1 Introduction
2 Preliminaries of HCF and OHCFG Lifetime Models
3 An Especial Case of OHCFG Model
3.1 Some Properties of the OHCEE Distribution
3.2 Quantiles
3.3 Moments and Moment Generating Function
\(\mu ^{\prime }_{r}\)
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\(\lambda _1=0.5,\,\lambda _2=0.5\)
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\(\lambda _1=0.5,\,\lambda _2=1.5\)
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\(\lambda _1=1.5,\,\lambda _2=0.5\)
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\(\lambda _1=1.5,\,\lambda _2=1.5\)
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\(\mu ^{\prime }_{1}\)
| 2.333524 | 1.185911 | 0.7778413 | 0.3953037 |
\(\mu ^{\prime }_{2}\)
| 6.861136 | 1.943436 | 0.7623485 | 0.2159373 |
\(\mu ^{\prime }_{3}\)
| 22.67089 | 3.760179 | 0.8396628 | 0.1392659 |
\(\mu ^{\prime }_{4}\)
| 81.05204 | 8.144519 | 1.000642 | 0.1005496 |
\(\mu ^{\prime }_{5}\)
| 307.6889 | 19.22541 | 1.266209 | 0.0791169 |
\(\mu ^{\prime }_{6}\)
| 1226.292 | 48.65189 | 1.682157 | 0.0667378 |
SD | 1.415802 | 0.537051 | 0.1573114 | 0.0596723 |
CV | 0.6067227 | 0.4528594 | 0.3953037 | 0.1509531 |
CS | 0.0312526 | 0.4615376 | 0.0312529 | 0.4615372 |
CK | 2.320697 | 2.680524 | 2.320698 | 2.680524 |
\(\mu ^{\prime }_{r}\)
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\(a=0.3,\,\lambda _1=0.5\)
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\(a=0.5,\,\lambda _1=1\)
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\(a=0.8,\,\lambda _1=1.2\)
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\(a=1,\,\lambda _1=1.8\)
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---|---|---|---|---|
\(\mu ^{\prime }_{1}\)
| 1.860842 | 0.943458 | 0.8113764 | 0.5553856 |
\(\mu ^{\prime }_{2}\)
| 4.789929 | 1.225457 | 0.8961562 | 0.4156781 |
\(\mu ^{\prime }_{3}\)
| 14.49713 | 1.866463 | 1.153372 | 0.360613 |
\(\mu ^{\prime }_{4}\)
| 48.74443 | 3.152546 | 1.639693 | 0.344453 |
\(\mu ^{\prime }_{5}\)
| 176.9999 | 5.743568 | 2.507746 | 0.3532041 |
\(\mu ^{\prime }_{6}\)
| 682.5668 | 11.10332 | 4.062111 | 0.3830325 |
SD | 1.327196 | 0.335344 | 0.2378245 | 0.1072249 |
CV | 0.7132234 | 0.3554414 | 0.2931124 | 0.1930639 |
CS | 0.4214693 | 0.39924 | 0.3476608 | 0.303294 |
CK | 2.488223 | 2.460495 | 2.404912 | 2.366611 |
3.4 Order Statistics
4 Infernce Procedure
4.1 Maximum Likelihood Estimation
4.2 Stress–Strength Parameter Estimation
4.3 Bootstrap Estimation
4.4 Bayesian Estimation
4.4.1 Lindleys Approximation
loss function | Bayes estimator | Posterior risk |
---|---|---|
\(L_1=SELF=(\theta -d)^2\)
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\(E(\theta |x)\)
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\(Var(\theta |x)\)
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\(L_2=WSELF=\frac{(\theta -d)^2}{\theta }\)
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\((E(\theta ^{-1}|x))^{-1}\)
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\(E(\theta |x)-(E(\theta ^{-1}|x))^{-1}\)
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\(L_3=MSELF=\left( 1-\frac{d}{\theta }\right) ^2\)
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\(\frac{E(\theta ^{-1}|x)}{E(\theta ^{-2}|x)}\)
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\(1-\frac{E(\theta ^{-1}|x)^2}{E(\theta ^{-2}|x)}\)
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\(L_4=PLF=\frac{(\theta -d)^2}{d}\)
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\(\sqrt{E(\theta ^2|x)}\)
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\(2\left( \sqrt{E(\theta ^2|x)}-E(\theta |x)\right) \)
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\(L_5=KLF=\left( \sqrt{\frac{d}{\theta }-\sqrt{\frac{\theta }{d}}}\right) \)
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\(\sqrt{\frac{E(\theta |x)}{E(\theta ^{-1}|x)}}\)
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\(2\left( \sqrt{E(\theta |x)E(\theta ^{-1}|x)}-1\right) \)
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5 Algorithm and a Simulation Study
5.1 Algorithm
5.2 Monte Carlo Simulation Study
\(\hbox {a}=0.3\)
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\(\lambda _1=0.5\)
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\(\lambda _2=1.5\)
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n | 30 | 0.8189 (0.1738) | 0.0865 (0.0544) | 132.3321 (2.3249) |
50 | 0.7216 (0.1095) | 0.0519 (0.0187) | 40.7226 (1.3271) | |
100 | 0.6768 (0.1396) | 0.0275 (\(-\)0.0182) | 15.1627 (0.9064) | |
200 | 0.5902 (0.1135) | 0.0172 (\(-\)0.0303) | 4.4592 (0.6561) |
\(\hbox {a}=0.5\)
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\(\lambda _1=1\)
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\(\lambda _2=2\)
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n | 30 | 1.0717 (0.1182) | 0.4595 (0.1103) | 353.7897 (4.8048) |
50 | 0.8683 (0.0302) | 0.2869 (0.0377) | 165.9069 (3.2168) | |
100 | 0.7678 (0.0574) | 0.1643 (\(-\)0.0512) | 65.0995 (2.2592) | |
200 | 0.6099 (0.0405) | 0.1014 (\(-\)0.0595) | 17.2459 (1.2489) |
\(\hbox {a}=0.8\)
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\(\lambda _1=0.5\)
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\(\lambda _2=1.5\)
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n | 30 | 1.2207 (\(-\)0.0169) | 0.0873 (0.0515) | 106.8476 (2.3648) |
50 | 1.0508 (\(-\)0.0300) | 0.0538 (0.0134) | 54.7730 (1.7901) | |
100 | 0.8062 (\(-\)0.0321) | 0.0304 (\(-\)0.0151) | 17.3773 (1.1130) | |
200 | 0.6423 (\(-\)0.0515) | 0.0196 (-0.0180) | 4.9051 (0.6426) |
\(\hbox {a}=1\)
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\(\lambda _1=1.5\)
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\(\lambda _2=2\)
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n | 30 | 1.4442 (\(-\)0.0288) | 1.0211 (0.1839) | 297.3460 (4.7064) |
50 | 1.1301 (\(-\)0.0819) | 0.7082 (0.0773) | 163.7686 (3.5522) | |
100 | 0.8579 (\(-\)0.0866) | 0.4011 (\(-\)0.0087) | 64.1250 (2.0630) | |
200 | 0.6941 (\(-\)0.1346) | 0.2789 (\(-\)0.0210) | 26.7817 (1.2582) |
\(\hbox {a}=2\)
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\(\lambda _1=2\)
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\(\lambda _2=1.5\)
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n | 30 | 2.3294 (\(-\)0.1836 ) | 2.1399 (0.5090) | 223.8817 (3.3442) |
50 | 1.6476 (\(-\)0.2089) | 1.4322 (0.3452) | 57.4308 (1.7158) | |
100 | 1.1889 (\(-\)0.2213) | 0.8673 (0.2137) | 12.2516 (0.6404) | |
200 | 0.9679 (\(-\)0.2788) | 0.6181 (0.2014) | 3.7672 (0.2618) |
6 Practical Data Applications
Parametric bootstrap | Non-parametric bootstrap | |||
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Point estimation | CI | Point estimation | CI | |
a | 2.735 | (0.464, 4.430) | 2.641 | (0.672,4.063) |
\(\lambda _1\)
| 0.501 | (0.248, 0.967) | 0.497 | (0.324, 0.847) |
\(\lambda _2\)
| 0.910 | (0.136, 3.224) | 0.927 | (0.188, 2.080) |
Model | MLEs of parameters (s.e) | Log-likelihood | AIC | BIC |
\(A^*\)
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\(W^*\)
| K.S | P |
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OHCEE | \({\hat{a}}=2.58\) (0.96) | \(-\) 97.91 | 201.83 | 208.26 | 0.30 | 0.04 | 0.06 | 0.95 |
\({\hat{\lambda }}_1=0.24\,(0.14)\)
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\({\hat{\lambda }}_2= 2.08\,(2.02)\)
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HCE |
\({\hat{a}}=3.69\,(0.67)\)
| \(-\) 99.81 | 203.63 | 207.92 | 0.45 | 0.07 | 0.10 | 0.51 |
\({\hat{\lambda }}=0.89\,(0.09)\)
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Weibull |
\({\hat{\alpha }}=1.62\,(0.16)\)
| \(-\) 100.31 | 204.63 | 208.92 | 0.64 | 0.09 | 0.10 | 0.41 |
\({\hat{\beta }}=2.30\,(0.18)\)
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Gamma |
\({\hat{\alpha }}=1.9\,(0.31)\)
| \(-\) 102.83 | 209.66 | 213.95 | 1.16 | 0.2 | 0.58 | 0.84 |
\({\hat{\theta }}=0.91\,(0.17)\)
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GE |
\({\hat{\alpha }}=1.89\,(0.34)\)
| \(-\) 103.54 | 211.09 | 215.37 | 1.31 | 0.23 | 0.14 | 0.13 |
\({\hat{\lambda }}=0.69\,(0.09)\)
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Data | Service times | |||
---|---|---|---|---|
Bayes |
\(\widehat{\lambda _1}\)
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\(\widehat{\lambda _2}\)
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Loss functions | Estimate | Risk | Estimate | Risk |
SELF | 0.242722 | 0.00022 | 2.140396 | 0.04963 |
WSELF | 0.243575 | 0.00085 | 2.165938 | 0.02554 |
MSELF | 0.244380 | 0.00330 | 2.195038 | 0.01343 |
PLF | 0.242265 | 0.00091 | 2.128769 | 0.02325 |
KLF | 0.243148 | 0.00350 | 2.153129 | 0.01182 |