The Exponentiated Burr XII Power Series Distribution: Properties and Applications
Abstract
:1. Introduction and Motivation
2. Construction of the New Family
3. Special Sub Models
3.1. EBXII Logarithmic Distribution
3.2. EBXII Binomial Distribution
3.3. EBXII Poisson Distribution
3.4. EBXII Geometric Distribution
4. Mathematical Properties
Moments
5. Estimation
6. Simulation Studies
- Generate
- Solve the following nonlinear equation for given parameters values,
6.1. Simulation Study 1
6.2. Simulation Study 2
7. Data Analysis
7.1. Stress Data
7.2. Service Times Data
7.3. Failure Data
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Distribution | Parameter Space | |||
---|---|---|---|---|
Poisson | ||||
Geometric | 1 | (0,1) | ||
Binomial | ||||
Logarithmic | (0,1) |
Parameters | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
2, 0.5, 0.5, 0.5 | 2.1928 | 0.5310 | 0.3114 | 2.1858 | 2.1284 | 0.5115 | 0.3696 | 2.0686 | 2.0594 | 0.5008 | 0.4321 | 2.0351 |
(0.3739) | (0.1468) | (0.2383) | (0.5159) | (0.2951) | (0.1055) | (0.2005) | (0.4276) | (0.1405) | (0.0677) | (0.1259) | (0.2414) | |
1, 0.5, 3, 0.75 | 1.0621 | 0.6203 | 2.9386 | 0.6986 | 1.0569 | 0.5545 | 2.9642 | 0.7734 | 0.9722 | 0.5219 | 2.9723 | 0.7567 |
(0.4210) | (0.2078) | (0.3679) | (0.3324) | (0.3474) | (0.1529) | (0.2411) | (0.2188) | (0.2352) | (0.0731) | (0.1793) | (0.1538) | |
3, 0.1, 3, 0.1 | 2.9904 | 0.1044 | 3.0617 | 0.1154 | 2.9902 | 0.1002 | 2.9978 | 0.0785 | 3.0087 | 0.1005 | 3.0027 | 0.1022 |
(0.1310) | (0.0117) | (0.1578) | (0.1754) | (0.1053) | (0.0078) | (0.1181) | (0.1633) | (0.0805) | (0.0053) | (0.0942) | (0.1170) | |
5, 0.9, 2, 0.9 | 4.9810 | 0.9545 | 2.1140 | 0.8496 | 5.0138 | 0.9169 | 2.0942 | 0.8699 | 4.9976 | 0.9104 | 1.9875 | 0.8711 |
(0.7592) | (0.1832) | (0.6073) | (0.9044) | (0.2847) | (0.0882) | (0.4431) | (0.4365) | (0.1819) | (0.0703) | (0.2970) | (0.1816) | |
5, 5, 5, 0.99 | 4.9483 | 4.9838 | 5.0854 | 0.9572 | 4.9415 | 5.0126 | 5.1212 | 0.9842 | 4.9743 | 4.9785 | 5.0246 | 0.9954 |
(0.3309) | (0.2440) | (0.4827) | (0.1362) | (0.3526) | (0.2423) | (0.5678) | (0.1036) | (0.1308) | (0.1446) | (0.2212) | (0.0209) | |
10, 0.7, 0.5, 0.3 | 10.0795 | 0.7367 | 0.4899 | 0.2615 | 10.0686 | 0.7103 | 0.4901 | 0.2925 | 10.0355 | 0.7077 | 0.4974 | 0.3026 |
(0.3178) | (0.2787) | (0.1950) | (0.5675) | (0.2825) | (0.2398) | (0.1558) | (0.4701) | (0.1563) | (0.2049) | (0.1341) | (0.2793) | |
0.8, 2, 0.2, 0.5 | 1.0035 | 1.8689 | 0.2907 | 0.4027 | 0.8862 | 1.9841 | 0.2169 | 0.5169 | 0.8488 | 1.9911 | 0.2104 | 0.5121 |
(0.3488) | (0.4761) | (0.2087) | (0.4911) | (0.1938) | (0.3719) | (0.0883) | (0.2861) | (0.1275) | (0.2608) | (0.0616) | (0.2586) |
Model | |||||||||
---|---|---|---|---|---|---|---|---|---|
EBXII-G | 0.1837 | −2.3736 | 2.8794 | 0.7734 | 102.2356 | 212.4712 | 0.0835 | 0.8597 | 0.1276 |
(0.0531) | (1.2987) | (0.0852) | (0.0573) | ||||||
BXII-G | −6.5779 | 0.7905 | 3.8292 | 103.7589 | 213.5178 | 0.0907 | 1.3308 | 0.1920 | |
(9.3669) | (0.2420) | (1.6013) | |||||||
EBXII-L | 0.1466 | −16.6902 | 3.5208 | 0.7453 | 101.0149 | 210.0298 | 0.0761 | 0.6064 | 0.0821 |
(0.0244) | (5.1283) | (0.0233) | (0.1067) | ||||||
EBXII-P | 0.2237 | −1.6098 | 2.8432 | 0.6581 | 103.4967 | 214.9934 | 0.0850 | 1.2765 | 0.2155 |
(0.0592) | (0.8678) | (0.0996) | (0.1445) | ||||||
BurrXII | 1.1736 | 1.6327 | 108.5477 | 221.0955 | 0.1143 | 2.8873 | 0.3072 | ||
(0.0983) | (0.1638) |
Model | |||||||||
---|---|---|---|---|---|---|---|---|---|
EBXII-G | 0.3573 | −69.4175 | 1.3364 | 2.6885 | 102.0361 | 212.0721 | 0.0935 | 0.6693 | 0.0803 |
(0.4481) | (7.6873) | (0.7013) | (1.7470) | ||||||
BXII-G | −47.7605 | 1.0344 | 3.6365 | 102.3057 | 210.6114 | 0.0967 | 0.7748 | 0.0861 | |
(4.0336) | (0.1569) | (0.3727) | |||||||
EBXII-L | 0.2364 | 2.9710 | 3.0922 | 99.5671 | 207.1343 | 0.1067 | 0.4639 | 0.0860 | |
(0.0461) | (0.4027) | (0.1331) | (0.1867) | ||||||
EBXII-P | 1.0879 | −3.6258 | 1.4626 | 1.5307 | 108.0832 | 224.1664 | 0.1498 | 1.7402 | 0.2545 |
(0.6565) | (1.1587) | (0.4915) | (0.6687) | ||||||
BurrXII | 2.1335 | 0.6151 | 116.1148 | 236.2296 | 0.2545 | 10.4160 | 1.3521 | ||
(0.2804) | (0.0982) |
Model | |||||||||
---|---|---|---|---|---|---|---|---|---|
EBXII-G | 324.1786 | −59.7180 | 0.2562 | 7.4612 | 1178.5170 | 2365.0330 | 0.0410 | 0.5110 | 0.0592 |
(0.5597) | (0.1066) | (0.0085) | (0.1257) | ||||||
BXII-G | −349.9338 | 0.8401 | 1.7449 | 1180.261 | 2366.5210 | 0.0413 | 0.5832 | 0.0678 | |
(1.2261) | (0.2915) | (0.5884) | |||||||
EBXII-L | 377.6428 | −267.0753 | 0.3544 | 5.6072 | 1189.0280 | 2386.0560 | 0.1026 | 3.9962 | 0.7387 |
(0.1188) | (0.1809) | (0.0226) | (0.2711) | ||||||
EBXII-P | 68.2377 | −4.1156 | 0.3268 | 3.9270 | 1186.3710 | 2380.7430 | 0.0645 | 1.4650 | 0.1768 |
(2.2356) | (0.9638) | (0.0319) | (0.3690) | ||||||
BurrXII | 53.7460 | 0.0047 | 1335.4150 | 2674.8300 | 0.3697 | 44.8020 | 9.2016 | ||
(1.6200) | (0.0004) |
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Nasir, A.; Yousof, H.M.; Jamal, F.; Korkmaz, M.Ç. The Exponentiated Burr XII Power Series Distribution: Properties and Applications. Stats 2019, 2, 15-31. https://doi.org/10.3390/stats2010002
Nasir A, Yousof HM, Jamal F, Korkmaz MÇ. The Exponentiated Burr XII Power Series Distribution: Properties and Applications. Stats. 2019; 2(1):15-31. https://doi.org/10.3390/stats2010002
Chicago/Turabian StyleNasir, Arslan, Haitham M. Yousof, Farrukh Jamal, and Mustafa Ç. Korkmaz. 2019. "The Exponentiated Burr XII Power Series Distribution: Properties and Applications" Stats 2, no. 1: 15-31. https://doi.org/10.3390/stats2010002