mathbf{0.3}$ | $\mathit{c}=\mathbf{3.0}$ | ||||||||||||

$\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | ||||

MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | ||||

$\mathit{n}$ | $\mathit{m}$ | $\mathit{T}$ | CS | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB |

50 | 25 | 1.5 | I | 1.977050 | 0.429075 | 1.871590 | 1.976570 | 0.433322 | 1.871140 | 1.994750 | 0.433887 | 1.887400 | 1.805450 | 0.427747 | 1.718180 |

0.152675 | 0.005345 | 0.136822 | 0.143822 | 0.005071 | 0.128888 | 0.147255 | 0.005089 | 0.131779 | 0.143169 | 0.004918 | 0.126959 | ||||

0.158370 | 0.139925 | 0.158370 | 0.153716 | 0.135749 | 0.153716 | 0.155389 | 0.135938 | 0.155290 | 0.153725 | 0.134238 | 0.152880 | ||||

II | 2.053630 | 0.414787 | 1.944080 | 2.050720 | 0.418692 | 1.941330 | 2.066000 | 0.419134 | 1.955010 | 1.903300 | 0.414335 | 1.809640 | |||

0.151215 | 0.004597 | 0.135513 | 0.143380 | 0.004280 | 0.128491 | 0.149333 | 0.004277 | 0.133523 | 0.110752 | 0.004321 | 0.099826 | ||||

0.156124 | 0.132388 | 0.156124 | 0.152255 | 0.127740 | 0.152255 | 0.154842 | 0.127697 | 0.154697 | 0.137237 | 0.128351 | 0.137559 | ||||

III | 2.025980 | 0.419955 | 1.917910 | 2.023880 | 0.423842 | 1.915920 | 2.039530 | 0.424308 | 1.929930 | 1.873620 | 0.419251 | 1.781670 | |||

0.152622 | 0.004831 | 0.136774 | 0.144647 | 0.004539 | 0.129627 | 0.149686 | 0.004542 | 0.133884 | 0.121450 | 0.004521 | 0.108915 | ||||

0.157028 | 0.134920 | 0.157028 | 0.153041 | 0.130625 | 0.153041 | 0.155232 | 0.130661 | 0.155107 | 0.143021 | 0.130511 | 0.143011 | ||||

3.0 | I | 2.004100 | 0.423050 | 1.897200 | 2.002700 | 0.426715 | 1.895870 | 2.017500 | 0.427165 | 1.909130 | 1.860620 | 0.422271 | 1.768910 | ||

0.128141 | 0.004183 | 0.114835 | 0.121832 | 0.003959 | 0.109182 | 0.125548 | 0.003964 | 0.112319 | 0.108912 | 0.003923 | 0.097280 | ||||

0.143928 | 0.124885 | 0.143928 | 0.140460 | 0.121305 | 0.140460 | 0.142201 | 0.121366 | 0.142101 | 0.135117 | 0.120924 | 0.134821 | ||||

II | 2.055290 | 0.414485 | 1.945660 | 2.052350 | 0.418400 | 1.942870 | 2.067640 | 0.418842 | 1.956570 | 1.904690 | 0.414042 | 1.810970 | |||

0.151250 | 0.004599 | 0.135545 | 0.143408 | 0.004279 | 0.128517 | 0.149410 | 0.004276 | 0.133590 | 0.110381 | 0.004323 | 0.099510 | ||||

0.155656 | 0.131949 | 0.155656 | 0.151795 | 0.127270 | 0.151795 | 0.154406 | 0.127222 | 0.154259 | 0.136486 | 0.127934 | 0.136826 | ||||

III | 2.038000 | 0.417241 | 1.929290 | 2.035580 | 0.421056 | 1.927000 | 2.050610 | 0.421498 | 1.940460 | 1.890710 | 0.416697 | 1.797580 | |||

0.140065 | 0.004334 | 0.125521 | 0.132927 | 0.004053 | 0.119124 | 0.138094 | 0.004053 | 0.123490 | 0.107059 | 0.004073 | 0.096216 | ||||

0.149735 | 0.127824 | 0.149735 | 0.146074 | 0.123535 | 0.146074 | 0.148425 | 0.123518 | 0.148292 | 0.134168 | 0.123925 | 0.134239 | ||||

40 | 1.5 | I | 2.000340 | 0.423111 | 1.893640 | 1.999790 | 0.425992 | 1.893110 | 2.011500 | 0.426359 | 1.903610 | 1.886360 | 0.422357 | 1.791760 | |

0.111063 | 0.003724 | 0.099530 | 0.106826 | 0.003582 | 0.095733 | 0.108861 | 0.003588 | 0.097451 | 0.100772 | 0.003531 | 0.090041 | ||||

0.134640 | 0.117433 | 0.134640 | 0.132068 | 0.114918 | 0.132068 | 0.133161 | 0.114993 | 0.133098 | 0.129239 | 0.114357 | 0.129013 | ||||

II | 2.007770 | 0.421330 | 1.900680 | 2.007150 | 0.423735 | 1.900090 | 2.016770 | 0.424036 | 1.908700 | 1.913330 | 0.420757 | 1.816240 | |||

0.100979 | 0.003310 | 0.090494 | 0.097717 | 0.003194 | 0.087571 | 0.099502 | 0.003197 | 0.089079 | 0.089680 | 0.003168 | 0.080385 | ||||

0.127933 | 0.110975 | 0.127933 | 0.125903 | 0.108915 | 0.125903 | 0.126838 | 0.108962 | 0.126786 | 0.122031 | 0.108589 | 0.121994 | ||||

III | 1.997610 | 0.423665 | 1.891050 | 1.997080 | 0.426512 | 1.890550 | 2.008680 | 0.426876 | 1.900940 | 1.884800 | 0.422910 | 1.790210 | |||

0.112349 | 0.003783 | 0.100683 | 0.108106 | 0.003641 | 0.096880 | 0.110106 | 0.003648 | 0.098569 | 0.102132 | 0.003588 | 0.091266 | ||||

0.135467 | 0.118320 | 0.135467 | 0.132905 | 0.115873 | 0.132905 | 0.133933 | 0.115955 | 0.133874 | 0.130031 | 0.115234 | 0.129841 | ||||

3.0 | I | 2.007740 | 0.420786 | 1.900650 | 2.007050 | 0.423150 | 1.899990 | 2.016360 | 0.423440 | 1.908330 | 1.91607 | 0.420275 | 1.818680 | ||

0.087140 | 0.002837 | 0.078091 | 0.084424 | 0.002733 | 0.075658 | 0.086130 | 0.002735 | 0.077099 | 0.07683 | 0.002723 | 0.068861 | ||||

0.118321 | 0.102508 | 0.118321 | 0.116513 | 0.100575 | 0.116513 | 0.117455 | 0.100604 | 0.117401 | 0.11273 | 0.100422 | 0.112671 | ||||

II | 2.026000 | 0.417566 | 1.917930 | 2.024970 | 0.419959 | 1.916960 | 2.034260 | 0.420243 | 1.925270 | 1.934050 | 0.417145 | 1.835700 | |||

0.090797 | 0.002865 | 0.081368 | 0.087935 | 0.002744 | 0.078804 | 0.090087 | 0.002743 | 0.080624 | 0.075998 | 0.002755 | 0.068332 | ||||

0.120531 | 0.103441 | 0.120531 | 0.118691 | 0.101279 | 0.118691 | 0.119834 | 0.101279 | 0.119770 | 0.112821 | 0.101414 | 0.112894 | ||||

III | 2.013390 | 0.419833 | 1.905990 | 2.012600 | 0.422217 | 1.905240 | 2.021960 | 0.422507 | 1.913620 | 1.921140 | 0.419344 | 1.823510 | |||

0.089442 | 0.002880 | 0.080155 | 0.086625 | 0.002769 | 0.077630 | 0.088483 | 0.002771 | 0.079201 | 0.077621 | 0.002764 | 0.069637 | ||||

0.119775 | 0.103457 | 0.119775 | 0.117935 | 0.101397 | 0.117935 | 0.118965 | 0.101412 | 0.118907 | 0.113161 | 0.101369 | 0.113146 | ||||

100 | 50 | 1.5 | I | 2.005890 | 0.421389 | 1.898890 | 2.005420 | 0.423743 | 1.898450 | 2.014830 | 0.424039 | 1.906880 | 1.913570 | 0.420802 | 1.816360 |

0.093472 | 0.003087 | 0.083766 | 0.090583 | 0.002987 | 0.081177 | 0.092077 | 0.002991 | 0.082439 | 0.084919 | 0.002958 | 0.076017 | ||||

0.122868 | 0.106780 | 0.122868 | 0.120976 | 0.104866 | 0.120976 | 0.121850 | 0.104913 | 0.121799 | 0.118039 | 0.104535 | 0.117925 | ||||

II | 2.040250 | 0.414756 | 1.931420 | 2.039280 | 0.416727 | 1.930500 | 2.046830 | 0.416957 | 1.937260 | 1.964950 | 0.414450 | 1.864060 | |||

0.086360 | 0.002627 | 0.077393 | 0.084086 | 0.002525 | 0.075355 | 0.085972 | 0.002523 | 0.076951 | 0.071526 | 0.002546 | 0.064485 | ||||

0.115452 | 0.098160 | 0.115452 | 0.114006 | 0.096388 | 0.114006 | 0.114966 | 0.096376 | 0.114913 | 0.108126 | 0.096608 | 0.108308 | ||||

III | 2.025150 | 0.417510 | 1.917130 | 2.024410 | 0.419527 | 1.916420 | 2.032240 | 0.419770 | 1.923440 | 1.947390 | 0.417122 | 1.847580 | |||

0.085481 | 0.002689 | 0.076605 | 0.083199 | 0.002595 | 0.074560 | 0.084821 | 0.002595 | 0.075932 | 0.073654 | 0.002599 | 0.066225 | ||||

0.115726 | 0.099310 | 0.115726 | 0.114230 | 0.097581 | 0.114230 | 0.115121 | 0.097586 | 0.115070 | 0.109475 | 0.097638 | 0.109552 | ||||

3.0 | I | 2.008910 | 0.420059 | 1.901750 | 2.008400 | 0.421970 | 1.901270 | 2.015870 | 0.422204 | 1.907960 | 1.935070 | 0.419643 | 1.835710 | ||

0.074111 | 0.002395 | 0.066415 | 0.072244 | 0.002322 | 0.064742 | 0.073451 | 0.002323 | 0.065763 | 0.066236 | 0.002317 | 0.059421 | ||||

0.108838 | 0.094142 | 0.108838 | 0.107497 | 0.092722 | 0.107497 | 0.108193 | 0.092746 | 0.108154 | 0.104660 | 0.092590 | 0.104632 | ||||

II | 2.036370 | 0.415353 | 1.927740 | 2.035460 | 0.417313 | 1.926880 | 2.042970 | 0.417542 | 1.933610 | 1.961470 | 0.415036 | 1.860750 | |||

0.083229 | 0.002552 | 0.074587 | 0.081056 | 0.002455 | 0.072639 | 0.082836 | 0.002454 | 0.074145 | 0.069497 | 0.002473 | 0.062619 | ||||

0.113293 | 0.096552 | 0.113293 | 0.111882 | 0.094808 | 0.111882 | 0.112836 | 0.094794 | 0.112783 | 0.106206 | 0.095033 | 0.106354 | ||||

III | 2.029630 | 0.416493 | 1.921360 | 2.028820 | 0.418433 | 1.920600 | 2.036280 | 0.418662 | 1.927280 | 1.95534 | 0.416155 | 1.854910 | |||

0.080508 | 0.002506 | 0.072149 | 0.078438 | 0.002415 | 0.070293 | 0.080072 | 0.002414 | 0.071676 | 0.06825 | 0.002428 | 0.061435 | ||||

0.112363 | 0.096154 | 0.112363 | 0.110976 | 0.094466 | 0.110976 | 0.111886 | 0.094457 | 0.111835 | 0.10561 | 0.094640 | 0.105746 | ||||

80 | 1.5 | I | 2.008290 | 0.419757 | 1.901170 | 2.008040 | 0.421271 | 1.900930 | 2.013970 | 0.421460 | 1.906240 | 1.949640 | 0.419389 | 1.848700 | |

0.063250 | 0.002067 | 0.056682 | 0.062005 | 0.002021 | 0.055567 | 0.062702 | 0.002023 | 0.056156 | 0.058751 | 0.002012 | 0.052666 | ||||

0.100523 | 0.087109 | 0.100523 | 0.099541 | 0.086130 | 0.099541 | 0.099975 | 0.086157 | 0.099951 | 0.098002 | 0.085934 | 0.097958 | ||||

II | 2.014560 | 0.418259 | 1.907100 | 2.014310 | 0.419501 | 1.906860 | 2.019120 | 0.419654 | 1.911170 | 1.966790 | 0.417977 | 1.864350 | |||

0.054446 | 0.001752 | 0.048793 | 0.053558 | 0.001716 | 0.047997 | 0.054141 | 0.001717 | 0.048490 | 0.050204 | 0.001715 | 0.045060 | ||||

0.092964 | 0.080253 | 0.092964 | 0.092222 | 0.079429 | 0.092222 | 0.092619 | 0.079438 | 0.092596 | 0.090293 | 0.079390 | 0.090311 | ||||

III | 2.011240 | 0.419161 | 1.903950 | 2.010960 | 0.420669 | 1.903700 | 2.016840 | 0.420857 | 1.908960 | 1.952970 | 0.418806 | 1.851830 | |||

0.061850 | 0.002012 | 0.055428 | 0.060640 | 0.001966 | 0.054343 | 0.061359 | 0.001967 | 0.054952 | 0.057118 | 0.001959 | 0.051217 | ||||

0.099054 | 0.085737 | 0.099054 | 0.098094 | 0.084719 | 0.098094 | 0.098567 | 0.084737 | 0.098540 | 0.096190 | 0.084603 | 0.096170 | ||||

3.0 | I | 2.016080 | 0.417773 | 1.908540 | 2.015790 | 0.418997 | 1.908260 | 2.020480 | 0.419146 | 1.912470 | 1.969350 | 0.417515 | 1.866720 | ||

0.049171 | 0.001576 | 0.044065 | 0.048389 | 0.001540 | 0.043364 | 0.048989 | 0.001540 | 0.043872 | 0.044788 | 0.001544 | 0.040222 | ||||

0.088458 | 0.076284 | 0.088458 | 0.087770 | 0.075457 | 0.087770 | 0.088198 | 0.075457 | 0.088174 | 0.085473 | 0.075504 | 0.085513 | ||||

II | 2.021400 | 0.416910 | 1.913580 | 2.021050 | 0.418120 | 1.913250 | 2.025670 | 0.418266 | 1.917380 | 1.975300 | 0.416672 | 1.872320 | |||

0.052228 | 0.001645 | 0.046804 | 0.051396 | 0.001606 | 0.046059 | 0.052083 | 0.001605 | 0.046640 | 0.046877 | 0.001613 | 0.042142 | ||||

0.089795 | 0.077108 | 0.089795 | 0.089104 | 0.076259 | 0.089104 | 0.089552 | 0.076256 | 0.089527 | 0.086606 | 0.076340 | 0.086652 | ||||

III | 2.012830 | 0.418386 | 1.905460 | 2.012560 | 0.419603 | 1.905200 | 2.017240 | 0.419751 | 1.909400 | 1.966270 | 0.418122 | 1.863800 | |||

0.049647 | 0.001594 | 0.044492 | 0.048858 | 0.001559 | 0.043785 | 0.049433 | 0.001559 | 0.044271 | 0.045489 | 0.001561 | 0.040839 | ||||

0.088777 | 0.076631 | 0.088777 | 0.088088 | 0.075839 | 0.088088 | 0.088484 | 0.075844 | 0.088462 | 0.086083 | 0.075840 | 0.086101 |

**Table A2.**Empirical Bayes estimates of $\eta $, $S\left(x\right)$, and $h\left(x\right)$ (at time $x=1.2$) with their MSEs and RABs.

Empirical Bayes Estimate | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

BLINEX | ||||||||||||

BSE | $\mathit{c}=-\mathbf{0.3}$ | $\mathit{c}=\mathbf{3.0}$ | ||||||||||

$\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\overline{\widehat{\mathit{S}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{h}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | ||||

MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | MSE | ||||

$\mathit{n}$ | $\mathit{m}$ | $\mathit{T}$ | CS | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB | RAB |

50 | 25 | 1.5 | I | 1.945050 | 0.420809 | 1.841300 | 1.932720 | 0.349998 | 1.824340 | 1.834250 | 0.432910 | 1.745690 |

0.148182 | 0.005897 | 0.132795 | 0.151826 | 0.021981 | 0.136390 | 0.144394 | 0.004954 | 0.128605 | ||||

0.155780 | 0.144480 | 0.155780 | 0.160234 | 0.291955 | 0.160752 | 0.153521 | 0.136140 | 0.153096 | ||||

II | 2.049740 | 0.413219 | 1.940410 | 2.053720 | 0.390717 | 1.941960 | 1.920290 | 0.414891 | 1.825760 | |||

0.162071 | 0.004511 | 0.145242 | 0.169236 | 0.006833 | 0.151926 | 0.114708 | 0.004641 | 0.103605 | ||||

0.158790 | 0.130590 | 0.158790 | 0.162230 | 0.163743 | 0.162341 | 0.139314 | 0.132548 | 0.139770 | ||||

III | 2.013250 | 0.416318 | 1.905860 | 2.011740 | 0.379299 | 1.901320 | 1.893610 | 0.421276 | 1.800690 | |||

0.159461 | 0.004925 | 0.142903 | 0.165017 | 0.010804 | 0.148156 | 0.124839 | 0.004753 | 0.112253 | ||||

0.157956 | 0.135153 | 0.157956 | 0.161362 | 0.203428 | 0.161569 | 0.144523 | 0.133767 | 0.144703 | ||||

3.0 | I | 1.942470 | 0.407031 | 1.838860 | 1.900590 | 0.289687 | 1.791260 | 1.894680 | 0.432888 | 1.801660 | ||

0.136838 | 0.004068 | 0.122629 | 0.162728 | 0.029203 | 0.148680 | 0.106873 | 0.004487 | 0.095823 | ||||

0.149141 | 0.123577 | 0.149141 | 0.165343 | 0.348686 | 0.167277 | 0.133133 | 0.129918 | 0.133102 | ||||

II | 2.051190 | 0.412902 | 1.941780 | 2.055160 | 0.390135 | 1.943310 | 1.921800 | 0.414615 | 1.827200 | |||

0.161975 | 0.004535 | 0.145156 | 0.168870 | 0.007063 | 0.151576 | 0.114427 | 0.004633 | 0.103374 | ||||

0.158156 | 0.130330 | 0.158156 | 0.161499 | 0.165360 | 0.161603 | 0.138630 | 0.131975 | 0.139109 | ||||

III | 2.018860 | 0.411926 | 1.911170 | 2.011360 | 0.364995 | 1.900390 | 1.912060 | 0.419837 | 1.817940 | |||

0.148538 | 0.004245 | 0.133115 | 0.158030 | 0.010572 | 0.142238 | 0.109445 | 0.004396 | 0.098626 | ||||

0.151835 | 0.126563 | 0.151835 | 0.157404 | 0.205068 | 0.157834 | 0.135154 | 0.128330 | 0.135446 | ||||

40 | 1.5 | I | 1.948080 | 0.410970 | 1.844170 | 1.917920 | 0.317866 | 1.809340 | 1.912030 | 0.430903 | 1.816450 | |

0.106463 | 0.003999 | 0.095408 | 0.114252 | 0.020614 | 0.103203 | 0.101958 | 0.003594 | 0.091374 | ||||

0.132250 | 0.121824 | 0.132250 | 0.139596 | 0.299050 | 0.140504 | 0.129534 | 0.115952 | 0.129508 | ||||

II | 1.951440 | 0.408380 | 1.847350 | 1.916530 | 0.311434 | 1.807900 | 1.937010 | 0.430089 | 1.839100 | |||

0.099383 | 0.003474 | 0.089063 | 0.109623 | 0.019699 | 0.099375 | 0.090662 | 0.003305 | 0.081447 | ||||

0.127471 | 0.114151 | 0.127471 | 0.136625 | 0.296695 | 0.137800 | 0.122276 | 0.111448 | 0.122376 | ||||

III | 1.952030 | 0.413119 | 1.847910 | 1.926830 | 0.330837 | 1.818430 | 1.908340 | 0.430331 | 1.812850 | |||

0.108002 | 0.004059 | 0.096788 | 0.113875 | 0.018180 | 0.102637 | 0.103609 | 0.003629 | 0.092842 | ||||

0.133132 | 0.122187 | 0.133132 | 0.138902 | 0.277586 | 0.139568 | 0.130642 | 0.116646 | 0.130627 | ||||

3.0 | I | 1.921950 | 0.400770 | 1.819430 | 1.862270 | 0.255898 | 1.753630 | 1.946970 | 0.434895 | 1.848630 | ||

0.092571 | 0.002959 | 0.082958 | 0.119059 | 0.034949 | 0.109993 | 0.075773 | 0.003170 | 0.068102 | ||||

0.123789 | 0.106173 | 0.123789 | 0.143270 | 0.400192 | 0.145845 | 0.111513 | 0.109046 | 0.111600 | ||||

II | 1.961720 | 0.402672 | 1.857070 | 1.919750 | 0.293153 | 1.810190 | 1.959490 | 0.427888 | 1.860300 | |||

0.092356 | 0.002926 | 0.082766 | 0.107848 | 0.021643 | 0.098337 | 0.076462 | 0.003015 | 0.068911 | ||||

0.122769 | 0.105208 | 0.122769 | 0.135281 | 0.314475 | 0.136793 | 0.112563 | 0.106685 | 0.112746 | ||||

III | 1.936870 | 0.401986 | 1.833550 | 1.884670 | 0.272109 | 1.775740 | 1.949760 | 0.432315 | 1.851220 | |||

0.093625 | 0.002973 | 0.083903 | 0.115659 | 0.029572 | 0.106317 | 0.077106 | 0.003141 | 0.069353 | ||||

0.123707 | 0.106245 | 0.123707 | 0.140078 | 0.364342 | 0.142210 | 0.112363 | 0.108430 | 0.112510 | ||||

100 | 50 | 1.5 | I | 1.949950 | 0.408753 | 1.84594 | 1.916090 | 0.313411 | 1.807590 | 1.936600 | 0.430001 | 1.838590 |

0.089794 | 0.003310 | 0.08047 | 0.098121 | 0.019307 | 0.088864 | 0.085949 | 0.003028 | 0.077128 | ||||

0.120880 | 0.111438 | 0.12088 | 0.128682 | 0.292066 | 0.129686 | 0.118405 | 0.106057 | 0.118455 | ||||

II | 2.020040 | 0.410110 | 1.912290 | 2.009490 | 0.371991 | 1.899580 | 1.976530 | 0.417777 | 1.875190 | |||

0.089742 | 0.002564 | 0.080424 | 0.093812 | 0.005076 | 0.084386 | 0.072814 | 0.002682 | 0.065684 | ||||

0.116747 | 0.097045 | 0.116747 | 0.120207 | 0.149887 | 0.120534 | 0.108724 | 0.099020 | 0.108925 | ||||

III | 1.995590 | 0.410856 | 1.889140 | 1.979220 | 0.358340 | 1.870030 | 1.961540 | 0.421963 | 1.861210 | |||

0.086057 | 0.002713 | 0.077121 | 0.090261 | 0.007886 | 0.081230 | 0.074918 | 0.002691 | 0.067435 | ||||

0.115549 | 0.100044 | 0.115549 | 0.119522 | 0.184521 | 0.119941 | 0.110066 | 0.099321 | 0.110211 | ||||

3.0 | I | 1.920320 | 0.399955 | 1.817890 | 1.858790 | 0.254445 | 1.750400 | 1.963210 | 0.434703 | 1.863090 | ||

0.078849 | 0.002585 | 0.070662 | 0.103482 | 0.034019 | 0.095900 | 0.065675 | 0.002705 | 0.059056 | ||||

0.114500 | 0.099010 | 0.114500 | 0.134261 | 0.399018 | 0.136937 | 0.103654 | 0.100855 | 0.103738 | ||||

II | 2.015610 | 0.410621 | 1.908090 | 2.004810 | 0.371826 | 1.895100 | 1.973060 | 0.418446 | 1.871900 | |||

0.086057 | 0.002500 | 0.077121 | 0.089923 | 0.005104 | 0.080877 | 0.070700 | 0.002598 | 0.063743 | ||||

0.114315 | 0.095707 | 0.114315 | 0.117674 | 0.150510 | 0.117994 | 0.106749 | 0.097245 | 0.106935 | ||||

III | 1.994250 | 0.408469 | 1.887870 | 1.973090 | 0.347123 | 1.863710 | 1.970540 | 0.422008 | 1.869600 | |||

0.082005 | 0.002494 | 0.073490 | 0.088284 | 0.008996 | 0.079699 | 0.069194 | 0.002563 | 0.062350 | ||||

0.113145 | 0.096223 | 0.113145 | 0.118652 | 0.197952 | 0.119272 | 0.106020 | 0.097243 | 0.106203 | ||||

80 | 1.5 | I | 1.940600 | 0.405031 | 1.837080 | 1.896840 | 0.296741 | 1.788770 | 1.969970 | 0.430635 | 1.868500 | |

0.061065 | 0.002300 | 0.054724 | 0.071052 | 0.019984 | 0.064955 | 0.059369 | 0.002094 | 0.053326 | ||||

0.100366 | 0.092444 | 0.100366 | 0.110652 | 0.305950 | 0.112101 | 0.098091 | 0.088377 | 0.098141 | ||||

II | 1.944150 | 0.403079 | 1.840440 | 1.897840 | 0.292191 | 1.789620 | 1.985970 | 0.429717 | 1.883110 | |||

0.053441 | 0.001982 | 0.047892 | 0.064108 | 0.019849 | 0.058849 | 0.050778 | 0.001815 | 0.045659 | ||||

0.093563 | 0.086209 | 0.093563 | 0.105010 | 0.309314 | 0.106631 | 0.090462 | 0.082179 | 0.090554 | ||||

III | 1.951450 | 0.406182 | 1.847360 | 1.913440 | 0.310227 | 1.805240 | 1.971420 | 0.428703 | 1.869790 | |||

0.059517 | 0.002207 | 0.053337 | 0.067355 | 0.016471 | 0.061333 | 0.057786 | 0.002018 | 0.051912 | ||||

0.098184 | 0.090446 | 0.098184 | 0.106501 | 0.275523 | 0.107666 | 0.096425 | 0.086231 | 0.096483 | ||||

3.0 | I | 1.914540 | 0.395729 | 1.812420 | 1.844000 | 0.235823 | 1.735680 | 1.995160 | 0.434726 | 1.892020 | ||

0.053750 | 0.001983 | 0.048169 | 0.078187 | 0.037936 | 0.073462 | 0.044912 | 0.001863 | 0.040437 | ||||

0.094968 | 0.086851 | 0.094968 | 0.117158 | 0.437717 | 0.120360 | 0.085111 | 0.083738 | 0.085243 | ||||

II | 1.944870 | 0.400265 | 1.841130 | 1.893170 | 0.279579 | 1.784600 | 1.995650 | 0.429536 | 1.892230 | |||

0.053861 | 0.001850 | 0.048268 | 0.068330 | 0.022407 | 0.063117 | 0.047182 | 0.001803 | 0.042488 | ||||

0.093806 | 0.082546 | 0.093806 | 0.108367 | 0.334236 | 0.110379 | 0.086234 | 0.081991 | 0.086343 | ||||

III | 1.922590 | 0.398700 | 1.820040 | 1.860250 | 0.256237 | 1.752120 | 1.989640 | 0.433403 | 1.886700 | |||

0.053524 | 0.001883 | 0.047966 | 0.073927 | 0.030694 | 0.069028 | 0.045466 | 0.001843 | 0.040903 | ||||

0.094728 | 0.084164 | 0.094728 | 0.113737 | 0.389377 | 0.116441 | 0.085619 | 0.083385 | 0.085720 |

## References

- Ghitany, M.E.; Tuan, V.K.; Balakrishnan, N. Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data. J. Stat. Comput. Simul.
**2014**, 84, 96–106. [Google Scholar] [CrossRef] - Rastogi, M.K.; Tripathi, Y.M. Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring. J. Appl. Stat.
**2014**, 41, 2375–2405. [Google Scholar] [CrossRef] - Kohansal, A. Large Estimation of the stress-strength reliability of progressively censored inverted exponentiated Rayleigh distributions. J. Appl. Math. Inform.
**2017**, 13, 49–76. [Google Scholar] [CrossRef] - Kayal, T.; Tripathi, Y.M.; Rastogi, M.K. Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring. Commun. Stat.-Theory Methods
**2018**, 47, 1615–1640. [Google Scholar] [CrossRef] - Maurya, R.K.; Tripathi, Y.M.; Rastogi, M.K. Estimation and prediction for a progressively first-failure censored inverted exponentiated Rayleigh distribution. J. Stat. Theory Pract.
**2019**, 13, 1–48. [Google Scholar] [CrossRef] - Rao, G.S.; Mbwambo, S. Exponentiated inverse Rayleigh distribution and an application to coating weights of iron sheets data. J. Probab. Stat.
**2019**, 2019, 7519429. [Google Scholar] [CrossRef] - Rao, G.S.; Mbwambo, S.; Josephat, P.K. Estimation of stress–strength reliability from exponentiated inverse Rayleigh distribution. Int. J. Reliab. Qual. Saf. Eng.
**2019**, 26, 1950005. [Google Scholar] [CrossRef] - Gao, S.; Yu, J.; Gui, W. Pivotal inference for the inverted exponentiated Rayleigh distribution based on progressive type-II censored data. Am. J. Math. Manag. Sci.
**2020**, 39, 315–328. [Google Scholar] [CrossRef] - Panahi, H.; Moradi, N. Estimation of the inverted exponentiated Rayleigh distribution based on adaptive Type II progressive hybrid censored sample. J. Comput. Appl. Math.
**2020**, 364, 112–345. [Google Scholar] [CrossRef] - Fan, J.; Gui, W. Statistical inference of inverted exponentiated rayleigh distribution under joint progressively type-II censoring. Entropy
**2022**, 24, 171. [Google Scholar] [CrossRef] - Anwar, S.; Lone, S.A.; Khan, A.; Almutlak, S. Stress-strength reliability estimation for the inverted exponentiated Rayleigh distribution under unified progressive hybrid censoring with application. Electron. Res. Arch.
**2023**, 31, 4011–4033. [Google Scholar] [CrossRef] - Chalabi, L. High-resolution sea clutter modelling using compound inverted exponentiated Rayleigh distribution. Remote Sens. Lett.
**2023**, 14, 33–441. [Google Scholar] [CrossRef] - Epstein, B. Truncated life tests in the exponential case. Ann. Math. Stat.
**1954**, 25, 555–564. [Google Scholar] [CrossRef] - Kundu, D.; Joarder, A. Analysis of Type-II progressively hybrid censored data. Comput. Stat. Data Anal.
**2006**, 50, 2509–2528. [Google Scholar] [CrossRef] - Lin, C.T.; Huang, Y.L.; Balakrishnan, N. Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring. J. Stat. Comput. Simul.
**2011**, 81, 873–882. [Google Scholar] [CrossRef] - Lin, C.T.; Huang, Y.L. On progressive hybrid censored exponential distribution. J. Stat. Comput. Simul.
**2012**, 82, 689–709. [Google Scholar] [CrossRef] - Gürünlü Alma, Ö.; Arabi Belaghi, R. On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data. J. Stat. Comput. Simul.
**2016**, 86, 569–596. [Google Scholar] [CrossRef] - Kayal, T.; Tripathi, Y.M.; Rastogi, M.K.; Asgharzadeh, A. Inference for Burr XII distribution under Type I progressive hybrid censoring. Commun.-Stat.-Simul. Comput.
**2017**, 46, 7447–7465. [Google Scholar] [CrossRef] - Asl, M.N.; Belaghi, R.A.; Bevrani, H. Classical and Bayesian inferential approaches using Lomax model under progressively type-I hybrid censoring. J. Comput. Appl. Math.
**2018**, 343, 397–412. [Google Scholar] [CrossRef] - Basu, S.; Singh, S.K.; Singh, U. Bayesian inference using product of spacings function for Progressive hybrid Type-I censoring scheme. Statistics
**2018**, 52, 345–363. [Google Scholar] [CrossRef] - Goyal, T.; Rai, P.K.; Maurya, S.K. Bayesian estimation for GDUS exponential distribution under type-I progressive hybrid censoring. Ann. Data Sci.
**2020**, 7, 307–345. [Google Scholar] [CrossRef] - Berger, J.O. Statistical Decision Theory and Bayesian Analysis; Springer: New York, NY, USA, 1985. [Google Scholar]
- Wang, F.K.; Keats, J.B.; Zimmer, W.J. Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data. Microelectron. Reliab.
**1996**, 36, 359–362. [Google Scholar] [CrossRef] - Zimmer, W.J.; Keats, J.B.; Wang, F.K. The Burr XII distribution in reliability analysis. J. Qual. Technol.
**1998**, 30, 386–394. [Google Scholar] [CrossRef] - Nassar, M.M.; Eissa, F.H. Bayesian estimation for the exponentiated Weibull model. Commun. Stat.-Theory Methods
**2005**, 33, 2343–2362. [Google Scholar] [CrossRef] - Kim, C.; Jung, J.; Chung, Y. Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring. Stat. Pap.
**2011**, 52, 53–70. [Google Scholar] [CrossRef] - Calabria, R.; Pulcini, G. Point estimation under asymmetric loss functions for left-truncated exponential samples. Commun. Stat.-Theory Methods
**1996**, 25, 585–600. [Google Scholar] [CrossRef] - EL-Sagheer, R.M.; Eliwa, M.S.; Alqahtani, K.M.; EL-Morshedy, M. Asymmetric randomly censored mortality distribution: Bayesian framework and parametric bootstrap with application to COVID-19 data. J. Math.
**2022**, 2022, 1–14. [Google Scholar] [CrossRef] - Zellner, A. Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions; Springer: New York, NY, USA, 1994; pp. 377–390. [Google Scholar]
- Ahmadi, J.; Jozani, M.J.; Marchand, É.; Parsian, A. Bayes estimation based on k-record data from a general class of distributions under balanced type loss functions. J. Stat. Plan. Inference
**2009**, 139, 1180–1189. [Google Scholar] [CrossRef] - Jafari, J.M.; Marchand, É.; Parsian, A. Bayesian and Robust Bayesian analysis under a general class of balanced loss functions. Stat. Pap.
**2012**, 53, 51–60. [Google Scholar] [CrossRef] - Oksuz, K.; Cam, B.C.; Akbas, E.; Kalkan, S. A ranking-based, balanced loss function unifying classification and localisation in object detection. Adv. Neural Inf. Process.
**2020**, 33, 15534–15545. [Google Scholar] - Yousef, M.M.; Almetwally, E.M. Multi stress-strength reliability based on progressive first failure for Kumaraswamy model: Bayesian and non-Bayesian estimation. Symmetry
**2021**, 13, 2120. [Google Scholar] [CrossRef] - Benkhaled, A.; Hamdaoui, A.; Almutiry, W.; Alshahrani, M.; Terbeche, M. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function. Open Math.
**2022**, 20, 1–11. [Google Scholar] [CrossRef] - Xie, Z.; Shu, C.; Fu, Y.; Zhou, J.; Chen, D. Balanced Loss Function for Accurate Surface Defect Segmentation. Appl. Sci.
**2023**, 13, 826. [Google Scholar] [CrossRef] - Yan, S.; Gendai, G. Bayes estimation for reliability indexes of cold standby system. J. N. China Electr. Power Univ.
**2003**, 30, 96–99. [Google Scholar] - Shi, Y.; Shi, X.; Xu, Y. Approximate confidence limits of the reliability performances for a cold standby series system. J. Appl. Math. Comput.
**2005**, 19, 439–445. [Google Scholar] [CrossRef] - Petrone, S.; Rizzelli, S.; Rousseau, J.; Scricciolo, C. Empirical Bayes methods in classical and Bayesian inference. Metron
**2014**, 72, 201–215. [Google Scholar] [CrossRef] - Gross, A.J.; Clark, V.A. Survival Distributions: Reliability Applications in the Biomedical Sciences; Wiley: New York, NY, USA, 1975; Volume 11. [Google Scholar]

**Figure 1.**Plots of PDF, CDF, reliability function, and failure rate function of IER distribution for different parameter values.

**Figure 3.**The procedure for creating order statistics for Type-I PHCS when ${X}_{k:m:n}<T<{X}_{k+1:m:n}$.

**Figure 4.**Histogram and Empirical CDF (Red color) against PDF and CDF (Blue color) of IER distribution for the above dataset.

I | II | III |
---|---|---|

${R}_{1}=n-m,$ | ${R}_{m}=n-m,$ | ${R}_{1}=\cdots ={R}_{n-m}=1,$ |

${R}_{2}=\cdots ={R}_{m}=0$ | ${R}_{1}=\cdots ={R}_{m-1}=0$ | ${R}_{n-m+1}=\cdots ={R}_{m}=0$ |

n | m | T | CS | Censored Data |
---|---|---|---|---|

20 | 16 | 1.75 | I | 1.1, 1.5, 1.6, 1.6, 1.7, 1.7, 1.7 |

II | 1.1, 1.2, 1.3, 1.4, 1.4, 1.5, 1.6, 1.6, 1.7, 1.7, 1.7 | |||

III | 1.1, 1.3, 1.4, 1.6, 1.7, 1.7, 1.7 | |||

2.75 | I | 1.1, 1.5, 1.6, 1.6, 1.7, 1.7, 1.7, 1.8, 1.8, 1.9, 2., 2.2, 2.3, 2.7 | ||

II | 1.1, 1.2, 1.3, 1.4, 1.4, 1.5, 1.6, 1.6, 1.7, 1.7, 1.7, 1.8, 1.8, 1.9, 2., 2.2 | |||

III | 1.1, 1.3, 1.4, 1.6, 1.7, 1.7, 1.7, 1.8, 1.8, 1.9, 2., 2.2, 2.3, 2.7 |

**Table 3.**Based on the data given in Table 2, MLEs and Bayes estimates of $\eta $, $S\left(x\right)$, and $h\left(x\right)$ (at time $x=1.35$).

Bayes Estimate | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

BLINEX | |||||||||||||||

MLE | BSE | $\mathit{c}=-\mathbf{0.3}$ | $\mathit{c}=\mathbf{3.0}$ | ||||||||||||

$\mathit{n}$ | $\mathit{m}$ | $\mathit{T}$ | CS | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\overline{\widehat{\mathit{\eta}}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ |

20 | 16 | 1.75 | I | 2.7417 | 0.8685 | 0.6415 | 2.7927 | 0.8669 | 0.6535 | 2.8917 | 0.8671 | 0.6585 | 2.0788 | 0.8651 | 0.6102 |

II | 4.0743 | 0.811 | 0.9533 | 4.0484 | 0.8129 | 0.9473 | 4.1803 | 0.8131 | 0.9541 | 2.9538 | 0.8108 | 0.8855 | |||

III | 2.6557 | 0.8724 | 0.6214 | 2.7102 | 0.8706 | 0.6342 | 2.8035 | 0.8707 | 0.6389 | 2.0368 | 0.8689 | 0.5933 | |||

2.75 | I | 3.0085 | 0.8567 | 0.704 | 3.0288 | 0.8562 | 0.7087 | 3.0873 | 0.8563 | 0.7118 | 2.5149 | 0.8551 | 0.6804 | ||

II | 4.0476 | 0.8121 | 0.9471 | 4.0304 | 0.8134 | 0.9431 | 4.1208 | 0.8136 | 0.9478 | 3.1973 | 0.8119 | 0.8988 | |||

III | 2.956 | 0.859 | 0.6917 | 2.9777 | 0.8584 | 0.6968 | 3.0343 | 0.8585 | 0.6997 | 2.4816 | 0.8574 | 0.6694 |

**Table 4.**Based on the data given in Table 2, empirical Bayes estimates of $\eta $, $S\left(x\right)$, and $h\left(x\right)$ (at time $x=1.35$).

Empirical Bayes Estimate | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

BLINEX | ||||||||||||

BSE | $\mathit{c}=-\mathbf{0.3}$ | $\mathit{c}=\mathbf{3.0}$ | ||||||||||

$\mathit{n}$ | $\mathit{m}$ | $\mathit{T}$ | CS | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{S}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\widehat{\mathit{h}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ |

20 | 16 | 1.75 | I | 2.807 | 0.8819 | 0.6568 | 2.9111 | 0.9207 | 0.7037 | 2.0477 | 0.8612 | 1.6901 |

II | 4.0769 | 0.8022 | 0.9539 | 4.2253 | 0.7776 | 0.9378 | 3.0151 | 0.8112 | 2.3031 | |||

III | 2.7315 | 0.8877 | 0.6391 | 2.8332 | 0.9323 | 0.6927 | 2.0056 | 0.8641 | 1.6541 | |||

2.75 | I | 2.9914 | 0.8416 | 0.6999 | 3.045 | 0.8005 | 0.6595 | 2.5354 | 0.8601 | 1.8589 | ||

II | 3.968 | 0.7864 | 0.9285 | 4.084 | 0.7087 | 0.86 | 3.2721 | 0.8196 | 2.3308 | |||

III | 2.9392 | 0.8435 | 0.6877 | 2.9898 | 0.8015 | 0.6459 | 2.5015 | 0.8625 | 1.8363 |

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**MDPI and ACS Style**

Hashem, A.F.; Alyami, S.A.; Yousef, M.M.
Utilizing Empirical Bayes Estimation to Assess Reliability in Inverted Exponentiated Rayleigh Distribution with Progressive Hybrid Censored Medical Data. *Axioms* **2023**, *12*, 872.
https://doi.org/10.3390/axioms12090872

**AMA Style**

Hashem AF, Alyami SA, Yousef MM.
Utilizing Empirical Bayes Estimation to Assess Reliability in Inverted Exponentiated Rayleigh Distribution with Progressive Hybrid Censored Medical Data. *Axioms*. 2023; 12(9):872.
https://doi.org/10.3390/axioms12090872

**Chicago/Turabian Style**

Hashem, Atef F., Salem A. Alyami, and Manal M. Yousef.
2023. "Utilizing Empirical Bayes Estimation to Assess Reliability in Inverted Exponentiated Rayleigh Distribution with Progressive Hybrid Censored Medical Data" *Axioms* 12, no. 9: 872.
https://doi.org/10.3390/axioms12090872